Functions
subroutine zbdt01 (M, N, KD, A, LDA, Q, LDQ, D, E, PT, LDPT, WORK, RWORK, RESID)
ZBDT01
subroutine zbdt02 (M, N, B, LDB, C, LDC, U, LDU, WORK, RWORK, RESID)
ZBDT02
subroutine zbdt03 (UPLO, N, KD, D, E, U, LDU, S, VT, LDVT, WORK, RESID)
ZBDT03
subroutine zchkbb (NSIZES, MVAL, NVAL, NWDTHS, KK, NTYPES, DOTYPE, NRHS, ISEED, THRESH, NOUNIT, A, LDA, AB, LDAB, BD, BE, Q, LDQ, P, LDP, C, LDC, CC, WORK, LWORK, RWORK, RESULT, INFO)
ZCHKBB
subroutine zchkbd (NSIZES, MVAL, NVAL, NTYPES, DOTYPE, NRHS, ISEED, THRESH, A, LDA, BD, BE, S1, S2, X, LDX, Y, Z, Q, LDQ, PT, LDPT, U, VT, WORK, LWORK, RWORK, NOUT, INFO)
ZCHKBD
subroutine zchkbk (NIN, NOUT)
ZCHKBK
subroutine zchkbl (NIN, NOUT)
ZCHKBL
subroutine zchkec (THRESH, TSTERR, NIN, NOUT)
ZCHKEC
program zchkee
ZCHKEE
subroutine zchkgg (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, TSTDIF, THRSHN, NOUNIT, A, LDA, B, H, T, S1, S2, P1, P2, U, LDU, V, Q, Z, ALPHA1, BETA1, ALPHA3, BETA3, EVECTL, EVECTR, WORK, LWORK, RWORK, LLWORK, RESULT, INFO)
ZCHKGG
subroutine zchkgk (NIN, NOUT)
ZCHKGK
subroutine zchkgl (NIN, NOUT)
ZCHKGL
subroutine zchkhb (NSIZES, NN, NWDTHS, KK, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, SD, SE, U, LDU, WORK, LWORK, RWORK, RESULT, INFO)
ZCHKHB
subroutine zchkhs (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, H, T1, T2, U, LDU, Z, UZ, W1, W3, EVECTL, EVECTR, EVECTY, EVECTX, UU, TAU, WORK, NWORK, RWORK, IWORK, SELECT, RESULT, INFO)
ZCHKHS
subroutine zchkst (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, AP, SD, SE, D1, D2, D3, D4, D5, WA1, WA2, WA3, WR, U, LDU, V, VP, TAU, Z, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, RESULT, INFO)
ZCHKST
subroutine zckcsd (NM, MVAL, PVAL, QVAL, NMATS, ISEED, THRESH, MMAX, X, XF, U1, U2, V1T, V2T, THETA, IWORK, WORK, RWORK, NIN, NOUT, INFO)
ZCKCSD
subroutine zckglm (NN, NVAL, MVAL, PVAL, NMATS, ISEED, THRESH, NMAX, A, AF, B, BF, X, WORK, RWORK, NIN, NOUT, INFO)
ZCKGLM
subroutine zckgqr (NM, MVAL, NP, PVAL, NN, NVAL, NMATS, ISEED, THRESH, NMAX, A, AF, AQ, AR, TAUA, B, BF, BZ, BT, BWK, TAUB, WORK, RWORK, NIN, NOUT, INFO)
ZCKGQR
subroutine zckgsv (NM, MVAL, PVAL, NVAL, NMATS, ISEED, THRESH, NMAX, A, AF, B, BF, U, V, Q, ALPHA, BETA, R, IWORK, WORK, RWORK, NIN, NOUT, INFO)
ZCKGSV
subroutine zcklse (NN, MVAL, PVAL, NVAL, NMATS, ISEED, THRESH, NMAX, A, AF, B, BF, X, WORK, RWORK, NIN, NOUT, INFO)
ZCKLSE
subroutine zcsdts (M, P, Q, X, XF, LDX, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, THETA, IWORK, WORK, LWORK, RWORK, RESULT)
ZCSDTS
subroutine zdrges (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, S, T, Q, LDQ, Z, ALPHA, BETA, WORK, LWORK, RWORK, RESULT, BWORK, INFO)
ZDRGES
subroutine zdrges3 (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, S, T, Q, LDQ, Z, ALPHA, BETA, WORK, LWORK, RWORK, RESULT, BWORK, INFO)
ZDRGES3
subroutine zdrgev (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, S, T, Q, LDQ, Z, QE, LDQE, ALPHA, BETA, ALPHA1, BETA1, WORK, LWORK, RWORK, RESULT, INFO)
ZDRGEV
subroutine zdrgev3 (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, S, T, Q, LDQ, Z, QE, LDQE, ALPHA, BETA, ALPHA1, BETA1, WORK, LWORK, RWORK, RESULT, INFO)
ZDRGEV3
subroutine zdrgsx (NSIZE, NCMAX, THRESH, NIN, NOUT, A, LDA, B, AI, BI, Z, Q, ALPHA, BETA, C, LDC, S, WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO)
ZDRGSX
subroutine zdrgvx (NSIZE, THRESH, NIN, NOUT, A, LDA, B, AI, BI, ALPHA, BETA, VL, VR, ILO, IHI, LSCALE, RSCALE, S, DTRU, DIF, DIFTRU, WORK, LWORK, RWORK, IWORK, LIWORK, RESULT, BWORK, INFO)
ZDRGVX
subroutine zdrvbd (NSIZES, MM, NN, NTYPES, DOTYPE, ISEED, THRESH, A, LDA, U, LDU, VT, LDVT, ASAV, USAV, VTSAV, S, SSAV, E, WORK, LWORK, RWORK, IWORK, NOUNIT, INFO)
ZDRVBD
subroutine zdrves (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, H, HT, W, WT, VS, LDVS, RESULT, WORK, NWORK, RWORK, IWORK, BWORK, INFO)
ZDRVES
subroutine zdrvev (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, H, W, W1, VL, LDVL, VR, LDVR, LRE, LDLRE, RESULT, WORK, NWORK, RWORK, IWORK, INFO)
ZDRVEV
subroutine zdrvsg (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, LDB, D, Z, LDZ, AB, BB, AP, BP, WORK, NWORK, RWORK, LRWORK, IWORK, LIWORK, RESULT, INFO)
ZDRVSG
subroutine zdrvst (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, D1, D2, D3, WA1, WA2, WA3, U, LDU, V, TAU, Z, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, RESULT, INFO)
ZDRVST
subroutine zdrvsx (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NIUNIT, NOUNIT, A, LDA, H, HT, W, WT, WTMP, VS, LDVS, VS1, RESULT, WORK, LWORK, RWORK, BWORK, INFO)
ZDRVSX
subroutine zdrvvx (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NIUNIT, NOUNIT, A, LDA, H, W, W1, VL, LDVL, VR, LDVR, LRE, LDLRE, RCONDV, RCNDV1, RCDVIN, RCONDE, RCNDE1, RCDEIN, SCALE, SCALE1, RESULT, WORK, NWORK, RWORK, INFO)
ZDRVVX
subroutine zerrbd (PATH, NUNIT)
ZERRBD
subroutine zerrec (PATH, NUNIT)
ZERREC
subroutine zerred (PATH, NUNIT)
ZERRED
subroutine zerrgg (PATH, NUNIT)
ZERRGG
subroutine zerrhs (PATH, NUNIT)
ZERRHS
subroutine zerrst (PATH, NUNIT)
ZERRST
subroutine zget02 (TRANS, M, N, NRHS, A, LDA, X, LDX, B, LDB, RWORK, RESID)
ZGET02
subroutine zget10 (M, N, A, LDA, B, LDB, WORK, RWORK, RESULT)
ZGET10
subroutine zget22 (TRANSA, TRANSE, TRANSW, N, A, LDA, E, LDE, W, WORK, RWORK, RESULT)
ZGET22
subroutine zget23 (COMP, ISRT, BALANC, JTYPE, THRESH, ISEED, NOUNIT, N, A, LDA, H, W, W1, VL, LDVL, VR, LDVR, LRE, LDLRE, RCONDV, RCNDV1, RCDVIN, RCONDE, RCNDE1, RCDEIN, SCALE, SCALE1, RESULT, WORK, LWORK, RWORK, INFO)
ZGET23
subroutine zget24 (COMP, JTYPE, THRESH, ISEED, NOUNIT, N, A, LDA, H, HT, W, WT, WTMP, VS, LDVS, VS1, RCDEIN, RCDVIN, NSLCT, ISLCT, ISRT, RESULT, WORK, LWORK, RWORK, BWORK, INFO)
ZGET24
subroutine zget35 (RMAX, LMAX, NINFO, KNT, NIN)
ZGET35
subroutine zget36 (RMAX, LMAX, NINFO, KNT, NIN)
ZGET36
subroutine zget37 (RMAX, LMAX, NINFO, KNT, NIN)
ZGET37
subroutine zget38 (RMAX, LMAX, NINFO, KNT, NIN)
ZGET38
subroutine zget51 (ITYPE, N, A, LDA, B, LDB, U, LDU, V, LDV, WORK, RWORK, RESULT)
ZGET51
subroutine zget52 (LEFT, N, A, LDA, B, LDB, E, LDE, ALPHA, BETA, WORK, RWORK, RESULT)
ZGET52
subroutine zget54 (N, A, LDA, B, LDB, S, LDS, T, LDT, U, LDU, V, LDV, WORK, RESULT)
ZGET54
subroutine zglmts (N, M, P, A, AF, LDA, B, BF, LDB, D, DF, X, U, WORK, LWORK, RWORK, RESULT)
ZGLMTS
subroutine zgqrts (N, M, P, A, AF, Q, R, LDA, TAUA, B, BF, Z, T, BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT)
ZGQRTS
subroutine zgrqts (M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T, BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT)
ZGRQTS
subroutine zgsvts3 (M, P, N, A, AF, LDA, B, BF, LDB, U, LDU, V, LDV, Q, LDQ, ALPHA, BETA, R, LDR, IWORK, WORK, LWORK, RWORK, RESULT)
ZGSVTS3
subroutine zhbt21 (UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK, RWORK, RESULT)
ZHBT21
subroutine zhet21 (ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V, LDV, TAU, WORK, RWORK, RESULT)
ZHET21
subroutine zhet22 (ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU, V, LDV, TAU, WORK, RWORK, RESULT)
ZHET22
subroutine zhpt21 (ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP, TAU, WORK, RWORK, RESULT)
ZHPT21
subroutine zhst01 (N, ILO, IHI, A, LDA, H, LDH, Q, LDQ, WORK, LWORK, RWORK, RESULT)
ZHST01
subroutine zlarfy (UPLO, N, V, INCV, TAU, C, LDC, WORK)
ZLARFY
subroutine zlarhs (PATH, XTYPE, UPLO, TRANS, M, N, KL, KU, NRHS, A, LDA, X, LDX, B, LDB, ISEED, INFO)
ZLARHS
subroutine zlatm4 (ITYPE, N, NZ1, NZ2, RSIGN, AMAGN, RCOND, TRIANG, IDIST, ISEED, A, LDA)
ZLATM4
logical function zlctes (Z, D)
ZLCTES
logical function zlctsx (ALPHA, BETA)
ZLCTSX
subroutine zlsets (M, P, N, A, AF, LDA, B, BF, LDB, C, CF, D, DF, X, WORK, LWORK, RWORK, RESULT)
ZLSETS
subroutine zsbmv (UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZSBMV
subroutine zsgt01 (ITYPE, UPLO, N, M, A, LDA, B, LDB, Z, LDZ, D, WORK, RWORK, RESULT)
ZSGT01
logical function zslect (Z)
ZSLECT
subroutine zstt21 (N, KBAND, AD, AE, SD, SE, U, LDU, WORK, RWORK, RESULT)
ZSTT21
subroutine zstt22 (N, M, KBAND, AD, AE, SD, SE, U, LDU, WORK, LDWORK, RWORK, RESULT)
ZSTT22
subroutine zunt01 (ROWCOL, M, N, U, LDU, WORK, LWORK, RWORK, RESID)
ZUNT01
subroutine zunt03 (RC, MU, MV, N, K, U, LDU, V, LDV, WORK, LWORK, RWORK, RESULT, INFO)
ZUNT03
Detailed Description
This is the group of complex16 LAPACK TESTING EIG routines.
Function Documentation
subroutine zbdt01 (integer M, integer N, integer KD, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldpt, * ) PT, integer LDPT, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision RESID)
ZBDT01
Purpose:

ZBDT01 reconstructs a general matrix A from its bidiagonal form A = Q * B * P' where Q (m by min(m,n)) and P' (min(m,n) by n) are unitary matrices and B is bidiagonal. The test ratio to test the reduction is RESID = norm( A  Q * B * PT ) / ( n * norm(A) * EPS ) where PT = P' and EPS is the machine precision.
Parameters:

M
M is INTEGER The number of rows of the matrices A and Q.
NN is INTEGER The number of columns of the matrices A and P'.
KDKD is INTEGER If KD = 0, B is diagonal and the array E is not referenced. If KD = 1, the reduction was performed by xGEBRD; B is upper bidiagonal if M >= N, and lower bidiagonal if M < N. If KD = 1, the reduction was performed by xGBBRD; B is always upper bidiagonal.
AA is COMPLEX*16 array, dimension (LDA,N) The m by n matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
QQ is COMPLEX*16 array, dimension (LDQ,N) The m by min(m,n) unitary matrix Q in the reduction A = Q * B * P'.
LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).
DD is DOUBLE PRECISION array, dimension (min(M,N)) The diagonal elements of the bidiagonal matrix B.
EE is DOUBLE PRECISION array, dimension (min(M,N)1) The superdiagonal elements of the bidiagonal matrix B if m >= n, or the subdiagonal elements of B if m < n.
PTPT is COMPLEX*16 array, dimension (LDPT,N) The min(m,n) by n unitary matrix P' in the reduction A = Q * B * P'.
LDPTLDPT is INTEGER The leading dimension of the array PT. LDPT >= max(1,min(M,N)).
WORKWORK is COMPLEX*16 array, dimension (M+N)
RWORKRWORK is DOUBLE PRECISION array, dimension (M)
RESIDRESID is DOUBLE PRECISION The test ratio: norm(A  Q * B * P') / ( n * norm(A) * EPS )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zbdt02 (integer M, integer N, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision RESID)
ZBDT02
Purpose:

ZBDT02 tests the change of basis C = U' * B by computing the residual RESID = norm( B  U * C ) / ( max(m,n) * norm(B) * EPS ), where B and C are M by N matrices, U is an M by M orthogonal matrix, and EPS is the machine precision.
Parameters:

M
M is INTEGER The number of rows of the matrices B and C and the order of the matrix Q.
NN is INTEGER The number of columns of the matrices B and C.
BB is COMPLEX*16 array, dimension (LDB,N) The m by n matrix B.
LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,M).
CC is COMPLEX*16 array, dimension (LDC,N) The m by n matrix C, assumed to contain U' * B.
LDCLDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).
UU is COMPLEX*16 array, dimension (LDU,M) The m by m orthogonal matrix U.
LDULDU is INTEGER The leading dimension of the array U. LDU >= max(1,M).
WORKWORK is COMPLEX*16 array, dimension (M)
RWORKRWORK is DOUBLE PRECISION array, dimension (M)
RESIDRESID is DOUBLE PRECISION RESID = norm( B  U * C ) / ( max(m,n) * norm(B) * EPS ),
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zbdt03 (character UPLO, integer N, integer KD, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldu, * ) U, integer LDU, double precision, dimension( * ) S, complex*16, dimension( ldvt, * ) VT, integer LDVT, complex*16, dimension( * ) WORK, double precision RESID)
ZBDT03
Purpose:

ZBDT03 reconstructs a bidiagonal matrix B from its SVD: S = U' * B * V where U and V are orthogonal matrices and S is diagonal. The test ratio to test the singular value decomposition is RESID = norm( B  U * S * VT ) / ( n * norm(B) * EPS ) where VT = V' and EPS is the machine precision.
Parameters:

UPLO
UPLO is CHARACTER*1 Specifies whether the matrix B is upper or lower bidiagonal. = 'U': Upper bidiagonal = 'L': Lower bidiagonal
NN is INTEGER The order of the matrix B.
KDKD is INTEGER The bandwidth of the bidiagonal matrix B. If KD = 1, the matrix B is bidiagonal, and if KD = 0, B is diagonal and E is not referenced. If KD is greater than 1, it is assumed to be 1, and if KD is less than 0, it is assumed to be 0.
DD is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the bidiagonal matrix B.
EE is DOUBLE PRECISION array, dimension (N1) The (n1) superdiagonal elements of the bidiagonal matrix B if UPLO = 'U', or the (n1) subdiagonal elements of B if UPLO = 'L'.
UU is COMPLEX*16 array, dimension (LDU,N) The n by n orthogonal matrix U in the reduction B = U'*A*P.
LDULDU is INTEGER The leading dimension of the array U. LDU >= max(1,N)
SS is DOUBLE PRECISION array, dimension (N) The singular values from the SVD of B, sorted in decreasing order.
VTVT is COMPLEX*16 array, dimension (LDVT,N) The n by n orthogonal matrix V' in the reduction B = U * S * V'.
LDVTLDVT is INTEGER The leading dimension of the array VT.
WORKWORK is COMPLEX*16 array, dimension (2*N)
RESIDRESID is DOUBLE PRECISION The test ratio: norm(B  U * S * V') / ( n * norm(A) * EPS )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zchkbb (integer NSIZES, integer, dimension( * ) MVAL, integer, dimension( * ) NVAL, integer NWDTHS, integer, dimension( * ) KK, integer NTYPES, logical, dimension( * ) DOTYPE, integer NRHS, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) BD, double precision, dimension( * ) BE, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldp, * ) P, integer LDP, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( ldc, * ) CC, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( * ) RESULT, integer INFO)
ZCHKBB
Purpose:

ZCHKBB tests the reduction of a general complex rectangular band matrix to real bidiagonal form. ZGBBRD factors a general band matrix A as Q B P* , where * means conjugate transpose, B is upper bidiagonal, and Q and P are unitary; ZGBBRD can also overwrite a given matrix C with Q* C . For each pair of matrix dimensions (M,N) and each selected matrix type, an M by N matrix A and an M by NRHS matrix C are generated. The problem dimensions are as follows A: M x N Q: M x M P: N x N B: min(M,N) x min(M,N) C: M x NRHS For each generated matrix, 4 tests are performed: (1)  A  Q B PT  / ( A max(M,N) ulp ), PT = P' (2)  I  Q' Q  / ( M ulp ) (3)  I  PT PT'  / ( N ulp ) (4)  Y  Q' C  / ( Y max(M,NRHS) ulp ), where Y = Q' C. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: The possible matrix types are (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (3), but multiplied by SQRT( overflow threshold ) (7) Same as (3), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U D V, where U and V are orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U D V, where U and V are orthogonal and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U D V, where U and V are orthogonal and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Rectangular matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold )
Parameters:

NSIZES
NSIZES is INTEGER The number of values of M and N contained in the vectors MVAL and NVAL. The matrix sizes are used in pairs (M,N). If NSIZES is zero, ZCHKBB does nothing. NSIZES must be at least zero.
MVALMVAL is INTEGER array, dimension (NSIZES) The values of the matrix row dimension M.
NVALNVAL is INTEGER array, dimension (NSIZES) The values of the matrix column dimension N.
NWDTHSNWDTHS is INTEGER The number of bandwidths to use. If it is zero, ZCHKBB does nothing. It must be at least zero.
KKKK is INTEGER array, dimension (NWDTHS) An array containing the bandwidths to be used for the band matrices. The values must be at least zero.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, ZCHKBB does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
NRHSNRHS is INTEGER The number of columns in the "righthand side" matrix C. If NRHS = 0, then the operations on the righthand side will not be tested. NRHS must be at least 0.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZCHKBB to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
AA is DOUBLE PRECISION array, dimension (LDA, max(NN)) Used to hold the matrix A.
LDALDA is INTEGER The leading dimension of A. It must be at least 1 and at least max( NN ).
ABAB is DOUBLE PRECISION array, dimension (LDAB, max(NN)) Used to hold A in band storage format.
LDABLDAB is INTEGER The leading dimension of AB. It must be at least 2 (not 1!) and at least max( KK )+1.
BDBD is DOUBLE PRECISION array, dimension (max(NN)) Used to hold the diagonal of the bidiagonal matrix computed by ZGBBRD.
BEBE is DOUBLE PRECISION array, dimension (max(NN)) Used to hold the offdiagonal of the bidiagonal matrix computed by ZGBBRD.
QQ is COMPLEX*16 array, dimension (LDQ, max(NN)) Used to hold the unitary matrix Q computed by ZGBBRD.
LDQLDQ is INTEGER The leading dimension of Q. It must be at least 1 and at least max( NN ).
PP is COMPLEX*16 array, dimension (LDP, max(NN)) Used to hold the unitary matrix P computed by ZGBBRD.
LDPLDP is INTEGER The leading dimension of P. It must be at least 1 and at least max( NN ).
CC is COMPLEX*16 array, dimension (LDC, max(NN)) Used to hold the matrix C updated by ZGBBRD.
LDCLDC is INTEGER The leading dimension of U. It must be at least 1 and at least max( NN ).
CCCC is COMPLEX*16 array, dimension (LDC, max(NN)) Used to hold a copy of the matrix C.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The number of entries in WORK. This must be at least max( LDA+1, max(NN)+1 )*max(NN).
RWORKRWORK is DOUBLE PRECISION array, dimension (max(NN))
RESULTRESULT is DOUBLE PRECISION array, dimension (4) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
INFOINFO is INTEGER If 0, then everything ran OK.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. NTESTT The total number of tests performed so far. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far. COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL Square roots of the previous 2 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zchkbd (integer NSIZES, integer, dimension( * ) MVAL, integer, dimension( * ) NVAL, integer NTYPES, logical, dimension( * ) DOTYPE, integer NRHS, integer, dimension( 4 ) ISEED, double precision THRESH, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) BD, double precision, dimension( * ) BE, double precision, dimension( * ) S1, double precision, dimension( * ) S2, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( ldx, * ) Y, complex*16, dimension( ldx, * ) Z, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldpt, * ) PT, integer LDPT, complex*16, dimension( ldpt, * ) U, complex*16, dimension( ldpt, * ) VT, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer NOUT, integer INFO)
ZCHKBD
Purpose:

ZCHKBD checks the singular value decomposition (SVD) routines. ZGEBRD reduces a complex general m by n matrix A to real upper or lower bidiagonal form by an orthogonal transformation: Q' * A * P = B (or A = Q * B * P'). The matrix B is upper bidiagonal if m >= n and lower bidiagonal if m < n. ZUNGBR generates the orthogonal matrices Q and P' from ZGEBRD. Note that Q and P are not necessarily square. ZBDSQR computes the singular value decomposition of the bidiagonal matrix B as B = U S V'. It is called three times to compute 1) B = U S1 V', where S1 is the diagonal matrix of singular values and the columns of the matrices U and V are the left and right singular vectors, respectively, of B. 2) Same as 1), but the singular values are stored in S2 and the singular vectors are not computed. 3) A = (UQ) S (P'V'), the SVD of the original matrix A. In addition, ZBDSQR has an option to apply the left orthogonal matrix U to a matrix X, useful in least squares applications. For each pair of matrix dimensions (M,N) and each selected matrix type, an M by N matrix A and an M by NRHS matrix X are generated. The problem dimensions are as follows A: M x N Q: M x min(M,N) (but M x M if NRHS > 0) P: min(M,N) x N B: min(M,N) x min(M,N) U, V: min(M,N) x min(M,N) S1, S2 diagonal, order min(M,N) X: M x NRHS For each generated matrix, 14 tests are performed: Test ZGEBRD and ZUNGBR (1)  A  Q B PT  / ( A max(M,N) ulp ), PT = P' (2)  I  Q' Q  / ( M ulp ) (3)  I  PT PT'  / ( N ulp ) Test ZBDSQR on bidiagonal matrix B (4)  B  U S1 VT  / ( B min(M,N) ulp ), VT = V' (5)  Y  U Z  / ( Y max(min(M,N),k) ulp ), where Y = Q' X and Z = U' Y. (6)  I  U' U  / ( min(M,N) ulp ) (7)  I  VT VT'  / ( min(M,N) ulp ) (8) S1 contains min(M,N) nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) (9) 0 if the true singular values of B are within THRESH of those in S1. 2*THRESH if they are not. (Tested using DSVDCH) (10)  S1  S2  / ( S1 ulp ), where S2 is computed without computing U and V. Test ZBDSQR on matrix A (11)  A  (QU) S (VT PT)  / ( A max(M,N) ulp ) (12)  X  (QU) Z  / ( X max(M,k) ulp ) (13)  I  (QU)'(QU)  / ( M ulp ) (14)  I  (VT PT) (PT'VT')  / ( N ulp ) The possible matrix types are (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (3), but multiplied by SQRT( overflow threshold ) (7) Same as (3), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U D V, where U and V are orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U D V, where U and V are orthogonal and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U D V, where U and V are orthogonal and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Rectangular matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold ) Special case: (16) A bidiagonal matrix with random entries chosen from a logarithmic distribution on [ulp^2,ulp^(2)] (I.e., each entry is e^x, where x is chosen uniformly on [ 2 log(ulp), 2 log(ulp) ] .) For *this* type: (a) ZGEBRD is not called to reduce it to bidiagonal form. (b) the bidiagonal is min(M,N) x min(M,N); if M<N, the matrix will be lower bidiagonal, otherwise upper. (c) only tests 58 and 14 are performed. A subset of the full set of matrix types may be selected through the logical array DOTYPE.
Parameters:

NSIZES
NSIZES is INTEGER The number of values of M and N contained in the vectors MVAL and NVAL. The matrix sizes are used in pairs (M,N).
MVALMVAL is INTEGER array, dimension (NM) The values of the matrix row dimension M.
NVALNVAL is INTEGER array, dimension (NM) The values of the matrix column dimension N.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, ZCHKBD does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrices are in A and B. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
NRHSNRHS is INTEGER The number of columns in the "righthand side" matrices X, Y, and Z, used in testing ZBDSQR. If NRHS = 0, then the operations on the righthand side will not be tested. NRHS must be at least 0.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The values of ISEED are changed on exit, and can be used in the next call to ZCHKBD to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0. Note that the expected value of the test ratios is O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100.
AA is COMPLEX*16 array, dimension (LDA,NMAX) where NMAX is the maximum value of N in NVAL.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,MMAX), where MMAX is the maximum value of M in MVAL.
BDBD is DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j))))
BEBE is DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j))))
S1S1 is DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j))))
S2S2 is DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j))))
XX is COMPLEX*16 array, dimension (LDX,NRHS)
LDXLDX is INTEGER The leading dimension of the arrays X, Y, and Z. LDX >= max(1,MMAX).
YY is COMPLEX*16 array, dimension (LDX,NRHS)
ZZ is COMPLEX*16 array, dimension (LDX,NRHS)
QQ is COMPLEX*16 array, dimension (LDQ,MMAX)
LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,MMAX).
PTPT is COMPLEX*16 array, dimension (LDPT,NMAX)
LDPTLDPT is INTEGER The leading dimension of the arrays PT, U, and V. LDPT >= max(1, max(min(MVAL(j),NVAL(j)))).
UU is COMPLEX*16 array, dimension (LDPT,max(min(MVAL(j),NVAL(j))))
VTVT is COMPLEX*16 array, dimension (LDPT,max(min(MVAL(j),NVAL(j))))
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The number of entries in WORK. This must be at least 3(M+N) and M(M + max(M,N,k) + 1) + N*min(M,N) for all pairs (M,N)=(MM(j),NN(j))
RWORKRWORK is DOUBLE PRECISION array, dimension (5*max(min(M,N)))
NOUTNOUT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
INFOINFO is INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some MM(j) < 0 3: Some NN(j) < 0 4: NTYPES < 0 6: NRHS < 0 8: THRESH < 0 11: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ). 17: LDB < 1 or LDB < MMAX. 21: LDQ < 1 or LDQ < MMAX. 23: LDP < 1 or LDP < MNMAX. 27: LWORK too small. If ZLATMR, CLATMS, ZGEBRD, ZUNGBR, or ZBDSQR, returns an error code, the absolute value of it is returned.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. MMAX Largest value in NN. NMAX Largest value in NN. MNMIN min(MM(j), NN(j)) (the dimension of the bidiagonal matrix.) MNMAX The maximum value of MNMIN for j=1,...,NSIZES. NFAIL The number of tests which have exceeded THRESH COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. RTOVFL, RTUNFL Square roots of the previous 2 values. ULP, ULPINV Finest relative precision and its inverse. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zchkbk (integer NIN, integer NOUT)
ZCHKBK
Purpose:

ZCHKBK tests ZGEBAK, a routine for backward transformation of the computed right or left eigenvectors if the original matrix was preprocessed by balance subroutine ZGEBAL.
Parameters:

NIN
NIN is INTEGER The logical unit number for input. NIN > 0.
NOUTNOUT is INTEGER The logical unit number for output. NOUT > 0.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zchkbl (integer NIN, integer NOUT)
ZCHKBL
Purpose:

ZCHKBL tests ZGEBAL, a routine for balancing a general complex matrix and isolating some of its eigenvalues.
Parameters:

NIN
NIN is INTEGER The logical unit number for input. NIN > 0.
NOUTNOUT is INTEGER The logical unit number for output. NOUT > 0.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zchkec (double precision THRESH, logical TSTERR, integer NIN, integer NOUT)
ZCHKEC
Purpose:

ZCHKEC tests eigen condition estimation routines ZTRSYL, CTREXC, CTRSNA, CTRSEN In all cases, the routine runs through a fixed set of numerical examples, subjects them to various tests, and compares the test results to a threshold THRESH. In addition, ZTRSNA and CTRSEN are tested by reading in precomputed examples from a file (on input unit NIN). Output is written to output unit NOUT.
Parameters:

THRESH
THRESH is DOUBLE PRECISION Threshold for residual tests. A computed test ratio passes the threshold if it is less than THRESH.
TSTERRTSTERR is LOGICAL Flag that indicates whether error exits are to be tested.
NINNIN is INTEGER The logical unit number for input.
NOUTNOUT is INTEGER The logical unit number for output.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 September 2012
program zchkee ()
ZCHKEE
Purpose:

ZCHKEE tests the COMPLEX*16 LAPACK subroutines for the matrix eigenvalue problem. The test paths in this version are NEP (Nonsymmetric Eigenvalue Problem): Test ZGEHRD, ZUNGHR, ZHSEQR, ZTREVC, ZHSEIN, and ZUNMHR SEP (Hermitian Eigenvalue Problem): Test ZHETRD, ZUNGTR, ZSTEQR, ZSTERF, ZSTEIN, ZSTEDC, and drivers ZHEEV(X), ZHBEV(X), ZHPEV(X), ZHEEVD, ZHBEVD, ZHPEVD SVD (Singular Value Decomposition): Test ZGEBRD, ZUNGBR, and ZBDSQR and the drivers ZGESVD, ZGESDD ZEV (Nonsymmetric Eigenvalue/eigenvector Driver): Test ZGEEV ZES (Nonsymmetric Schur form Driver): Test ZGEES ZVX (Nonsymmetric Eigenvalue/eigenvector Expert Driver): Test ZGEEVX ZSX (Nonsymmetric Schur form Expert Driver): Test ZGEESX ZGG (Generalized Nonsymmetric Eigenvalue Problem): Test ZGGHD3, ZGGBAL, ZGGBAK, ZHGEQZ, and ZTGEVC ZGS (Generalized Nonsymmetric Schur form Driver): Test ZGGES ZGV (Generalized Nonsymmetric Eigenvalue/eigenvector Driver): Test ZGGEV ZGX (Generalized Nonsymmetric Schur form Expert Driver): Test ZGGESX ZXV (Generalized Nonsymmetric Eigenvalue/eigenvector Expert Driver): Test ZGGEVX ZSG (Hermitian Generalized Eigenvalue Problem): Test ZHEGST, ZHEGV, ZHEGVD, ZHEGVX, ZHPGST, ZHPGV, ZHPGVD, ZHPGVX, ZHBGST, ZHBGV, ZHBGVD, and ZHBGVX ZHB (Hermitian Band Eigenvalue Problem): Test ZHBTRD ZBB (Band Singular Value Decomposition): Test ZGBBRD ZEC (Eigencondition estimation): Test ZTRSYL, ZTREXC, ZTRSNA, and ZTRSEN ZBL (Balancing a general matrix) Test ZGEBAL ZBK (Back transformation on a balanced matrix) Test ZGEBAK ZGL (Balancing a matrix pair) Test ZGGBAL ZGK (Back transformation on a matrix pair) Test ZGGBAK GLM (Generalized Linear Regression Model): Tests ZGGGLM GQR (Generalized QR and RQ factorizations): Tests ZGGQRF and ZGGRQF GSV (Generalized Singular Value Decomposition): Tests ZGGSVD, ZGGSVP, ZTGSJA, ZLAGS2, ZLAPLL, and ZLAPMT CSD (CS decomposition): Tests ZUNCSD LSE (Constrained Linear Least Squares): Tests ZGGLSE Each test path has a different set of inputs, but the data sets for the driver routines xEV, xES, xVX, and xSX can be concatenated in a single input file. The first line of input should contain one of the 3character path names in columns 13. The number of remaining lines depends on what is found on the first line. The number of matrix types used in testing is often controllable from the input file. The number of matrix types for each path, and the test routine that describes them, is as follows: Path name(s) Types Test routine ZHS or NEP 21 ZCHKHS ZST or SEP 21 ZCHKST (routines) 18 ZDRVST (drivers) ZBD or SVD 16 ZCHKBD (routines) 5 ZDRVBD (drivers) ZEV 21 ZDRVEV ZES 21 ZDRVES ZVX 21 ZDRVVX ZSX 21 ZDRVSX ZGG 26 ZCHKGG (routines) ZGS 26 ZDRGES ZGX 5 ZDRGSX ZGV 26 ZDRGEV ZXV 2 ZDRGVX ZSG 21 ZDRVSG ZHB 15 ZCHKHB ZBB 15 ZCHKBB ZEC  ZCHKEC ZBL  ZCHKBL ZBK  ZCHKBK ZGL  ZCHKGL ZGK  ZCHKGK GLM 8 ZCKGLM GQR 8 ZCKGQR GSV 8 ZCKGSV CSD 3 ZCKCSD LSE 8 ZCKLSE  NEP input file: line 2: NN, INTEGER Number of values of N. line 3: NVAL, INTEGER array, dimension (NN) The values for the matrix dimension N. line 4: NPARMS, INTEGER Number of values of the parameters NB, NBMIN, NX, NS, and MAXB. line 5: NBVAL, INTEGER array, dimension (NPARMS) The values for the blocksize NB. line 6: NBMIN, INTEGER array, dimension (NPARMS) The values for the minimum blocksize NBMIN. line 7: NXVAL, INTEGER array, dimension (NPARMS) The values for the crossover point NX. line 8: INMIN, INTEGER array, dimension (NPARMS) LAHQR vs TTQRE crossover point, >= 11 line 9: INWIN, INTEGER array, dimension (NPARMS) recommended deflation window size line 10: INIBL, INTEGER array, dimension (NPARMS) nibble crossover point line 11: ISHFTS, INTEGER array, dimension (NPARMS) number of simultaneous shifts) line 12: IACC22, INTEGER array, dimension (NPARMS) select structured matrix multiply: 0, 1 or 2) line 13: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. To have all of the test ratios printed, use THRESH = 0.0 . line 14: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 14 was 2: line 15: INTEGER array, dimension (4) Four integer values for the random number seed. lines 15EOF: The remaining lines occur in sets of 1 or 2 and allow the user to specify the matrix types. Each line contains a 3character path name in columns 13, and the number of matrix types must be the first nonblank item in columns 480. If the number of matrix types is at least 1 but is less than the maximum number of possible types, a second line will be read to get the numbers of the matrix types to be used. For example, NEP 21 requests all of the matrix types for the nonsymmetric eigenvalue problem, while NEP 4 9 10 11 12 requests only matrices of type 9, 10, 11, and 12. The valid 3character path names are 'NEP' or 'ZHS' for the nonsymmetric eigenvalue routines.  SEP or ZSG input file: line 2: NN, INTEGER Number of values of N. line 3: NVAL, INTEGER array, dimension (NN) The values for the matrix dimension N. line 4: NPARMS, INTEGER Number of values of the parameters NB, NBMIN, and NX. line 5: NBVAL, INTEGER array, dimension (NPARMS) The values for the blocksize NB. line 6: NBMIN, INTEGER array, dimension (NPARMS) The values for the minimum blocksize NBMIN. line 7: NXVAL, INTEGER array, dimension (NPARMS) The values for the crossover point NX. line 8: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 9: TSTCHK, LOGICAL Flag indicating whether or not to test the LAPACK routines. line 10: TSTDRV, LOGICAL Flag indicating whether or not to test the driver routines. line 11: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 12: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 12 was 2: line 13: INTEGER array, dimension (4) Four integer values for the random number seed. lines 13EOF: Lines specifying matrix types, as for NEP. The valid 3character path names are 'SEP' or 'ZST' for the Hermitian eigenvalue routines and driver routines, and 'ZSG' for the routines for the Hermitian generalized eigenvalue problem.  SVD input file: line 2: NN, INTEGER Number of values of M and N. line 3: MVAL, INTEGER array, dimension (NN) The values for the matrix row dimension M. line 4: NVAL, INTEGER array, dimension (NN) The values for the matrix column dimension N. line 5: NPARMS, INTEGER Number of values of the parameter NB, NBMIN, NX, and NRHS. line 6: NBVAL, INTEGER array, dimension (NPARMS) The values for the blocksize NB. line 7: NBMIN, INTEGER array, dimension (NPARMS) The values for the minimum blocksize NBMIN. line 8: NXVAL, INTEGER array, dimension (NPARMS) The values for the crossover point NX. line 9: NSVAL, INTEGER array, dimension (NPARMS) The values for the number of right hand sides NRHS. line 10: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 11: TSTCHK, LOGICAL Flag indicating whether or not to test the LAPACK routines. line 12: TSTDRV, LOGICAL Flag indicating whether or not to test the driver routines. line 13: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 14: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 14 was 2: line 15: INTEGER array, dimension (4) Four integer values for the random number seed. lines 15EOF: Lines specifying matrix types, as for NEP. The 3character path names are 'SVD' or 'ZBD' for both the SVD routines and the SVD driver routines.  ZEV and ZES data files: line 1: 'ZEV' or 'ZES' in columns 1 to 3. line 2: NSIZES, INTEGER Number of sizes of matrices to use. Should be at least 0 and at most 20. If NSIZES = 0, no testing is done (although the remaining 3 lines are still read). line 3: NN, INTEGER array, dimension(NSIZES) Dimensions of matrices to be tested. line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs These integer parameters determine how blocking is done (see ILAENV for details) NB : block size NBMIN : minimum block size NX : minimum dimension for blocking NS : number of shifts in xHSEQR NBCOL : minimum column dimension for blocking line 5: THRESH, REAL The test threshold against which computed residuals are compared. Should generally be in the range from 10. to 20. If it is 0., all test case data will be printed. line 6: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 6 was 2: line 7: INTEGER array, dimension (4) Four integer values for the random number seed. lines 8 and following: Lines specifying matrix types, as for NEP. The 3character path name is 'ZEV' to test CGEEV, or 'ZES' to test CGEES.  The ZVX data has two parts. The first part is identical to ZEV, and the second part consists of test matrices with precomputed solutions. line 1: 'ZVX' in columns 13. line 2: NSIZES, INTEGER If NSIZES = 0, no testing of randomly generated examples is done, but any precomputed examples are tested. line 3: NN, INTEGER array, dimension(NSIZES) line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs line 5: THRESH, REAL line 6: NEWSD, INTEGER If line 6 was 2: line 7: INTEGER array, dimension (4) lines 8 and following: The first line contains 'ZVX' in columns 13 followed by the number of matrix types, possibly with a second line to specify certain matrix types. If the number of matrix types = 0, no testing of randomly generated examples is done, but any precomputed examples are tested. remaining lines : Each matrix is stored on 1+N+N**2 lines, where N is its dimension. The first line contains the dimension N and ISRT (two integers). ISRT indicates whether the last N lines are sorted by increasing real part of the eigenvalue (ISRT=0) or by increasing imaginary part (ISRT=1). The next N**2 lines contain the matrix rowwise, one entry per line. The last N lines correspond to each eigenvalue. Each of these last N lines contains 4 real values: the real part of the eigenvalues, the imaginary part of the eigenvalue, the reciprocal condition number of the eigenvalues, and the reciprocal condition number of the vector eigenvector. The end of data is indicated by dimension N=0. Even if no data is to be tested, there must be at least one line containing N=0.  The ZSX data is like ZVX. The first part is identical to ZEV, and the second part consists of test matrices with precomputed solutions. line 1: 'ZSX' in columns 13. line 2: NSIZES, INTEGER If NSIZES = 0, no testing of randomly generated examples is done, but any precomputed examples are tested. line 3: NN, INTEGER array, dimension(NSIZES) line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs line 5: THRESH, REAL line 6: NEWSD, INTEGER If line 6 was 2: line 7: INTEGER array, dimension (4) lines 8 and following: The first line contains 'ZSX' in columns 13 followed by the number of matrix types, possibly with a second line to specify certain matrix types. If the number of matrix types = 0, no testing of randomly generated examples is done, but any precomputed examples are tested. remaining lines : Each matrix is stored on 3+N**2 lines, where N is its dimension. The first line contains the dimension N, the dimension M of an invariant subspace, and ISRT. The second line contains M integers, identifying the eigenvalues in the invariant subspace (by their position in a list of eigenvalues ordered by increasing real part (if ISRT=0) or by increasing imaginary part (if ISRT=1)). The next N**2 lines contain the matrix rowwise. The last line contains the reciprocal condition number for the average of the selected eigenvalues, and the reciprocal condition number for the corresponding right invariant subspace. The end of data in indicated by a line containing N=0, M=0, and ISRT = 0. Even if no data is to be tested, there must be at least one line containing N=0, M=0 and ISRT=0.  ZGG input file: line 2: NN, INTEGER Number of values of N. line 3: NVAL, INTEGER array, dimension (NN) The values for the matrix dimension N. line 4: NPARMS, INTEGER Number of values of the parameters NB, NBMIN, NBCOL, NS, and MAXB. line 5: NBVAL, INTEGER array, dimension (NPARMS) The values for the blocksize NB. line 6: NBMIN, INTEGER array, dimension (NPARMS) The values for NBMIN, the minimum row dimension for blocks. line 7: NSVAL, INTEGER array, dimension (NPARMS) The values for the number of shifts. line 8: MXBVAL, INTEGER array, dimension (NPARMS) The values for MAXB, used in determining minimum blocksize. line 9: IACC22, INTEGER array, dimension (NPARMS) select structured matrix multiply: 1 or 2) line 10: NBCOL, INTEGER array, dimension (NPARMS) The values for NBCOL, the minimum column dimension for blocks. line 11: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 12: TSTCHK, LOGICAL Flag indicating whether or not to test the LAPACK routines. line 13: TSTDRV, LOGICAL Flag indicating whether or not to test the driver routines. line 14: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 15: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 15 was 2: line 16: INTEGER array, dimension (4) Four integer values for the random number seed. lines 17EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'ZGG' for the generalized eigenvalue problem routines and driver routines.  ZGS and ZGV input files: line 1: 'ZGS' or 'ZGV' in columns 1 to 3. line 2: NN, INTEGER Number of values of N. line 3: NVAL, INTEGER array, dimension(NN) Dimensions of matrices to be tested. line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs These integer parameters determine how blocking is done (see ILAENV for details) NB : block size NBMIN : minimum block size NX : minimum dimension for blocking NS : number of shifts in xHGEQR NBCOL : minimum column dimension for blocking line 5: THRESH, REAL The test threshold against which computed residuals are compared. Should generally be in the range from 10. to 20. If it is 0., all test case data will be printed. line 6: TSTERR, LOGICAL Flag indicating whether or not to test the error exits. line 7: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 17 was 2: line 7: INTEGER array, dimension (4) Four integer values for the random number seed. lines 7EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'ZGS' for the generalized eigenvalue problem routines and driver routines.  ZGX input file: line 1: 'ZGX' in columns 1 to 3. line 2: N, INTEGER Value of N. line 3: NB, NBMIN, NX, NS, NBCOL, INTEGERs These integer parameters determine how blocking is done (see ILAENV for details) NB : block size NBMIN : minimum block size NX : minimum dimension for blocking NS : number of shifts in xHGEQR NBCOL : minimum column dimension for blocking line 4: THRESH, REAL The test threshold against which computed residuals are compared. Should generally be in the range from 10. to 20. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 5: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 6: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 6 was 2: line 7: INTEGER array, dimension (4) Four integer values for the random number seed. If line 2 was 0: line 7EOF: Precomputed examples are tested. remaining lines : Each example is stored on 3+2*N*N lines, where N is its dimension. The first line contains the dimension (a single integer). The next line contains an integer k such that only the last k eigenvalues will be selected and appear in the leading diagonal blocks of $A$ and $B$. The next N*N lines contain the matrix A, one element per line. The next N*N lines contain the matrix B. The last line contains the reciprocal of the eigenvalue cluster condition number and the reciprocal of the deflating subspace (associated with the selected eigencluster) condition number. The end of data is indicated by dimension N=0. Even if no data is to be tested, there must be at least one line containing N=0.  ZXV input files: line 1: 'ZXV' in columns 1 to 3. line 2: N, INTEGER Value of N. line 3: NB, NBMIN, NX, NS, NBCOL, INTEGERs These integer parameters determine how blocking is done (see ILAENV for details) NB : block size NBMIN : minimum block size NX : minimum dimension for blocking NS : number of shifts in xHGEQR NBCOL : minimum column dimension for blocking line 4: THRESH, REAL The test threshold against which computed residuals are compared. Should generally be in the range from 10. to 20. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 5: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 6: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 6 was 2: line 7: INTEGER array, dimension (4) Four integer values for the random number seed. If line 2 was 0: line 7EOF: Precomputed examples are tested. remaining lines : Each example is stored on 3+2*N*N lines, where N is its dimension. The first line contains the dimension (a single integer). The next N*N lines contain the matrix A, one element per line. The next N*N lines contain the matrix B. The next line contains the reciprocals of the eigenvalue condition numbers. The last line contains the reciprocals of the eigenvector condition numbers. The end of data is indicated by dimension N=0. Even if no data is to be tested, there must be at least one line containing N=0.  ZHB input file: line 2: NN, INTEGER Number of values of N. line 3: NVAL, INTEGER array, dimension (NN) The values for the matrix dimension N. line 4: NK, INTEGER Number of values of K. line 5: KVAL, INTEGER array, dimension (NK) The values for the matrix dimension K. line 6: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 7 was 2: line 8: INTEGER array, dimension (4) Four integer values for the random number seed. lines 8EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'ZHB'.  ZBB input file: line 2: NN, INTEGER Number of values of M and N. line 3: MVAL, INTEGER array, dimension (NN) The values for the matrix row dimension M. line 4: NVAL, INTEGER array, dimension (NN) The values for the matrix column dimension N. line 4: NK, INTEGER Number of values of K. line 5: KVAL, INTEGER array, dimension (NK) The values for the matrix bandwidth K. line 6: NPARMS, INTEGER Number of values of the parameter NRHS line 7: NSVAL, INTEGER array, dimension (NPARMS) The values for the number of right hand sides NRHS. line 8: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 9: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 9 was 2: line 10: INTEGER array, dimension (4) Four integer values for the random number seed. lines 10EOF: Lines specifying matrix types, as for SVD. The 3character path name is 'ZBB'.  ZEC input file: line 2: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. lines 3EOF: Input for testing the eigencondition routines consists of a set of specially constructed test cases and their solutions. The data format is not intended to be modified by the user.  ZBL and ZBK input files: line 1: 'ZBL' in columns 13 to test CGEBAL, or 'ZBK' in columns 13 to test CGEBAK. The remaining lines consist of specially constructed test cases.  ZGL and ZGK input files: line 1: 'ZGL' in columns 13 to test ZGGBAL, or 'ZGK' in columns 13 to test ZGGBAK. The remaining lines consist of specially constructed test cases.  GLM data file: line 1: 'GLM' in columns 1 to 3. line 2: NN, INTEGER Number of values of M, P, and N. line 3: MVAL, INTEGER array, dimension(NN) Values of M (row dimension). line 4: PVAL, INTEGER array, dimension(NN) Values of P (row dimension). line 5: NVAL, INTEGER array, dimension(NN) Values of N (column dimension), note M <= N <= M+P. line 6: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 8: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 8 was 2: line 9: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'GLM' for the generalized linear regression model routines.  GQR data file: line 1: 'GQR' in columns 1 to 3. line 2: NN, INTEGER Number of values of M, P, and N. line 3: MVAL, INTEGER array, dimension(NN) Values of M. line 4: PVAL, INTEGER array, dimension(NN) Values of P. line 5: NVAL, INTEGER array, dimension(NN) Values of N. line 6: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 8: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 8 was 2: line 9: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'GQR' for the generalized QR and RQ routines.  GSV data file: line 1: 'GSV' in columns 1 to 3. line 2: NN, INTEGER Number of values of M, P, and N. line 3: MVAL, INTEGER array, dimension(NN) Values of M (row dimension). line 4: PVAL, INTEGER array, dimension(NN) Values of P (row dimension). line 5: NVAL, INTEGER array, dimension(NN) Values of N (column dimension). line 6: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 8: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 8 was 2: line 9: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'GSV' for the generalized SVD routines.  CSD data file: line 1: 'CSD' in columns 1 to 3. line 2: NM, INTEGER Number of values of M, P, and N. line 3: MVAL, INTEGER array, dimension(NM) Values of M (row and column dimension of orthogonal matrix). line 4: PVAL, INTEGER array, dimension(NM) Values of P (row dimension of topleft block). line 5: NVAL, INTEGER array, dimension(NM) Values of N (column dimension of topleft block). line 6: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 8: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 8 was 2: line 9: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'CSD' for the CSD routine.  LSE data file: line 1: 'LSE' in columns 1 to 3. line 2: NN, INTEGER Number of values of M, P, and N. line 3: MVAL, INTEGER array, dimension(NN) Values of M. line 4: PVAL, INTEGER array, dimension(NN) Values of P. line 5: NVAL, INTEGER array, dimension(NN) Values of N, note P <= N <= P+M. line 6: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 8: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 8 was 2: line 9: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'GSV' for the generalized SVD routines.  NMAX is currently set to 132 and must be at least 12 for some of the precomputed examples, and LWORK = NMAX*(5*NMAX+20) in the parameter statements below. For SVD, we assume NRHS may be as big as N. The parameter NEED is set to 14 to allow for 14 NbyN matrices for ZGG.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zchkgg (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, logical TSTDIF, double precision THRSHN, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) B, complex*16, dimension( lda, * ) H, complex*16, dimension( lda, * ) T, complex*16, dimension( lda, * ) S1, complex*16, dimension( lda, * ) S2, complex*16, dimension( lda, * ) P1, complex*16, dimension( lda, * ) P2, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldu, * ) V, complex*16, dimension( ldu, * ) Q, complex*16, dimension( ldu, * ) Z, complex*16, dimension( * ) ALPHA1, complex*16, dimension( * ) BETA1, complex*16, dimension( * ) ALPHA3, complex*16, dimension( * ) BETA3, complex*16, dimension( ldu, * ) EVECTL, complex*16, dimension( ldu, * ) EVECTR, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) LLWORK, double precision, dimension( 15 ) RESULT, integer INFO)
ZCHKGG
Purpose:

ZCHKGG checks the nonsymmetric generalized eigenvalue problem routines. H H H ZGGHRD factors A and B as U H V and U T V , where means conjugate transpose, H is hessenberg, T is triangular and U and V are unitary. H H ZHGEQZ factors H and T as Q S Z and Q P Z , where P and S are upper triangular and Q and Z are unitary. It also computes the generalized eigenvalues (alpha(1),beta(1)),...,(alpha(n),beta(n)), where alpha(j)=S(j,j) and beta(j)=P(j,j)  thus, w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue problem det( A  w(j) B ) = 0 and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent problem det( m(j) A  B ) = 0 ZTGEVC computes the matrix L of left eigenvectors and the matrix R of right eigenvectors for the matrix pair ( S, P ). In the description below, l and r are left and right eigenvectors corresponding to the generalized eigenvalues (alpha,beta). When ZCHKGG is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 13 tests will be performed. The first twelve "test ratios" should be small  O(1). They will be compared with the threshold THRESH: H (1)  A  U H V  / ( A n ulp ) H (2)  B  U T V  / ( B n ulp ) H (3)  I  UU  / ( n ulp ) H (4)  I  VV  / ( n ulp ) H (5)  H  Q S Z  / ( H n ulp ) H (6)  T  Q P Z  / ( T n ulp ) H (7)  I  QQ  / ( n ulp ) H (8)  I  ZZ  / ( n ulp ) (9) max over all left eigenvalue/vector pairs (beta/alpha,l) of H  (beta A  alpha B) l  / ( ulp max( beta A, alpha B ) ) (10) max over all left eigenvalue/vector pairs (beta/alpha,l') of H  (beta H  alpha T) l'  / ( ulp max( beta H, alpha T ) ) where the eigenvectors l' are the result of passing Q to DTGEVC and back transforming (JOB='B'). (11) max over all right eigenvalue/vector pairs (beta/alpha,r) of  (beta A  alpha B) r  / ( ulp max( beta A, alpha B ) ) (12) max over all right eigenvalue/vector pairs (beta/alpha,r') of  (beta H  alpha T) r'  / ( ulp max( beta H, alpha T ) ) where the eigenvectors r' are the result of passing Z to DTGEVC and back transforming (JOB='B'). The last three test ratios will usually be small, but there is no mathematical requirement that they be so. They are therefore compared with THRESH only if TSTDIF is .TRUE. (13)  S(Q,Z computed)  S(Q,Z not computed)  / ( S ulp ) (14)  P(Q,Z computed)  P(Q,Z not computed)  / ( P ulp ) (15) max( alpha(Q,Z computed)  alpha(Q,Z not computed)/S , beta(Q,Z computed)  beta(Q,Z not computed)/P ) / ulp In addition, the normalization of L and R are checked, and compared with the threshold THRSHN. Test Matrices   The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) ( 0, 0 ) (a pair of zero matrices) (2) ( I, 0 ) (an identity and a zero matrix) (3) ( 0, I ) (an identity and a zero matrix) (4) ( I, I ) (a pair of identity matrices) t t (5) ( J , J ) (a pair of transposed Jordan blocks) t ( I 0 ) (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) ( 0 I ) ( 0 J ) and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N1)/2 (7) ( D, I ) where D is P*D1, P is a random unitary diagonal matrix (i.e., with random magnitude 1 entries on the diagonal), and D1=diag( 0, 1,..., N1 ) (i.e., a diagonal matrix with D1(1,1)=0, D1(2,2)=1, ..., D1(N,N)=N1.) (8) ( I, D ) (9) ( big*D, small*I ) where "big" is near overflow and small=1/big (10) ( small*D, big*I ) (11) ( big*I, small*D ) (12) ( small*I, big*D ) (13) ( big*D, big*I ) (14) ( small*D, small*I ) (15) ( D1, D2 ) where D1=P*diag( 0, 0, 1, ..., N3, 0 ) and D2=Q*diag( 0, N3, N4,..., 1, 0, 0 ), and P and Q are random unitary diagonal matrices. t t (16) U ( J , J ) V where U and V are random unitary matrices. (17) U ( T1, T2 ) V where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = P*( 0, 0, 1, ..., N3, 0 ) and diag(T2) = Q*( 0, N3, N4,..., 1, 0, 0 ) (18) U ( T1, T2 ) V diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) U ( T1, T2 ) V diag(T1)=( 0,0,1,1, 1d, ..., 1(N5)*d=s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) N5 (20) U ( T1, T2 ) V diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) U ( T1, T2 ) V diag(T1)=( 0, 0, 1, r1, r2, ..., r(N4), 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) where r1,..., r(N4) are random. (22) U ( big*T1, small*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) U ( small*T1, big*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) U ( small*T1, small*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) U ( big*T1, big*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) U ( T1, T2 ) V where T1 and T2 are random uppertriangular matrices.
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, ZCHKGG does nothing. It must be at least zero.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, ZCHKGG does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZCHKGG to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
TSTDIFTSTDIF is LOGICAL Specifies whether test ratios 1315 will be computed and compared with THRESH. = .FALSE.: Only test ratios 112 will be computed and tested. Ratios 1315 will be set to zero. = .TRUE.: All the test ratios 115 will be computed and tested.
THRSHNTHRSHN is DOUBLE PRECISION Threshold for reporting eigenvector normalization error. If the normalization of any eigenvector differs from 1 by more than THRSHN*ulp, then a special error message will be printed. (This is handled separately from the other tests, since only a compiler or programming error should cause an error message, at least if THRSHN is at least 510.)
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
AA is COMPLEX*16 array, dimension (LDA, max(NN)) Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
LDALDA is INTEGER The leading dimension of A, B, H, T, S1, P1, S2, and P2. It must be at least 1 and at least max( NN ).
BB is COMPLEX*16 array, dimension (LDA, max(NN)) Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
HH is COMPLEX*16 array, dimension (LDA, max(NN)) The upper Hessenberg matrix computed from A by ZGGHRD.
TT is COMPLEX*16 array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by ZGGHRD.
S1S1 is COMPLEX*16 array, dimension (LDA, max(NN)) The Schur (upper triangular) matrix computed from H by ZHGEQZ when Q and Z are also computed.
S2S2 is COMPLEX*16 array, dimension (LDA, max(NN)) The Schur (upper triangular) matrix computed from H by ZHGEQZ when Q and Z are not computed.
P1P1 is COMPLEX*16 array, dimension (LDA, max(NN)) The upper triangular matrix computed from T by ZHGEQZ when Q and Z are also computed.
P2P2 is COMPLEX*16 array, dimension (LDA, max(NN)) The upper triangular matrix computed from T by ZHGEQZ when Q and Z are not computed.
UU is COMPLEX*16 array, dimension (LDU, max(NN)) The (left) unitary matrix computed by ZGGHRD.
LDULDU is INTEGER The leading dimension of U, V, Q, Z, EVECTL, and EVEZTR. It must be at least 1 and at least max( NN ).
VV is COMPLEX*16 array, dimension (LDU, max(NN)) The (right) unitary matrix computed by ZGGHRD.
QQ is COMPLEX*16 array, dimension (LDU, max(NN)) The (left) unitary matrix computed by ZHGEQZ.
ZZ is COMPLEX*16 array, dimension (LDU, max(NN)) The (left) unitary matrix computed by ZHGEQZ.
ALPHA1ALPHA1 is COMPLEX*16 array, dimension (max(NN))
BETA1BETA1 is COMPLEX*16 array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by ZHGEQZ when Q, Z, and the full Schur matrices are computed.
ALPHA3ALPHA3 is COMPLEX*16 array, dimension (max(NN))
BETA3BETA3 is COMPLEX*16 array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by ZHGEQZ when neither Q, Z, nor the Schur matrices are computed.
EVECTLEVECTL is COMPLEX*16 array, dimension (LDU, max(NN)) The (lower triangular) left eigenvector matrix for the matrices in S1 and P1.
EVECTREVECTR is COMPLEX*16 array, dimension (LDU, max(NN)) The (upper triangular) right eigenvector matrix for the matrices in S1 and P1.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The number of entries in WORK. This must be at least max( 4*N, 2 * N**2, 1 ), for all N=NN(j).
RWORKRWORK is DOUBLE PRECISION array, dimension (2*max(NN))
LLWORKLLWORK is LOGICAL array, dimension (max(NN))
RESULTRESULT is DOUBLE PRECISION array, dimension (15) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zchkgk (integer NIN, integer NOUT)
ZCHKGK
Purpose:

ZCHKGK tests ZGGBAK, a routine for backward balancing of a matrix pair (A, B).
Parameters:

NIN
NIN is INTEGER The logical unit number for input. NIN > 0.
NOUTNOUT is INTEGER The logical unit number for output. NOUT > 0.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zchkgl (integer NIN, integer NOUT)
ZCHKGL
Purpose:

ZCHKGL tests ZGGBAL, a routine for balancing a matrix pair (A, B).
Parameters:

NIN
NIN is INTEGER The logical unit number for input. NIN > 0.
NOUTNOUT is INTEGER The logical unit number for output. NOUT > 0.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zchkhb (integer NSIZES, integer, dimension( * ) NN, integer NWDTHS, integer, dimension( * ) KK, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) SD, double precision, dimension( * ) SE, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( * ) RESULT, integer INFO)
ZCHKHB
Purpose:

ZCHKHB tests the reduction of a Hermitian band matrix to tridiagonal from, used with the Hermitian eigenvalue problem. ZHBTRD factors a Hermitian band matrix A as U S U* , where * means conjugate transpose, S is symmetric tridiagonal, and U is unitary. ZHBTRD can use either just the lower or just the upper triangle of A; ZCHKHB checks both cases. When ZCHKHB is called, a number of matrix "sizes" ("n's"), a number of bandwidths ("k's"), and a number of matrix "types" are specified. For each size ("n"), each bandwidth ("k") less than or equal to "n", and each type of matrix, one matrix will be generated and used to test the hermitian banded reduction routine. For each matrix, a number of tests will be performed: (1)  A  V S V*  / ( A n ulp ) computed by ZHBTRD with UPLO='U' (2)  I  UU*  / ( n ulp ) (3)  A  V S V*  / ( A n ulp ) computed by ZHBTRD with UPLO='L' (4)  I  UU*  / ( n ulp ) The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (4), but multiplied by SQRT( overflow threshold ) (7) Same as (4), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U* D U, where U is unitary and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U* D U, where U is unitary and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U* D U, where U is unitary and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Hermitian matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold )
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, ZCHKHB does nothing. It must be at least zero.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero.
NWDTHSNWDTHS is INTEGER The number of bandwidths to use. If it is zero, ZCHKHB does nothing. It must be at least zero.
KKKK is INTEGER array, dimension (NWDTHS) An array containing the bandwidths to be used for the band matrices. The values must be at least zero.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, ZCHKHB does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZCHKHB to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
AA is COMPLEX*16 array, dimension (LDA, max(NN)) Used to hold the matrix whose eigenvalues are to be computed.
LDALDA is INTEGER The leading dimension of A. It must be at least 2 (not 1!) and at least max( KK )+1.
SDSD is DOUBLE PRECISION array, dimension (max(NN)) Used to hold the diagonal of the tridiagonal matrix computed by ZHBTRD.
SESE is DOUBLE PRECISION array, dimension (max(NN)) Used to hold the offdiagonal of the tridiagonal matrix computed by ZHBTRD.
UU is COMPLEX*16 array, dimension (LDU, max(NN)) Used to hold the unitary matrix computed by ZHBTRD.
LDULDU is INTEGER The leading dimension of U. It must be at least 1 and at least max( NN ).
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The number of entries in WORK. This must be at least max( LDA+1, max(NN)+1 )*max(NN).
RWORKRWORK is DOUBLE PRECISION array
RESULTRESULT is DOUBLE PRECISION array, dimension (4) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
INFOINFO is INTEGER If 0, then everything ran OK.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. NTESTT The total number of tests performed so far. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far. COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL Square roots of the previous 2 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zchkhs (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) H, complex*16, dimension( lda, * ) T1, complex*16, dimension( lda, * ) T2, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldu, * ) Z, complex*16, dimension( ldu, * ) UZ, complex*16, dimension( * ) W1, complex*16, dimension( * ) W3, complex*16, dimension( ldu, * ) EVECTL, complex*16, dimension( ldu, * ) EVECTR, complex*16, dimension( ldu, * ) EVECTY, complex*16, dimension( ldu, * ) EVECTX, complex*16, dimension( ldu, * ) UU, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer NWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, logical, dimension( * ) SELECT, double precision, dimension( 14 ) RESULT, integer INFO)
ZCHKHS
Purpose:

ZCHKHS checks the nonsymmetric eigenvalue problem routines. ZGEHRD factors A as U H U' , where ' means conjugate transpose, H is hessenberg, and U is unitary. ZUNGHR generates the unitary matrix U. ZUNMHR multiplies a matrix by the unitary matrix U. ZHSEQR factors H as Z T Z' , where Z is unitary and T is upper triangular. It also computes the eigenvalues, w(1), ..., w(n); we define a diagonal matrix W whose (diagonal) entries are the eigenvalues. ZTREVC computes the left eigenvector matrix L and the right eigenvector matrix R for the matrix T. The columns of L are the complex conjugates of the left eigenvectors of T. The columns of R are the right eigenvectors of T. L is lower triangular, and R is upper triangular. ZHSEIN computes the left eigenvector matrix Y and the right eigenvector matrix X for the matrix H. The columns of Y are the complex conjugates of the left eigenvectors of H. The columns of X are the right eigenvectors of H. Y is lower triangular, and X is upper triangular. When ZCHKHS is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 14 tests will be performed: (1)  A  U H U**H  / ( A n ulp ) (2)  I  UU**H  / ( n ulp ) (3)  H  Z T Z**H  / ( H n ulp ) (4)  I  ZZ**H  / ( n ulp ) (5)  A  UZ H (UZ)**H  / ( A n ulp ) (6)  I  UZ (UZ)**H  / ( n ulp ) (7)  T(Z computed)  T(Z not computed)  / ( T ulp ) (8)  W(Z computed)  W(Z not computed)  / ( W ulp ) (9)  TR  RW  / ( T R ulp ) (10)  L**H T  W**H L  / ( T L ulp ) (11)  HX  XW  / ( H X ulp ) (12)  Y**H H  W**H Y  / ( H Y ulp ) (13)  AX  XW  / ( A X ulp ) (14)  Y**H A  W**H Y  / ( A Y ulp ) The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A (transposed) Jordan block, with 1's on the diagonal. (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random complex angles. (ULP = (first number larger than 1)  1 ) (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random complex angles. (6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random complex angles. (7) Same as (4), but multiplied by SQRT( overflow threshold ) (8) Same as (4), but multiplied by SQRT( underflow threshold ) (9) A matrix of the form U' T U, where U is unitary and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (10) A matrix of the form U' T U, where U is unitary and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (11) A matrix of the form U' T U, where U is unitary and T has "clustered" entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (12) A matrix of the form U' T U, where U is unitary and T has complex eigenvalues randomly chosen from ULP < z < 1 and random O(1) entries in the upper triangle. (13) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (14) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (15) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (16) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has complex eigenvalues randomly chosen from ULP < z < 1 and random O(1) entries in the upper triangle. (17) Same as (16), but multiplied by SQRT( overflow threshold ) (18) Same as (16), but multiplied by SQRT( underflow threshold ) (19) Nonsymmetric matrix with random entries chosen from z < 1 (20) Same as (19), but multiplied by SQRT( overflow threshold ) (21) Same as (19), but multiplied by SQRT( underflow threshold )
NSIZES  INTEGER The number of sizes of matrices to use. If it is zero, ZCHKHS does nothing. It must be at least zero. Not modified. NN  INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. Not modified. NTYPES  INTEGER The number of elements in DOTYPE. If it is zero, ZCHKHS does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . Not modified. DOTYPE  LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. Not modified. ISEED  INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZCHKHS to continue the same random number sequence. Modified. THRESH  DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. Not modified. NOUNIT  INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) Not modified. A  COMPLEX*16 array, dimension (LDA,max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. Modified. LDA  INTEGER The leading dimension of A, H, T1 and T2. It must be at least 1 and at least max( NN ). Not modified. H  COMPLEX*16 array, dimension (LDA,max(NN)) The upper hessenberg matrix computed by ZGEHRD. On exit, H contains the Hessenberg form of the matrix in A. Modified. T1  COMPLEX*16 array, dimension (LDA,max(NN)) The Schur (="quasitriangular") matrix computed by ZHSEQR if Z is computed. On exit, T1 contains the Schur form of the matrix in A. Modified. T2  COMPLEX*16 array, dimension (LDA,max(NN)) The Schur matrix computed by ZHSEQR when Z is not computed. This should be identical to T1. Modified. LDU  INTEGER The leading dimension of U, Z, UZ and UU. It must be at least 1 and at least max( NN ). Not modified. U  COMPLEX*16 array, dimension (LDU,max(NN)) The unitary matrix computed by ZGEHRD. Modified. Z  COMPLEX*16 array, dimension (LDU,max(NN)) The unitary matrix computed by ZHSEQR. Modified. UZ  COMPLEX*16 array, dimension (LDU,max(NN)) The product of U times Z. Modified. W1  COMPLEX*16 array, dimension (max(NN)) The eigenvalues of A, as computed by a full Schur decomposition H = Z T Z'. On exit, W1 contains the eigenvalues of the matrix in A. Modified. W3  COMPLEX*16 array, dimension (max(NN)) The eigenvalues of A, as computed by a partial Schur decomposition (Z not computed, T only computed as much as is necessary for determining eigenvalues). On exit, W3 contains the eigenvalues of the matrix in A, possibly perturbed by ZHSEIN. Modified. EVECTL  COMPLEX*16 array, dimension (LDU,max(NN)) The conjugate transpose of the (upper triangular) left eigenvector matrix for the matrix in T1. Modified. EVEZTR  COMPLEX*16 array, dimension (LDU,max(NN)) The (upper triangular) right eigenvector matrix for the matrix in T1. Modified. EVECTY  COMPLEX*16 array, dimension (LDU,max(NN)) The conjugate transpose of the left eigenvector matrix for the matrix in H. Modified. EVECTX  COMPLEX*16 array, dimension (LDU,max(NN)) The right eigenvector matrix for the matrix in H. Modified. UU  COMPLEX*16 array, dimension (LDU,max(NN)) Details of the unitary matrix computed by ZGEHRD. Modified. TAU  COMPLEX*16 array, dimension (max(NN)) Further details of the unitary matrix computed by ZGEHRD. Modified. WORK  COMPLEX*16 array, dimension (NWORK) Workspace. Modified. NWORK  INTEGER The number of entries in WORK. NWORK >= 4*NN(j)*NN(j) + 2. RWORK  DOUBLE PRECISION array, dimension (max(NN)) Workspace. Could be equivalenced to IWORK, but not SELECT. Modified. IWORK  INTEGER array, dimension (max(NN)) Workspace. Modified. SELECT  LOGICAL array, dimension (max(NN)) Workspace. Could be equivalenced to IWORK, but not RWORK. Modified. RESULT  DOUBLE PRECISION array, dimension (14) The values computed by the fourteen tests described above. The values are currently limited to 1/ulp, to avoid overflow. Modified. INFO  INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 6: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 14: LDU < 1 or LDU < NMAX. 26: NWORK too small. If ZLATMR, CLATMS, or CLATME returns an error code, the absolute value of it is returned. If 1, then ZHSEQR could not find all the shifts. If 2, then the EISPACK code (for small blocks) failed. If >2, then 30*N iterations were not enough to find an eigenvalue or to decompose the problem. Modified.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. MTEST The number of tests defined: care must be taken that (1) the size of RESULT, (2) the number of tests actually performed, and (3) MTEST agree. NTEST The number of tests performed on this matrix so far. This should be less than MTEST, and equal to it by the last test. It will be less if any of the routines being tested indicates that it could not compute the matrices that would be tested. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far (computed by DLAFTS). COND, CONDS, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL, RTULP, RTULPI Square roots of the previous 4 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) KCONDS(j) Selects whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.)
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zchkst (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) AP, double precision, dimension( * ) SD, double precision, dimension( * ) SE, double precision, dimension( * ) D1, double precision, dimension( * ) D2, double precision, dimension( * ) D3, double precision, dimension( * ) D4, double precision, dimension( * ) D5, double precision, dimension( * ) WA1, double precision, dimension( * ) WA2, double precision, dimension( * ) WA3, double precision, dimension( * ) WR, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldu, * ) V, complex*16, dimension( * ) VP, complex*16, dimension( * ) TAU, complex*16, dimension( ldu, * ) Z, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, double precision, dimension( * ) RESULT, integer INFO)
ZCHKST
Purpose:

ZCHKST checks the Hermitian eigenvalue problem routines. ZHETRD factors A as U S U* , where * means conjugate transpose, S is real symmetric tridiagonal, and U is unitary. ZHETRD can use either just the lower or just the upper triangle of A; ZCHKST checks both cases. U is represented as a product of Householder transformations, whose vectors are stored in the first n1 columns of V, and whose scale factors are in TAU. ZHPTRD does the same as ZHETRD, except that A and V are stored in "packed" format. ZUNGTR constructs the matrix U from the contents of V and TAU. ZUPGTR constructs the matrix U from the contents of VP and TAU. ZSTEQR factors S as Z D1 Z* , where Z is the unitary matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal. D2 is the matrix of eigenvalues computed when Z is not computed. DSTERF computes D3, the matrix of eigenvalues, by the PWK method, which does not yield eigenvectors. ZPTEQR factors S as Z4 D4 Z4* , for a Hermitian positive definite tridiagonal matrix. D5 is the matrix of eigenvalues computed when Z is not computed. DSTEBZ computes selected eigenvalues. WA1, WA2, and WA3 will denote eigenvalues computed to high absolute accuracy, with different range options. WR will denote eigenvalues computed to high relative accuracy. ZSTEIN computes Y, the eigenvectors of S, given the eigenvalues. ZSTEDC factors S as Z D1 Z* , where Z is the unitary matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal ('I' option). It may also update an input unitary matrix, usually the output from ZHETRD/ZUNGTR or ZHPTRD/ZUPGTR ('V' option). It may also just compute eigenvalues ('N' option). ZSTEMR factors S as Z D1 Z* , where Z is the unitary matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal ('I' option). ZSTEMR uses the Relatively Robust Representation whenever possible. When ZCHKST is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the Hermitian eigenroutines. For each matrix, a number of tests will be performed: (1)  A  V S V*  / ( A n ulp ) ZHETRD( UPLO='U', ... ) (2)  I  UV*  / ( n ulp ) ZUNGTR( UPLO='U', ... ) (3)  A  V S V*  / ( A n ulp ) ZHETRD( UPLO='L', ... ) (4)  I  UV*  / ( n ulp ) ZUNGTR( UPLO='L', ... ) (58) Same as 14, but for ZHPTRD and ZUPGTR. (9)  S  Z D Z*  / ( S n ulp ) ZSTEQR('V',...) (10)  I  ZZ*  / ( n ulp ) ZSTEQR('V',...) (11)  D1  D2  / ( D1 ulp ) ZSTEQR('N',...) (12)  D1  D3  / ( D1 ulp ) DSTERF (13) 0 if the true eigenvalues (computed by sturm count) of S are within THRESH of those in D1. 2*THRESH if they are not. (Tested using DSTECH) For S positive definite, (14)  S  Z4 D4 Z4*  / ( S n ulp ) ZPTEQR('V',...) (15)  I  Z4 Z4*  / ( n ulp ) ZPTEQR('V',...) (16)  D4  D5  / ( 100 D4 ulp ) ZPTEQR('N',...) When S is also diagonally dominant by the factor gamma < 1, (17) max  D4(i)  WR(i)  / ( D4(i) omega ) , i omega = 2 (2n1) ULP (1 + 8 gamma**2) / (1  gamma)**4 DSTEBZ( 'A', 'E', ...) (18)  WA1  D3  / ( D3 ulp ) DSTEBZ( 'A', 'E', ...) (19) ( max { min  WA2(i)WA3(j)  } + i j max { min  WA3(i)WA2(j)  } ) / ( D3 ulp ) i j DSTEBZ( 'I', 'E', ...) (20)  S  Y WA1 Y*  / ( S n ulp ) DSTEBZ, ZSTEIN (21)  I  Y Y*  / ( n ulp ) DSTEBZ, ZSTEIN (22)  S  Z D Z*  / ( S n ulp ) ZSTEDC('I') (23)  I  ZZ*  / ( n ulp ) ZSTEDC('I') (24)  S  Z D Z*  / ( S n ulp ) ZSTEDC('V') (25)  I  ZZ*  / ( n ulp ) ZSTEDC('V') (26)  D1  D2  / ( D1 ulp ) ZSTEDC('V') and ZSTEDC('N') Test 27 is disabled at the moment because ZSTEMR does not guarantee high relatvie accuracy. (27) max  D6(i)  WR(i)  / ( D6(i) omega ) , i omega = 2 (2n1) ULP (1 + 8 gamma**2) / (1  gamma)**4 ZSTEMR('V', 'A') (28) max  D6(i)  WR(i)  / ( D6(i) omega ) , i omega = 2 (2n1) ULP (1 + 8 gamma**2) / (1  gamma)**4 ZSTEMR('V', 'I') Tests 29 through 34 are disable at present because ZSTEMR does not handle partial specturm requests. (29)  S  Z D Z*  / ( S n ulp ) ZSTEMR('V', 'I') (30)  I  ZZ*  / ( n ulp ) ZSTEMR('V', 'I') (31) ( max { min  WA2(i)WA3(j)  } + i j max { min  WA3(i)WA2(j)  } ) / ( D3 ulp ) i j ZSTEMR('N', 'I') vs. CSTEMR('V', 'I') (32)  S  Z D Z*  / ( S n ulp ) ZSTEMR('V', 'V') (33)  I  ZZ*  / ( n ulp ) ZSTEMR('V', 'V') (34) ( max { min  WA2(i)WA3(j)  } + i j max { min  WA3(i)WA2(j)  } ) / ( D3 ulp ) i j ZSTEMR('N', 'V') vs. CSTEMR('V', 'V') (35)  S  Z D Z*  / ( S n ulp ) ZSTEMR('V', 'A') (36)  I  ZZ*  / ( n ulp ) ZSTEMR('V', 'A') (37) ( max { min  WA2(i)WA3(j)  } + i j max { min  WA3(i)WA2(j)  } ) / ( D3 ulp ) i j ZSTEMR('N', 'A') vs. CSTEMR('V', 'A') The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (4), but multiplied by SQRT( overflow threshold ) (7) Same as (4), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U* D U, where U is unitary and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U* D U, where U is unitary and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U* D U, where U is unitary and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Hermitian matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold ) (16) Same as (8), but diagonal elements are all positive. (17) Same as (9), but diagonal elements are all positive. (18) Same as (10), but diagonal elements are all positive. (19) Same as (16), but multiplied by SQRT( overflow threshold ) (20) Same as (16), but multiplied by SQRT( underflow threshold ) (21) A diagonally dominant tridiagonal matrix with geometrically spaced diagonal entries 1, ..., ULP.
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, ZCHKST does nothing. It must be at least zero.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, ZCHKST does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZCHKST to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
AA is COMPLEX*16 array of dimension ( LDA , max(NN) ) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used.
LDALDA is INTEGER The leading dimension of A. It must be at least 1 and at least max( NN ).
APAP is COMPLEX*16 array of dimension( max(NN)*max(NN+1)/2 ) The matrix A stored in packed format.
SDSD is DOUBLE PRECISION array of dimension( max(NN) ) The diagonal of the tridiagonal matrix computed by ZHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A.
SESE is DOUBLE PRECISION array of dimension( max(NN) ) The offdiagonal of the tridiagonal matrix computed by ZHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A.
D1D1 is DOUBLE PRECISION array of dimension( max(NN) ) The eigenvalues of A, as computed by ZSTEQR simlutaneously with Z. On exit, the eigenvalues in D1 correspond with the matrix in A.
D2D2 is DOUBLE PRECISION array of dimension( max(NN) ) The eigenvalues of A, as computed by ZSTEQR if Z is not computed. On exit, the eigenvalues in D2 correspond with the matrix in A.
D3D3 is DOUBLE PRECISION array of dimension( max(NN) ) The eigenvalues of A, as computed by DSTERF. On exit, the eigenvalues in D3 correspond with the matrix in A.
D4D4 is DOUBLE PRECISION array of dimension( max(NN) ) The eigenvalues of A, as computed by ZPTEQR(V). ZPTEQR factors S as Z4 D4 Z4* On exit, the eigenvalues in D4 correspond with the matrix in A.
D5D5 is DOUBLE PRECISION array of dimension( max(NN) ) The eigenvalues of A, as computed by ZPTEQR(N) when Z is not computed. On exit, the eigenvalues in D4 correspond with the matrix in A.
WA1WA1 is DOUBLE PRECISION array of dimension( max(NN) ) All eigenvalues of A, computed to high absolute accuracy, with different range options. as computed by DSTEBZ.
WA2WA2 is DOUBLE PRECISION array of dimension( max(NN) ) Selected eigenvalues of A, computed to high absolute accuracy, with different range options. as computed by DSTEBZ. Choose random values for IL and IU, and ask for the ILth through IUth eigenvalues.
WA3WA3 is DOUBLE PRECISION array of dimension( max(NN) ) Selected eigenvalues of A, computed to high absolute accuracy, with different range options. as computed by DSTEBZ. Determine the values VL and VU of the ILth and IUth eigenvalues and ask for all eigenvalues in this range.
WRWR is DOUBLE PRECISION array of dimension( max(NN) ) All eigenvalues of A, computed to high absolute accuracy, with different options. as computed by DSTEBZ.
UU is COMPLEX*16 array of dimension( LDU, max(NN) ). The unitary matrix computed by ZHETRD + ZUNGTR.
LDULDU is INTEGER The leading dimension of U, Z, and V. It must be at least 1 and at least max( NN ).
VV is COMPLEX*16 array of dimension( LDU, max(NN) ). The Housholder vectors computed by ZHETRD in reducing A to tridiagonal form. The vectors computed with UPLO='U' are in the upper triangle, and the vectors computed with UPLO='L' are in the lower triangle. (As described in ZHETRD, the sub and superdiagonal are not set to 1, although the true Householder vector has a 1 in that position. The routines that use V, such as ZUNGTR, set those entries to 1 before using them, and then restore them later.)
VPVP is COMPLEX*16 array of dimension( max(NN)*max(NN+1)/2 ) The matrix V stored in packed format.
TAUTAU is COMPLEX*16 array of dimension( max(NN) ) The Householder factors computed by ZHETRD in reducing A to tridiagonal form.
ZZ is COMPLEX*16 array of dimension( LDU, max(NN) ). The unitary matrix of eigenvectors computed by ZSTEQR, ZPTEQR, and ZSTEIN.
WORKWORK is COMPLEX*16 array of dimension( LWORK )
LWORKLWORK is INTEGER The number of entries in WORK. This must be at least 1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2 where Nmax = max( NN(j), 2 ) and lg = log base 2.
IWORKIWORK is INTEGER array, Workspace.
LIWORKLIWORK is INTEGER The number of entries in IWORK. This must be at least 6 + 6*Nmax + 5 * Nmax * lg Nmax where Nmax = max( NN(j), 2 ) and lg = log base 2.
RWORKRWORK is DOUBLE PRECISION array
LRWORKLRWORK is INTEGER The number of entries in LRWORK (dimension( ??? )
RESULTRESULT is DOUBLE PRECISION array, dimension (26) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
INFOINFO is INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 5: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 23: LDU < 1 or LDU < NMAX. 29: LWORK too small. If ZLATMR, CLATMS, ZHETRD, ZUNGTR, ZSTEQR, DSTERF, or ZUNMC2 returns an error code, the absolute value of it is returned.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. NTESTT The total number of tests performed so far. NBLOCK Blocksize as returned by ENVIR. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far. COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL Square roots of the previous 2 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zckcsd (integer NM, integer, dimension( * ) MVAL, integer, dimension( * ) PVAL, integer, dimension( * ) QVAL, integer NMATS, integer, dimension( 4 ) ISEED, double precision THRESH, integer MMAX, complex*16, dimension( * ) X, complex*16, dimension( * ) XF, complex*16, dimension( * ) U1, complex*16, dimension( * ) U2, complex*16, dimension( * ) V1T, complex*16, dimension( * ) V2T, double precision, dimension( * ) THETA, integer, dimension( * ) IWORK, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer NIN, integer NOUT, integer INFO)
ZCKCSD
Purpose:

ZCKCSD tests ZUNCSD: the CSD for an MbyM unitary matrix X partitioned as [ X11 X12; X21 X22 ]. X11 is PbyQ.
Parameters:

NM
NM is INTEGER The number of values of M contained in the vector MVAL.
MVALMVAL is INTEGER array, dimension (NM) The values of the matrix row dimension M.
PVALPVAL is INTEGER array, dimension (NM) The values of the matrix row dimension P.
QVALQVAL is INTEGER array, dimension (NM) The values of the matrix column dimension Q.
NMATSNMATS is INTEGER The number of matrix types to be tested for each combination of matrix dimensions. If NMATS >= NTYPES (the maximum number of matrix types), then all the different types are generated for testing. If NMATS < NTYPES, another input line is read to get the numbers of the matrix types to be used.
ISEEDISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095, otherwise they will be reduced mod 4096, and ISEED(4) must be odd. On exit, the next seed in the random number sequence after all the test matrices have been generated.
THRESHTHRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0.
MMAXMMAX is INTEGER The maximum value permitted for M, used in dimensioning the work arrays.
XX is COMPLEX*16 array, dimension (MMAX*MMAX)
XFXF is COMPLEX*16 array, dimension (MMAX*MMAX)
U1U1 is COMPLEX*16 array, dimension (MMAX*MMAX)
U2U2 is COMPLEX*16 array, dimension (MMAX*MMAX)
V1TV1T is COMPLEX*16 array, dimension (MMAX*MMAX)
V2TV2T is COMPLEX*16 array, dimension (MMAX*MMAX)
THETATHETA is DOUBLE PRECISION array, dimension (MMAX)
IWORKIWORK is INTEGER array, dimension (MMAX)
WORKWORK is COMPLEX*16 array
RWORKRWORK is DOUBLE PRECISION array
NINNIN is INTEGER The unit number for input.
NOUTNOUT is INTEGER The unit number for output.
INFOINFO is INTEGER = 0 : successful exit > 0 : If ZLAROR returns an error code, the absolute value of it is returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zckglm (integer NN, integer, dimension( * ) NVAL, integer, dimension( * ) MVAL, integer, dimension( * ) PVAL, integer NMATS, integer, dimension( 4 ) ISEED, double precision THRESH, integer NMAX, complex*16, dimension( * ) A, complex*16, dimension( * ) AF, complex*16, dimension( * ) B, complex*16, dimension( * ) BF, complex*16, dimension( * ) X, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer NIN, integer NOUT, integer INFO)
ZCKGLM
Purpose:

ZCKGLM tests ZGGGLM  subroutine for solving generalized linear model problem.
Parameters:

NN
NN is INTEGER The number of values of N, M and P contained in the vectors NVAL, MVAL and PVAL.
NVALNVAL is INTEGER array, dimension (NN) The values of the matrix row dimension N.
MVALMVAL is INTEGER array, dimension (NN) The values of the matrix column dimension M.
PVALPVAL is INTEGER array, dimension (NN) The values of the matrix column dimension P.
NMATSNMATS is INTEGER The number of matrix types to be tested for each combination of matrix dimensions. If NMATS >= NTYPES (the maximum number of matrix types), then all the different types are generated for testing. If NMATS < NTYPES, another input line is read to get the numbers of the matrix types to be used.
ISEEDISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095, otherwise they will be reduced mod 4096, and ISEED(4) must be odd. On exit, the next seed in the random number sequence after all the test matrices have been generated.
THRESHTHRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESID >= THRESH. To have every test ratio printed, use THRESH = 0.
NMAXNMAX is INTEGER The maximum value permitted for M or N, used in dimensioning the work arrays.
AA is COMPLEX*16 array, dimension (NMAX*NMAX)
AFAF is COMPLEX*16 array, dimension (NMAX*NMAX)
BB is COMPLEX*16 array, dimension (NMAX*NMAX)
BFBF is COMPLEX*16 array, dimension (NMAX*NMAX)
XX is COMPLEX*16 array, dimension (4*NMAX)
RWORKRWORK is DOUBLE PRECISION array, dimension (NMAX)
WORKWORK is COMPLEX*16 array, dimension (NMAX*NMAX)
NINNIN is INTEGER The unit number for input.
NOUTNOUT is INTEGER The unit number for output.
INFOINFO is INTEGER = 0 : successful exit > 0 : If ZLATMS returns an error code, the absolute value of it is returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zckgqr (integer NM, integer, dimension( * ) MVAL, integer NP, integer, dimension( * ) PVAL, integer NN, integer, dimension( * ) NVAL, integer NMATS, integer, dimension( 4 ) ISEED, double precision THRESH, integer NMAX, complex*16, dimension( * ) A, complex*16, dimension( * ) AF, complex*16, dimension( * ) AQ, complex*16, dimension( * ) AR, complex*16, dimension( * ) TAUA, complex*16, dimension( * ) B, complex*16, dimension( * ) BF, complex*16, dimension( * ) BZ, complex*16, dimension( * ) BT, complex*16, dimension( * ) BWK, complex*16, dimension( * ) TAUB, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer NIN, integer NOUT, integer INFO)
ZCKGQR
Purpose:

ZCKGQR tests ZGGQRF: GQR factorization for NbyM matrix A and NbyP matrix B, ZGGRQF: GRQ factorization for MbyN matrix A and PbyN matrix B.
Parameters:

NM
NM is INTEGER The number of values of M contained in the vector MVAL.
MVALMVAL is INTEGER array, dimension (NM) The values of the matrix row(column) dimension M.
NPNP is INTEGER The number of values of P contained in the vector PVAL.
PVALPVAL is INTEGER array, dimension (NP) The values of the matrix row(column) dimension P.
NNNN is INTEGER The number of values of N contained in the vector NVAL.
NVALNVAL is INTEGER array, dimension (NN) The values of the matrix column(row) dimension N.
NMATSNMATS is INTEGER The number of matrix types to be tested for each combination of matrix dimensions. If NMATS >= NTYPES (the maximum number of matrix types), then all the different types are generated for testing. If NMATS < NTYPES, another input line is read to get the numbers of the matrix types to be used.
ISEEDISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095, otherwise they will be reduced mod 4096, and ISEED(4) must be odd. On exit, the next seed in the random number sequence after all the test matrices have been generated.
THRESHTHRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0.
NMAXNMAX is INTEGER The maximum value permitted for M or N, used in dimensioning the work arrays.
AA is COMPLEX*16 array, dimension (NMAX*NMAX)
AFAF is COMPLEX*16 array, dimension (NMAX*NMAX)
AQAQ is COMPLEX*16 array, dimension (NMAX*NMAX)
ARAR is COMPLEX*16 array, dimension (NMAX*NMAX)
TAUATAUA is COMPLEX*16 array, dimension (NMAX)
BB is COMPLEX*16 array, dimension (NMAX*NMAX)
BFBF is COMPLEX*16 array, dimension (NMAX*NMAX)
BZBZ is COMPLEX*16 array, dimension (NMAX*NMAX)
BTBT is COMPLEX*16 array, dimension (NMAX*NMAX)
BWKBWK is COMPLEX*16 array, dimension (NMAX*NMAX)
TAUBTAUB is COMPLEX*16 array, dimension (NMAX)
WORKWORK is COMPLEX*16 array, dimension (NMAX*NMAX)
RWORKRWORK is DOUBLE PRECISION array, dimension (NMAX)
NINNIN is INTEGER The unit number for input.
NOUTNOUT is INTEGER The unit number for output.
INFOINFO is INTEGER = 0 : successful exit > 0 : If ZLATMS returns an error code, the absolute value of it is returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zckgsv (integer NM, integer, dimension( * ) MVAL, integer, dimension( * ) PVAL, integer, dimension( * ) NVAL, integer NMATS, integer, dimension( 4 ) ISEED, double precision THRESH, integer NMAX, complex*16, dimension( * ) A, complex*16, dimension( * ) AF, complex*16, dimension( * ) B, complex*16, dimension( * ) BF, complex*16, dimension( * ) U, complex*16, dimension( * ) V, complex*16, dimension( * ) Q, double precision, dimension( * ) ALPHA, double precision, dimension( * ) BETA, complex*16, dimension( * ) R, integer, dimension( * ) IWORK, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer NIN, integer NOUT, integer INFO)
ZCKGSV
Purpose:

ZCKGSV tests ZGGSVD: the GSVD for MbyN matrix A and PbyN matrix B.
Parameters:

NM
NM is INTEGER The number of values of M contained in the vector MVAL.
MVALMVAL is INTEGER array, dimension (NM) The values of the matrix row dimension M.
PVALPVAL is INTEGER array, dimension (NP) The values of the matrix row dimension P.
NVALNVAL is INTEGER array, dimension (NN) The values of the matrix column dimension N.
NMATSNMATS is INTEGER The number of matrix types to be tested for each combination of matrix dimensions. If NMATS >= NTYPES (the maximum number of matrix types), then all the different types are generated for testing. If NMATS < NTYPES, another input line is read to get the numbers of the matrix types to be used.
ISEEDISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095, otherwise they will be reduced mod 4096, and ISEED(4) must be odd. On exit, the next seed in the random number sequence after all the test matrices have been generated.
THRESHTHRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0.
NMAXNMAX is INTEGER The maximum value permitted for M or N, used in dimensioning the work arrays.
AA is COMPLEX*16 array, dimension (NMAX*NMAX)
AFAF is COMPLEX*16 array, dimension (NMAX*NMAX)
BB is COMPLEX*16 array, dimension (NMAX*NMAX)
BFBF is COMPLEX*16 array, dimension (NMAX*NMAX)
UU is COMPLEX*16 array, dimension (NMAX*NMAX)
VV is COMPLEX*16 array, dimension (NMAX*NMAX)
QQ is COMPLEX*16 array, dimension (NMAX*NMAX)
ALPHAALPHA is DOUBLE PRECISION array, dimension (NMAX)
BETABETA is DOUBLE PRECISION array, dimension (NMAX)
RR is COMPLEX*16 array, dimension (NMAX*NMAX)
IWORKIWORK is INTEGER array, dimension (NMAX)
WORKWORK is COMPLEX*16 array, dimension (NMAX*NMAX)
RWORKRWORK is DOUBLE PRECISION array, dimension (NMAX)
NINNIN is INTEGER The unit number for input.
NOUTNOUT is INTEGER The unit number for output.
INFOINFO is INTEGER = 0 : successful exit > 0 : If ZLATMS returns an error code, the absolute value of it is returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2015
subroutine zcklse (integer NN, integer, dimension( * ) MVAL, integer, dimension( * ) PVAL, integer, dimension( * ) NVAL, integer NMATS, integer, dimension( 4 ) ISEED, double precision THRESH, integer NMAX, complex*16, dimension( * ) A, complex*16, dimension( * ) AF, complex*16, dimension( * ) B, complex*16, dimension( * ) BF, complex*16, dimension( * ) X, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer NIN, integer NOUT, integer INFO)
ZCKLSE
Purpose:

ZCKLSE tests ZGGLSE  a subroutine for solving linear equality constrained least square problem (LSE).
Parameters:

NN
NN is INTEGER The number of values of (M,P,N) contained in the vectors (MVAL, PVAL, NVAL).
MVALMVAL is INTEGER array, dimension (NN) The values of the matrix row(column) dimension M.
PVALPVAL is INTEGER array, dimension (NN) The values of the matrix row(column) dimension P.
NVALNVAL is INTEGER array, dimension (NN) The values of the matrix column(row) dimension N.
NMATSNMATS is INTEGER The number of matrix types to be tested for each combination of matrix dimensions. If NMATS >= NTYPES (the maximum number of matrix types), then all the different types are generated for testing. If NMATS < NTYPES, another input line is read to get the numbers of the matrix types to be used.
ISEEDISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095, otherwise they will be reduced mod 4096, and ISEED(4) must be odd. On exit, the next seed in the random number sequence after all the test matrices have been generated.
THRESHTHRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0.
NMAXNMAX is INTEGER The maximum value permitted for M or N, used in dimensioning the work arrays.
AA is COMPLEX*16 array, dimension (NMAX*NMAX)
AFAF is COMPLEX*16 array, dimension (NMAX*NMAX)
BB is COMPLEX*16 array, dimension (NMAX*NMAX)
BFBF is COMPLEX*16 array, dimension (NMAX*NMAX)
XX is COMPLEX*16 array, dimension (5*NMAX)
WORKWORK is COMPLEX*16 array, dimension (NMAX*NMAX)
RWORKRWORK is DOUBLE PRECISION array, dimension (NMAX)
NINNIN is INTEGER The unit number for input.
NOUTNOUT is INTEGER The unit number for output.
INFOINFO is INTEGER = 0 : successful exit > 0 : If ZLATMS returns an error code, the absolute value of it is returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zcsdts (integer M, integer P, integer Q, complex*16, dimension( ldx, * ) X, complex*16, dimension( ldx, * ) XF, integer LDX, complex*16, dimension( ldu1, * ) U1, integer LDU1, complex*16, dimension( ldu2, * ) U2, integer LDU2, complex*16, dimension( ldv1t, * ) V1T, integer LDV1T, complex*16, dimension( ldv2t, * ) V2T, integer LDV2T, double precision, dimension( * ) THETA, integer, dimension( * ) IWORK, complex*16, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 15 ) RESULT)
ZCSDTS
Purpose:

ZCSDTS tests ZUNCSD, which, given an MbyM partitioned unitary matrix X, Q MQ X = [ X11 X12 ] P , [ X21 X22 ] MP computes the CSD [ U1 ]**T * [ X11 X12 ] * [ V1 ] [ U2 ] [ X21 X22 ] [ V2 ] [ I 0 0  0 0 0 ] [ 0 C 0  0 S 0 ] [ 0 0 0  0 0 I ] = [] = [ D11 D12 ] . [ 0 0 0  I 0 0 ] [ D21 D22 ] [ 0 S 0  0 C 0 ] [ 0 0 I  0 0 0 ] and also SORCSD2BY1, which, given Q [ X11 ] P , [ X21 ] MP computes the 2by1 CSD [ I 0 0 ] [ 0 C 0 ] [ 0 0 0 ] [ U1 ]**T * [ X11 ] * V1 = [] = [ D11 ] , [ U2 ] [ X21 ] [ 0 0 0 ] [ D21 ] [ 0 S 0 ] [ 0 0 I ]
Parameters:

M
M is INTEGER The number of rows of the matrix X. M >= 0.
PP is INTEGER The number of rows of the matrix X11. P >= 0.
QQ is INTEGER The number of columns of the matrix X11. Q >= 0.
XX is COMPLEX*16 array, dimension (LDX,M) The MbyM matrix X.
XFXF is COMPLEX*16 array, dimension (LDX,M) Details of the CSD of X, as returned by ZUNCSD; see ZUNCSD for further details.
LDXLDX is INTEGER The leading dimension of the arrays X and XF. LDX >= max( 1,M ).
U1U1 is COMPLEX*16 array, dimension(LDU1,P) The PbyP unitary matrix U1.
LDU1LDU1 is INTEGER The leading dimension of the array U1. LDU >= max(1,P).
U2U2 is COMPLEX*16 array, dimension(LDU2,MP) The (MP)by(MP) unitary matrix U2.
LDU2LDU2 is INTEGER The leading dimension of the array U2. LDU >= max(1,MP).
V1TV1T is COMPLEX*16 array, dimension(LDV1T,Q) The QbyQ unitary matrix V1T.
LDV1TLDV1T is INTEGER The leading dimension of the array V1T. LDV1T >= max(1,Q).
V2TV2T is COMPLEX*16 array, dimension(LDV2T,MQ) The (MQ)by(MQ) unitary matrix V2T.
LDV2TLDV2T is INTEGER The leading dimension of the array V2T. LDV2T >= max(1,MQ).
THETATHETA is DOUBLE PRECISION array, dimension MIN(P,MP,Q,MQ) The CS values of X; the essentially diagonal matrices C and S are constructed from THETA; see subroutine ZUNCSD for details.
IWORKIWORK is INTEGER array, dimension (M)
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The dimension of the array WORK
RWORKRWORK is DOUBLE PRECISION array
RESULTRESULT is DOUBLE PRECISION array, dimension (15) The test ratios: First, the 2by2 CSD: RESULT(1) = norm( U1'*X11*V1  D11 ) / ( MAX(1,P,Q)*EPS2 ) RESULT(2) = norm( U1'*X12*V2  D12 ) / ( MAX(1,P,MQ)*EPS2 ) RESULT(3) = norm( U2'*X21*V1  D21 ) / ( MAX(1,MP,Q)*EPS2 ) RESULT(4) = norm( U2'*X22*V2  D22 ) / ( MAX(1,MP,MQ)*EPS2 ) RESULT(5) = norm( I  U1'*U1 ) / ( MAX(1,P)*ULP ) RESULT(6) = norm( I  U2'*U2 ) / ( MAX(1,MP)*ULP ) RESULT(7) = norm( I  V1T'*V1T ) / ( MAX(1,Q)*ULP ) RESULT(8) = norm( I  V2T'*V2T ) / ( MAX(1,MQ)*ULP ) RESULT(9) = 0 if THETA is in increasing order and all angles are in [0,pi/2]; = ULPINV otherwise. Then, the 2by1 CSD: RESULT(10) = norm( U1'*X11*V1  D11 ) / ( MAX(1,P,Q)*EPS2 ) RESULT(11) = norm( U2'*X21*V1  D21 ) / ( MAX(1,MP,Q)*EPS2 ) RESULT(12) = norm( I  U1'*U1 ) / ( MAX(1,P)*ULP ) RESULT(13) = norm( I  U2'*U2 ) / ( MAX(1,MP)*ULP ) RESULT(14) = norm( I  V1T'*V1T ) / ( MAX(1,Q)*ULP ) RESULT(15) = 0 if THETA is in increasing order and all angles are in [0,pi/2]; = ULPINV otherwise. ( EPS2 = MAX( norm( I  X'*X ) / M, ULP ). )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2015
subroutine zdrges (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) B, complex*16, dimension( lda, * ) S, complex*16, dimension( lda, * ) T, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldq, * ) Z, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 13 ) RESULT, logical, dimension( * ) BWORK, integer INFO)
ZDRGES
Purpose:

ZDRGES checks the nonsymmetric generalized eigenvalue (Schur form) problem driver ZGGES. ZGGES factors A and B as Q*S*Z' and Q*T*Z' , where ' means conjugate transpose, S and T are upper triangular (i.e., in generalized Schur form), and Q and Z are unitary. It also computes the generalized eigenvalues (alpha(j),beta(j)), j=1,...,n. Thus, w(j) = alpha(j)/beta(j) is a root of the characteristic equation det( A  w(j) B ) = 0 Optionally it also reorder the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms. When ZDRGES is called, a number of matrix "sizes" ("N's") and a number of matrix "TYPES" are specified. For each size ("N") and each TYPE of matrix, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following 13 tests will be performed and compared with the threshold THRESH except the tests (5), (11) and (13). (1)  A  Q S Z'  / ( A n ulp ) (no sorting of eigenvalues) (2)  B  Q T Z'  / ( B n ulp ) (no sorting of eigenvalues) (3)  I  QQ'  / ( n ulp ) (no sorting of eigenvalues) (4)  I  ZZ'  / ( n ulp ) (no sorting of eigenvalues) (5) if A is in Schur form (i.e. triangular form) (no sorting of eigenvalues) (6) if eigenvalues = diagonal elements of the Schur form (S, T), i.e., test the maximum over j of D(j) where: alpha(j)  S(j,j) beta(j)  T(j,j) D(j) =  +  max(alpha(j),S(j,j)) max(beta(j),T(j,j)) (no sorting of eigenvalues) (7)  (A,B)  Q (S,T) Z'  / ( (A,B) n ulp ) (with sorting of eigenvalues). (8)  I  QQ'  / ( n ulp ) (with sorting of eigenvalues). (9)  I  ZZ'  / ( n ulp ) (with sorting of eigenvalues). (10) if A is in Schur form (i.e. quasitriangular form) (with sorting of eigenvalues). (11) if eigenvalues = diagonal elements of the Schur form (S, T), i.e. test the maximum over j of D(j) where: alpha(j)  S(j,j) beta(j)  T(j,j) D(j) =  +  max(alpha(j),S(j,j)) max(beta(j),T(j,j)) (with sorting of eigenvalues). (12) if sorting worked and SDIM is the number of eigenvalues which were CELECTed. Test Matrices ============= The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) ( 0, 0 ) (a pair of zero matrices) (2) ( I, 0 ) (an identity and a zero matrix) (3) ( 0, I ) (an identity and a zero matrix) (4) ( I, I ) (a pair of identity matrices) t t (5) ( J , J ) (a pair of transposed Jordan blocks) t ( I 0 ) (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) ( 0 I ) ( 0 J ) and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N1)/2 (7) ( D, I ) where D is diag( 0, 1,..., N1 ) (a diagonal matrix with those diagonal entries.) (8) ( I, D ) (9) ( big*D, small*I ) where "big" is near overflow and small=1/big (10) ( small*D, big*I ) (11) ( big*I, small*D ) (12) ( small*I, big*D ) (13) ( big*D, big*I ) (14) ( small*D, small*I ) (15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N3, 0 ) and D2 is diag( 0, N3, N4,..., 1, 0, 0 ) t t (16) Q ( J , J ) Z where Q and Z are random orthogonal matrices. (17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N3, 0 ) and diag(T2) = ( 0, N3, N4,..., 1, 0, 0 ) (18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1d, ..., 1(N5)*d=s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) N5 (20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N4), 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) where r1,..., r(N4) are random. (22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) Q ( T1, T2 ) Z where T1 and T2 are random uppertriangular matrices.
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, DDRGES does nothing. NSIZES >= 0.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DDRGES does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A on input. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRGES to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. THRESH >= 0.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
AA is COMPLEX*16 array, dimension(LDA, max(NN)) Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
LDALDA is INTEGER The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ).
BB is COMPLEX*16 array, dimension(LDA, max(NN)) Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
SS is COMPLEX*16 array, dimension (LDA, max(NN)) The Schur form matrix computed from A by ZGGES. On exit, S contains the Schur form matrix corresponding to the matrix in A.
TT is COMPLEX*16 array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by ZGGES.
QQ is COMPLEX*16 array, dimension (LDQ, max(NN)) The (left) orthogonal matrix computed by ZGGES.
LDQLDQ is INTEGER The leading dimension of Q and Z. It must be at least 1 and at least max( NN ).
ZZ is COMPLEX*16 array, dimension( LDQ, max(NN) ) The (right) orthogonal matrix computed by ZGGES.
ALPHAALPHA is COMPLEX*16 array, dimension (max(NN))
BETABETA is COMPLEX*16 array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by ZGGES. ALPHA(k) / BETA(k) is the kth generalized eigenvalue of A and B.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The dimension of the array WORK. LWORK >= 3*N*N.
RWORKRWORK is DOUBLE PRECISION array, dimension ( 8*N ) Real workspace.
RESULTRESULT is DOUBLE PRECISION array, dimension (15) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
BWORKBWORK is LOGICAL array, dimension (N)
INFOINFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zdrges3 (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) B, complex*16, dimension( lda, * ) S, complex*16, dimension( lda, * ) T, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldq, * ) Z, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 13 ) RESULT, logical, dimension( * ) BWORK, integer INFO)
ZDRGES3
Purpose:

ZDRGES3 checks the nonsymmetric generalized eigenvalue (Schur form) problem driver ZGGES3. ZGGES3 factors A and B as Q*S*Z' and Q*T*Z' , where ' means conjugate transpose, S and T are upper triangular (i.e., in generalized Schur form), and Q and Z are unitary. It also computes the generalized eigenvalues (alpha(j),beta(j)), j=1,...,n. Thus, w(j) = alpha(j)/beta(j) is a root of the characteristic equation det( A  w(j) B ) = 0 Optionally it also reorder the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms. When ZDRGES3 is called, a number of matrix "sizes" ("N's") and a number of matrix "TYPES" are specified. For each size ("N") and each TYPE of matrix, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following 13 tests will be performed and compared with the threshold THRESH except the tests (5), (11) and (13). (1)  A  Q S Z'  / ( A n ulp ) (no sorting of eigenvalues) (2)  B  Q T Z'  / ( B n ulp ) (no sorting of eigenvalues) (3)  I  QQ'  / ( n ulp ) (no sorting of eigenvalues) (4)  I  ZZ'  / ( n ulp ) (no sorting of eigenvalues) (5) if A is in Schur form (i.e. triangular form) (no sorting of eigenvalues) (6) if eigenvalues = diagonal elements of the Schur form (S, T), i.e., test the maximum over j of D(j) where: alpha(j)  S(j,j) beta(j)  T(j,j) D(j) =  +  max(alpha(j),S(j,j)) max(beta(j),T(j,j)) (no sorting of eigenvalues) (7)  (A,B)  Q (S,T) Z'  / ( (A,B) n ulp ) (with sorting of eigenvalues). (8)  I  QQ'  / ( n ulp ) (with sorting of eigenvalues). (9)  I  ZZ'  / ( n ulp ) (with sorting of eigenvalues). (10) if A is in Schur form (i.e. quasitriangular form) (with sorting of eigenvalues). (11) if eigenvalues = diagonal elements of the Schur form (S, T), i.e. test the maximum over j of D(j) where: alpha(j)  S(j,j) beta(j)  T(j,j) D(j) =  +  max(alpha(j),S(j,j)) max(beta(j),T(j,j)) (with sorting of eigenvalues). (12) if sorting worked and SDIM is the number of eigenvalues which were CELECTed. Test Matrices ============= The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) ( 0, 0 ) (a pair of zero matrices) (2) ( I, 0 ) (an identity and a zero matrix) (3) ( 0, I ) (an identity and a zero matrix) (4) ( I, I ) (a pair of identity matrices) t t (5) ( J , J ) (a pair of transposed Jordan blocks) t ( I 0 ) (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) ( 0 I ) ( 0 J ) and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N1)/2 (7) ( D, I ) where D is diag( 0, 1,..., N1 ) (a diagonal matrix with those diagonal entries.) (8) ( I, D ) (9) ( big*D, small*I ) where "big" is near overflow and small=1/big (10) ( small*D, big*I ) (11) ( big*I, small*D ) (12) ( small*I, big*D ) (13) ( big*D, big*I ) (14) ( small*D, small*I ) (15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N3, 0 ) and D2 is diag( 0, N3, N4,..., 1, 0, 0 ) t t (16) Q ( J , J ) Z where Q and Z are random orthogonal matrices. (17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N3, 0 ) and diag(T2) = ( 0, N3, N4,..., 1, 0, 0 ) (18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1d, ..., 1(N5)*d=s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) N5 (20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N4), 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) where r1,..., r(N4) are random. (22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) Q ( T1, T2 ) Z where T1 and T2 are random uppertriangular matrices.
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, DDRGES3 does nothing. NSIZES >= 0.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DDRGES3 does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A on input. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRGES3 to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. THRESH >= 0.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
AA is COMPLEX*16 array, dimension(LDA, max(NN)) Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
LDALDA is INTEGER The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ).
BB is COMPLEX*16 array, dimension(LDA, max(NN)) Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
SS is COMPLEX*16 array, dimension (LDA, max(NN)) The Schur form matrix computed from A by ZGGES3. On exit, S contains the Schur form matrix corresponding to the matrix in A.
TT is COMPLEX*16 array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by ZGGES3.
QQ is COMPLEX*16 array, dimension (LDQ, max(NN)) The (left) orthogonal matrix computed by ZGGES3.
LDQLDQ is INTEGER The leading dimension of Q and Z. It must be at least 1 and at least max( NN ).
ZZ is COMPLEX*16 array, dimension( LDQ, max(NN) ) The (right) orthogonal matrix computed by ZGGES3.
ALPHAALPHA is COMPLEX*16 array, dimension (max(NN))
BETABETA is COMPLEX*16 array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by ZGGES3. ALPHA(k) / BETA(k) is the kth generalized eigenvalue of A and B.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The dimension of the array WORK. LWORK >= 3*N*N.
RWORKRWORK is DOUBLE PRECISION array, dimension ( 8*N ) Real workspace.
RESULTRESULT is DOUBLE PRECISION array, dimension (15) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
BWORKBWORK is LOGICAL array, dimension (N)
INFOINFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 February 2015
subroutine zdrgev (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) B, complex*16, dimension( lda, * ) S, complex*16, dimension( lda, * ) T, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldq, * ) Z, complex*16, dimension( ldqe, * ) QE, integer LDQE, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( * ) ALPHA1, complex*16, dimension( * ) BETA1, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( * ) RESULT, integer INFO)
ZDRGEV
Purpose:

ZDRGEV checks the nonsymmetric generalized eigenvalue problem driver routine ZGGEV. ZGGEV computes for a pair of nbyn nonsymmetric matrices (A,B) the generalized eigenvalues and, optionally, the left and right eigenvectors. A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A  w*B is singular. It is usually represented as the pair (alpha,beta), as there is reasonable interpretation for beta=0, and even for both being zero. A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A  wB) * r = 0. A left generalized eigenvector is a vector l such that l**H * (A  wB) = 0, where l**H is the conjugatetranspose of l. When ZDRGEV is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following tests will be performed and compared with the threshold THRESH. Results from ZGGEV: (1) max over all left eigenvalue/vector pairs (alpha/beta,l) of  VL**H * (beta A  alpha B) /( ulp max(beta A, alpha B) ) where VL**H is the conjugatetranspose of VL. (2)  VL(i)  1  / ulp and whether largest component real VL(i) denotes the ith column of VL. (3) max over all left eigenvalue/vector pairs (alpha/beta,r) of  (beta A  alpha B) * VR  / ( ulp max(beta A, alpha B) ) (4)  VR(i)  1  / ulp and whether largest component real VR(i) denotes the ith column of VR. (5) W(full) = W(partial) W(full) denotes the eigenvalues computed when both l and r are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and r, or only W and l are computed. (6) VL(full) = VL(partial) VL(full) denotes the left eigenvectors computed when both l and r are computed, and VL(partial) denotes the result when only l is computed. (7) VR(full) = VR(partial) VR(full) denotes the right eigenvectors computed when both l and r are also computed, and VR(partial) denotes the result when only l is computed. Test Matrices   The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) ( 0, 0 ) (a pair of zero matrices) (2) ( I, 0 ) (an identity and a zero matrix) (3) ( 0, I ) (an identity and a zero matrix) (4) ( I, I ) (a pair of identity matrices) t t (5) ( J , J ) (a pair of transposed Jordan blocks) t ( I 0 ) (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) ( 0 I ) ( 0 J ) and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N1)/2 (7) ( D, I ) where D is diag( 0, 1,..., N1 ) (a diagonal matrix with those diagonal entries.) (8) ( I, D ) (9) ( big*D, small*I ) where "big" is near overflow and small=1/big (10) ( small*D, big*I ) (11) ( big*I, small*D ) (12) ( small*I, big*D ) (13) ( big*D, big*I ) (14) ( small*D, small*I ) (15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N3, 0 ) and D2 is diag( 0, N3, N4,..., 1, 0, 0 ) t t (16) Q ( J , J ) Z where Q and Z are random orthogonal matrices. (17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N3, 0 ) and diag(T2) = ( 0, N3, N4,..., 1, 0, 0 ) (18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1d, ..., 1(N5)*d=s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) N5 (20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N4), 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) where r1,..., r(N4) are random. (22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) Q ( T1, T2 ) Z where T1 and T2 are random uppertriangular matrices.
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, ZDRGES does nothing. NSIZES >= 0.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, ZDRGEV does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZDRGES to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IERR not equal to 0.)
AA is COMPLEX*16 array, dimension(LDA, max(NN)) Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
LDALDA is INTEGER The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ).
BB is COMPLEX*16 array, dimension(LDA, max(NN)) Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
SS is COMPLEX*16 array, dimension (LDA, max(NN)) The Schur form matrix computed from A by ZGGEV. On exit, S contains the Schur form matrix corresponding to the matrix in A.
TT is COMPLEX*16 array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by ZGGEV.
QQ is COMPLEX*16 array, dimension (LDQ, max(NN)) The (left) eigenvectors matrix computed by ZGGEV.
LDQLDQ is INTEGER The leading dimension of Q and Z. It must be at least 1 and at least max( NN ).
ZZ is COMPLEX*16 array, dimension( LDQ, max(NN) ) The (right) orthogonal matrix computed by ZGGEV.
QEQE is COMPLEX*16 array, dimension( LDQ, max(NN) ) QE holds the computed right or left eigenvectors.
LDQELDQE is INTEGER The leading dimension of QE. LDQE >= max(1,max(NN)).
ALPHAALPHA is COMPLEX*16 array, dimension (max(NN))
BETABETA is COMPLEX*16 array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by ZGGEV. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the kth generalized eigenvalue of A and B.
ALPHA1ALPHA1 is COMPLEX*16 array, dimension (max(NN))
BETA1BETA1 is COMPLEX*16 array, dimension (max(NN)) Like ALPHAR, ALPHAI, BETA, these arrays contain the eigenvalues of A and B, but those computed when ZGGEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The number of entries in WORK. LWORK >= N*(N+1)
RWORKRWORK is DOUBLE PRECISION array, dimension (8*N) Real workspace.
RESULTRESULT is DOUBLE PRECISION array, dimension (2) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
INFOINFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zdrgev3 (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) B, complex*16, dimension( lda, * ) S, complex*16, dimension( lda, * ) T, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldq, * ) Z, complex*16, dimension( ldqe, * ) QE, integer LDQE, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( * ) ALPHA1, complex*16, dimension( * ) BETA1, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( * ) RESULT, integer INFO)
ZDRGEV3
Purpose:

ZDRGEV3 checks the nonsymmetric generalized eigenvalue problem driver routine ZGGEV3. ZGGEV3 computes for a pair of nbyn nonsymmetric matrices (A,B) the generalized eigenvalues and, optionally, the left and right eigenvectors. A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A  w*B is singular. It is usually represented as the pair (alpha,beta), as there is reasonable interpretation for beta=0, and even for both being zero. A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A  wB) * r = 0. A left generalized eigenvector is a vector l such that l**H * (A  wB) = 0, where l**H is the conjugatetranspose of l. When ZDRGEV3 is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following tests will be performed and compared with the threshold THRESH. Results from ZGGEV3: (1) max over all left eigenvalue/vector pairs (alpha/beta,l) of  VL**H * (beta A  alpha B) /( ulp max(beta A, alpha B) ) where VL**H is the conjugatetranspose of VL. (2)  VL(i)  1  / ulp and whether largest component real VL(i) denotes the ith column of VL. (3) max over all left eigenvalue/vector pairs (alpha/beta,r) of  (beta A  alpha B) * VR  / ( ulp max(beta A, alpha B) ) (4)  VR(i)  1  / ulp and whether largest component real VR(i) denotes the ith column of VR. (5) W(full) = W(partial) W(full) denotes the eigenvalues computed when both l and r are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and r, or only W and l are computed. (6) VL(full) = VL(partial) VL(full) denotes the left eigenvectors computed when both l and r are computed, and VL(partial) denotes the result when only l is computed. (7) VR(full) = VR(partial) VR(full) denotes the right eigenvectors computed when both l and r are also computed, and VR(partial) denotes the result when only l is computed. Test Matrices   The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) ( 0, 0 ) (a pair of zero matrices) (2) ( I, 0 ) (an identity and a zero matrix) (3) ( 0, I ) (an identity and a zero matrix) (4) ( I, I ) (a pair of identity matrices) t t (5) ( J , J ) (a pair of transposed Jordan blocks) t ( I 0 ) (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) ( 0 I ) ( 0 J ) and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N1)/2 (7) ( D, I ) where D is diag( 0, 1,..., N1 ) (a diagonal matrix with those diagonal entries.) (8) ( I, D ) (9) ( big*D, small*I ) where "big" is near overflow and small=1/big (10) ( small*D, big*I ) (11) ( big*I, small*D ) (12) ( small*I, big*D ) (13) ( big*D, big*I ) (14) ( small*D, small*I ) (15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N3, 0 ) and D2 is diag( 0, N3, N4,..., 1, 0, 0 ) t t (16) Q ( J , J ) Z where Q and Z are random orthogonal matrices. (17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N3, 0 ) and diag(T2) = ( 0, N3, N4,..., 1, 0, 0 ) (18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1d, ..., 1(N5)*d=s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) N5 (20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N4), 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) where r1,..., r(N4) are random. (22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) Q ( T1, T2 ) Z where T1 and T2 are random uppertriangular matrices.
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, ZDRGEV3 does nothing. NSIZES >= 0.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, ZDRGEV3 does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZDRGES to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IERR not equal to 0.)
AA is COMPLEX*16 array, dimension(LDA, max(NN)) Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
LDALDA is INTEGER The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ).
BB is COMPLEX*16 array, dimension(LDA, max(NN)) Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
SS is COMPLEX*16 array, dimension (LDA, max(NN)) The Schur form matrix computed from A by ZGGEV3. On exit, S contains the Schur form matrix corresponding to the matrix in A.
TT is COMPLEX*16 array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by ZGGEV3.
QQ is COMPLEX*16 array, dimension (LDQ, max(NN)) The (left) eigenvectors matrix computed by ZGGEV3.
LDQLDQ is INTEGER The leading dimension of Q and Z. It must be at least 1 and at least max( NN ).
ZZ is COMPLEX*16 array, dimension( LDQ, max(NN) ) The (right) orthogonal matrix computed by ZGGEV3.
QEQE is COMPLEX*16 array, dimension( LDQ, max(NN) ) QE holds the computed right or left eigenvectors.
LDQELDQE is INTEGER The leading dimension of QE. LDQE >= max(1,max(NN)).
ALPHAALPHA is COMPLEX*16 array, dimension (max(NN))
BETABETA is COMPLEX*16 array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by ZGGEV3. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the kth generalized eigenvalue of A and B.
ALPHA1ALPHA1 is COMPLEX*16 array, dimension (max(NN))
BETA1BETA1 is COMPLEX*16 array, dimension (max(NN)) Like ALPHAR, ALPHAI, BETA, these arrays contain the eigenvalues of A and B, but those computed when ZGGEV3 only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The number of entries in WORK. LWORK >= N*(N+1)
RWORKRWORK is DOUBLE PRECISION array, dimension (8*N) Real workspace.
RESULTRESULT is DOUBLE PRECISION array, dimension (2) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
INFOINFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 Febuary 2015
subroutine zdrgsx (integer NSIZE, integer NCMAX, double precision THRESH, integer NIN, integer NOUT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) B, complex*16, dimension( lda, * ) AI, complex*16, dimension( lda, * ) BI, complex*16, dimension( lda, * ) Z, complex*16, dimension( lda, * ) Q, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldc, * ) C, integer LDC, double precision, dimension( * ) S, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO)
ZDRGSX
Purpose:

ZDRGSX checks the nonsymmetric generalized eigenvalue (Schur form) problem expert driver ZGGESX. ZGGES factors A and B as Q*S*Z' and Q*T*Z' , where ' means conjugate transpose, S and T are upper triangular (i.e., in generalized Schur form), and Q and Z are unitary. It also computes the generalized eigenvalues (alpha(j),beta(j)), j=1,...,n. Thus, w(j) = alpha(j)/beta(j) is a root of the characteristic equation det( A  w(j) B ) = 0 Optionally it also reorders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms; computes a reciprocal condition number for the average of the selected eigenvalues; and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues. When ZDRGSX is called with NSIZE > 0, five (5) types of builtin matrix pairs are used to test the routine ZGGESX. When ZDRGSX is called with NSIZE = 0, it reads in test matrix data to test ZGGESX. (need more details on what kind of readin data are needed). For each matrix pair, the following tests will be performed and compared with the threshold THRESH except for the tests (7) and (9): (1)  A  Q S Z'  / ( A n ulp ) (2)  B  Q T Z'  / ( B n ulp ) (3)  I  QQ'  / ( n ulp ) (4)  I  ZZ'  / ( n ulp ) (5) if A is in Schur form (i.e. triangular form) (6) maximum over j of D(j) where: alpha(j)  S(j,j) beta(j)  T(j,j) D(j) =  +  max(alpha(j),S(j,j)) max(beta(j),T(j,j)) (7) if sorting worked and SDIM is the number of eigenvalues which were selected. (8) the estimated value DIF does not differ from the true values of Difu and Difl more than a factor 10*THRESH. If the estimate DIF equals zero the corresponding true values of Difu and Difl should be less than EPS*norm(A, B). If the true value of Difu and Difl equal zero, the estimate DIF should be less than EPS*norm(A, B). (9) If INFO = N+3 is returned by ZGGESX, the reordering "failed" and we check that DIF = PL = PR = 0 and that the true value of Difu and Difl is < EPS*norm(A, B). We count the events when INFO=N+3. For readin test matrices, the same tests are run except that the exact value for DIF (and PL) is input data. Additionally, there is one more test run for readin test matrices: (10) the estimated value PL does not differ from the true value of PLTRU more than a factor THRESH. If the estimate PL equals zero the corresponding true value of PLTRU should be less than EPS*norm(A, B). If the true value of PLTRU equal zero, the estimate PL should be less than EPS*norm(A, B). Note that for the builtin tests, a total of 10*NSIZE*(NSIZE1) matrix pairs are generated and tested. NSIZE should be kept small. SVD (routine ZGESVD) is used for computing the true value of DIF_u and DIF_l when testing the builtin test problems. Builtin Test Matrices ====================== All builtin test matrices are the 2 by 2 block of triangular matrices A = [ A11 A12 ] and B = [ B11 B12 ] [ A22 ] [ B22 ] where for different type of A11 and A22 are given as the following. A12 and B12 are chosen so that the generalized Sylvester equation A11*R  L*A22 = A12 B11*R  L*B22 = B12 have prescribed solution R and L. Type 1: A11 = J_m(1,1) and A_22 = J_k(1a,1). B11 = I_m, B22 = I_k where J_k(a,b) is the kbyk Jordan block with ``a'' on diagonal and ``b'' on superdiagonal. Type 2: A11 = (a_ij) = ( 2(.5sin(i)) ) and B11 = (b_ij) = ( 2(.5sin(ij)) ) for i=1,...,m, j=i,...,m A22 = (a_ij) = ( 2(.5sin(i+j)) ) and B22 = (b_ij) = ( 2(.5sin(ij)) ) for i=m+1,...,k, j=i,...,k Type 3: A11, A22 and B11, B22 are chosen as for Type 2, but each second diagonal block in A_11 and each third diagonal block in A_22 are made as 2 by 2 blocks. Type 4: A11 = ( 20(.5  sin(ij)) ) and B22 = ( 2(.5  sin(i+j)) ) for i=1,...,m, j=1,...,m and A22 = ( 20(.5  sin(i+j)) ) and B22 = ( 2(.5  sin(ij)) ) for i=m+1,...,k, j=m+1,...,k Type 5: (A,B) and have potentially close or common eigenvalues and very large departure from block diagonality A_11 is chosen as the m x m leading submatrix of A_1:  1 b   b 1   1+d b   b 1+d  A_1 =  d 1   1 d   d 1   1 d   1  and A_22 is chosen as the k x k leading submatrix of A_2:  1 b   b 1   1d b   b 1d  A_2 =  d 1+b   1b d   d 1+b   1+b d   1d  and matrix B are chosen as identity matrices (see DLATM5).
Parameters:

NSIZE
NSIZE is INTEGER The maximum size of the matrices to use. NSIZE >= 0. If NSIZE = 0, no builtin tests matrices are used, but readin test matrices are used to test DGGESX.
NCMAXNCMAX is INTEGER Maximum allowable NMAX for generating Kroneker matrix in call to ZLAKF2
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. THRESH >= 0.
NINNIN is INTEGER The FORTRAN unit number for reading in the data file of problems to solve.
NOUTNOUT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)
AA is COMPLEX*16 array, dimension (LDA, NSIZE) Used to store the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used.
LDALDA is INTEGER The leading dimension of A, B, AI, BI, Z and Q, LDA >= max( 1, NSIZE ). For the readin test, LDA >= max( 1, N ), N is the size of the test matrices.
BB is COMPLEX*16 array, dimension (LDA, NSIZE) Used to store the matrix whose eigenvalues are to be computed. On exit, B contains the last matrix actually used.
AIAI is COMPLEX*16 array, dimension (LDA, NSIZE) Copy of A, modified by ZGGESX.
BIBI is COMPLEX*16 array, dimension (LDA, NSIZE) Copy of B, modified by ZGGESX.
ZZ is COMPLEX*16 array, dimension (LDA, NSIZE) Z holds the left Schur vectors computed by ZGGESX.
QQ is COMPLEX*16 array, dimension (LDA, NSIZE) Q holds the right Schur vectors computed by ZGGESX.
ALPHAALPHA is COMPLEX*16 array, dimension (NSIZE)
BETABETA is COMPLEX*16 array, dimension (NSIZE) On exit, ALPHA/BETA are the eigenvalues.
CC is COMPLEX*16 array, dimension (LDC, LDC) Store the matrix generated by subroutine ZLAKF2, this is the matrix formed by Kronecker products used for estimating DIF.
LDCLDC is INTEGER The leading dimension of C. LDC >= max(1, LDA*LDA/2 ).
SS is DOUBLE PRECISION array, dimension (LDC) Singular values of C
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The dimension of the array WORK. LWORK >= 3*NSIZE*NSIZE/2
RWORKRWORK is DOUBLE PRECISION array, dimension (5*NSIZE*NSIZE/2  4)
IWORKIWORK is INTEGER array, dimension (LIWORK)
LIWORKLIWORK is INTEGER The dimension of the array IWORK. LIWORK >= NSIZE + 2.
BWORKBWORK is LOGICAL array, dimension (NSIZE)
INFOINFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zdrgvx (integer NSIZE, double precision THRESH, integer NIN, integer NOUT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) B, complex*16, dimension( lda, * ) AI, complex*16, dimension( lda, * ) BI, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( lda, * ) VL, complex*16, dimension( lda, * ) VR, integer ILO, integer IHI, double precision, dimension( * ) LSCALE, double precision, dimension( * ) RSCALE, double precision, dimension( * ) S, double precision, dimension( * ) DTRU, double precision, dimension( * ) DIF, double precision, dimension( * ) DIFTRU, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer LIWORK, double precision, dimension( 4 ) RESULT, logical, dimension( * ) BWORK, integer INFO)
ZDRGVX
Purpose:

ZDRGVX checks the nonsymmetric generalized eigenvalue problem expert driver ZGGEVX. ZGGEVX computes the generalized eigenvalues, (optionally) the left and/or right eigenvectors, (optionally) computes a balancing transformation to improve the conditioning, and (optionally) reciprocal condition numbers for the eigenvalues and eigenvectors. When ZDRGVX is called with NSIZE > 0, two types of test matrix pairs are generated by the subroutine DLATM6 and test the driver ZGGEVX. The test matrices have the known exact condition numbers for eigenvalues. For the condition numbers of the eigenvectors corresponding the first and last eigenvalues are also know ``exactly'' (see ZLATM6). For each matrix pair, the following tests will be performed and compared with the threshold THRESH. (1) max over all left eigenvalue/vector pairs (beta/alpha,l) of  l**H * (beta A  alpha B)  / ( ulp max( beta A, alpha B ) ) where l**H is the conjugate tranpose of l. (2) max over all right eigenvalue/vector pairs (beta/alpha,r) of  (beta A  alpha B) r  / ( ulp max( beta A, alpha B ) ) (3) The condition number S(i) of eigenvalues computed by ZGGEVX differs less than a factor THRESH from the exact S(i) (see ZLATM6). (4) DIF(i) computed by ZTGSNA differs less than a factor 10*THRESH from the exact value (for the 1st and 5th vectors only). Test Matrices ============= Two kinds of test matrix pairs (A, B) = inverse(YH) * (Da, Db) * inverse(X) are used in the tests: 1: Da = 1+a 0 0 0 0 Db = 1 0 0 0 0 0 2+a 0 0 0 0 1 0 0 0 0 0 3+a 0 0 0 0 1 0 0 0 0 0 4+a 0 0 0 0 1 0 0 0 0 0 5+a , 0 0 0 0 1 , and 2: Da = 1 1 0 0 0 Db = 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1+a 1+b 0 0 0 1 0 0 0 0 1b 1+a , 0 0 0 0 1 . In both cases the same inverse(YH) and inverse(X) are used to compute (A, B), giving the exact eigenvectors to (A,B) as (YH, X): YH: = 1 0 y y y X = 1 0 x x x 0 1 y y y 0 1 x x x 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1, 0 0 0 0 1 , where a, b, x and y will have all values independently of each other from { sqrt(sqrt(ULP)), 0.1, 1, 10, 1/sqrt(sqrt(ULP)) }.
Parameters:

NSIZE
NSIZE is INTEGER The number of sizes of matrices to use. NSIZE must be at least zero. If it is zero, no randomly generated matrices are tested, but any test matrices read from NIN will be tested. If it is not zero, then N = 5.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NINNIN is INTEGER The FORTRAN unit number for reading in the data file of problems to solve.
NOUTNOUT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
AA is COMPLEX*16 array, dimension (LDA, NSIZE) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used.
LDALDA is INTEGER The leading dimension of A, B, AI, BI, Ao, and Bo. It must be at least 1 and at least NSIZE.
BB is COMPLEX*16 array, dimension (LDA, NSIZE) Used to hold the matrix whose eigenvalues are to be computed. On exit, B contains the last matrix actually used.
AIAI is COMPLEX*16 array, dimension (LDA, NSIZE) Copy of A, modified by ZGGEVX.
BIBI is COMPLEX*16 array, dimension (LDA, NSIZE) Copy of B, modified by ZGGEVX.
ALPHAALPHA is COMPLEX*16 array, dimension (NSIZE)
BETABETA is COMPLEX*16 array, dimension (NSIZE) On exit, ALPHA/BETA are the eigenvalues.
VLVL is COMPLEX*16 array, dimension (LDA, NSIZE) VL holds the left eigenvectors computed by ZGGEVX.
VRVR is COMPLEX*16 array, dimension (LDA, NSIZE) VR holds the right eigenvectors computed by ZGGEVX.
ILOILO is INTEGER
IHIIHI is INTEGER
LSCALELSCALE is DOUBLE PRECISION array, dimension (N)
RSCALERSCALE is DOUBLE PRECISION array, dimension (N)
SS is DOUBLE PRECISION array, dimension (N)
DTRUDTRU is DOUBLE PRECISION array, dimension (N)
DIFDIF is DOUBLE PRECISION array, dimension (N)
DIFTRUDIFTRU is DOUBLE PRECISION array, dimension (N)
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER Leading dimension of WORK. LWORK >= 2*N*N + 2*N
RWORKRWORK is DOUBLE PRECISION array, dimension (6*N)
IWORKIWORK is INTEGER array, dimension (LIWORK)
LIWORKLIWORK is INTEGER Leading dimension of IWORK. LIWORK >= N+2.
RESULTRESULT is DOUBLE PRECISION array, dimension (4)
BWORKBWORK is LOGICAL array, dimension (N)
INFOINFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zdrvbd (integer NSIZES, integer, dimension( * ) MM, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldvt, * ) VT, integer LDVT, complex*16, dimension( lda, * ) ASAV, complex*16, dimension( ldu, * ) USAV, complex*16, dimension( ldvt, * ) VTSAV, double precision, dimension( * ) S, double precision, dimension( * ) SSAV, double precision, dimension( * ) E, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer NOUNIT, integer INFO)
ZDRVBD
Purpose:

ZDRVBD checks the singular value decomposition (SVD) driver ZGESVD and ZGESDD. ZGESVD and ZGESDD factors A = U diag(S) VT, where U and VT are unitary and diag(S) is diagonal with the entries of the array S on its diagonal. The entries of S are the singular values, nonnegative and stored in decreasing order. U and VT can be optionally not computed, overwritten on A, or computed partially. A is M by N. Let MNMIN = min( M, N ). S has dimension MNMIN. U can be M by M or M by MNMIN. VT can be N by N or MNMIN by N. When ZDRVBD is called, a number of matrix "sizes" (M's and N's) and a number of matrix "types" are specified. For each size (M,N) and each type of matrix, and for the minimal workspace as well as workspace adequate to permit blocking, an M x N matrix "A" will be generated and used to test the SVD routines. For each matrix, A will be factored as A = U diag(S) VT and the following 12 tests computed: Test for ZGESVD: (1)  A  U diag(S) VT  / ( A max(M,N) ulp ) (2)  I  U'U  / ( M ulp ) (3)  I  VT VT'  / ( N ulp ) (4) S contains MNMIN nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) (5)  U  Upartial  / ( M ulp ) where Upartial is a partially computed U. (6)  VT  VTpartial  / ( N ulp ) where VTpartial is a partially computed VT. (7)  S  Spartial  / ( MNMIN ulp S ) where Spartial is the vector of singular values from the partial SVD Test for ZGESDD: (1)  A  U diag(S) VT  / ( A max(M,N) ulp ) (2)  I  U'U  / ( M ulp ) (3)  I  VT VT'  / ( N ulp ) (4) S contains MNMIN nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) (5)  U  Upartial  / ( M ulp ) where Upartial is a partially computed U. (6)  VT  VTpartial  / ( N ulp ) where VTpartial is a partially computed VT. (7)  S  Spartial  / ( MNMIN ulp S ) where Spartial is the vector of singular values from the partial SVD Test for ZGESVJ: (1)  A  U diag(S) VT  / ( A max(M,N) ulp ) (2)  I  U'U  / ( M ulp ) (3)  I  VT VT'  / ( N ulp ) (4) S contains MNMIN nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) Test for ZGEJSV: (1)  A  U diag(S) VT  / ( A max(M,N) ulp ) (2)  I  U'U  / ( M ulp ) (3)  I  VT VT'  / ( N ulp ) (4) S contains MNMIN nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) Test for ZGESVDX( 'V', 'V', 'A' )/ZGESVDX( 'N', 'N', 'A' ) (1)  A  U diag(S) VT  / ( A max(M,N) ulp ) (2)  I  U'U  / ( M ulp ) (3)  I  VT VT'  / ( N ulp ) (4) S contains MNMIN nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) (5)  U  Upartial  / ( M ulp ) where Upartial is a partially computed U. (6)  VT  VTpartial  / ( N ulp ) where VTpartial is a partially computed VT. (7)  S  Spartial  / ( MNMIN ulp S ) where Spartial is the vector of singular values from the partial SVD Test for ZGESVDX( 'V', 'V', 'I' ) (8)  U' A VT'''  diag(S)  / ( A max(M,N) ulp ) (9)  I  U'U  / ( M ulp ) (10)  I  VT VT'  / ( N ulp ) Test for ZGESVDX( 'V', 'V', 'V' ) (11)  U' A VT'''  diag(S)  / ( A max(M,N) ulp ) (12)  I  U'U  / ( M ulp ) (13)  I  VT VT'  / ( N ulp ) The "sizes" are specified by the arrays MM(1:NSIZES) and NN(1:NSIZES); the value of each element pair (MM(j),NN(j)) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A matrix of the form U D V, where U and V are unitary and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (4) Same as (3), but multiplied by the underflowthreshold / ULP. (5) Same as (3), but multiplied by the overflowthreshold * ULP.
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, ZDRVBD does nothing. It must be at least zero.
MMMM is INTEGER array, dimension (NSIZES) An array containing the matrix "heights" to be used. For each j=1,...,NSIZES, if MM(j) is zero, then MM(j) and NN(j) will be ignored. The MM(j) values must be at least zero.
NNNN is INTEGER array, dimension (NSIZES) An array containing the matrix "widths" to be used. For each j=1,...,NSIZES, if NN(j) is zero, then MM(j) and NN(j) will be ignored. The NN(j) values must be at least zero.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, ZDRVBD does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrices are in A and B. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZDRVBD to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
AA is COMPLEX*16 array, dimension (LDA,max(NN)) Used to hold the matrix whose singular values are to be computed. On exit, A contains the last matrix actually used.
LDALDA is INTEGER The leading dimension of A. It must be at least 1 and at least max( MM ).
UU is COMPLEX*16 array, dimension (LDU,max(MM)) Used to hold the computed matrix of right singular vectors. On exit, U contains the last such vectors actually computed.
LDULDU is INTEGER The leading dimension of U. It must be at least 1 and at least max( MM ).
VTVT is COMPLEX*16 array, dimension (LDVT,max(NN)) Used to hold the computed matrix of left singular vectors. On exit, VT contains the last such vectors actually computed.
LDVTLDVT is INTEGER The leading dimension of VT. It must be at least 1 and at least max( NN ).
ASAVASAV is COMPLEX*16 array, dimension (LDA,max(NN)) Used to hold a different copy of the matrix whose singular values are to be computed. On exit, A contains the last matrix actually used.
USAVUSAV is COMPLEX*16 array, dimension (LDU,max(MM)) Used to hold a different copy of the computed matrix of right singular vectors. On exit, USAV contains the last such vectors actually computed.
VTSAVVTSAV is COMPLEX*16 array, dimension (LDVT,max(NN)) Used to hold a different copy of the computed matrix of left singular vectors. On exit, VTSAV contains the last such vectors actually computed.
SS is DOUBLE PRECISION array, dimension (max(min(MM,NN))) Contains the computed singular values.
SSAVSSAV is DOUBLE PRECISION array, dimension (max(min(MM,NN))) Contains another copy of the computed singular values.
EE is DOUBLE PRECISION array, dimension (max(min(MM,NN))) Workspace for ZGESVD.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The number of entries in WORK. This must be at least MAX(3*MIN(M,N)+MAX(M,N)**2,5*MIN(M,N),3*MAX(M,N)) for all pairs (M,N)=(MM(j),NN(j))
RWORKRWORK is DOUBLE PRECISION array, dimension ( 5*max(max(MM,NN)) )
IWORKIWORK is INTEGER array, dimension at least 8*min(M,N)
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
INFOINFO is INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some MM(j) < 0 3: Some NN(j) < 0 4: NTYPES < 0 7: THRESH < 0 10: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ). 12: LDU < 1 or LDU < MMAX. 14: LDVT < 1 or LDVT < NMAX, where NMAX is max( NN(j) ). 21: LWORK too small. If ZLATMS, or ZGESVD returns an error code, the absolute value of it is returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zdrves (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) H, complex*16, dimension( lda, * ) HT, complex*16, dimension( * ) W, complex*16, dimension( * ) WT, complex*16, dimension( ldvs, * ) VS, integer LDVS, double precision, dimension( 13 ) RESULT, complex*16, dimension( * ) WORK, integer NWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, logical, dimension( * ) BWORK, integer INFO)
ZDRVES
Purpose:

ZDRVES checks the nonsymmetric eigenvalue (Schur form) problem driver ZGEES. When ZDRVES is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 13 tests will be performed: (1) 0 if T is in Schur form, 1/ulp otherwise (no sorting of eigenvalues) (2)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (no sorting of eigenvalues). (3)  I  VS VS'  / ( n ulp ) (no sorting of eigenvalues). (4) 0 if W are eigenvalues of T 1/ulp otherwise (no sorting of eigenvalues) (5) 0 if T(with VS) = T(without VS), 1/ulp otherwise (no sorting of eigenvalues) (6) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise (no sorting of eigenvalues) (7) 0 if T is in Schur form, 1/ulp otherwise (with sorting of eigenvalues) (8)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (with sorting of eigenvalues). (9)  I  VS VS'  / ( n ulp ) (with sorting of eigenvalues). (10) 0 if W are eigenvalues of T 1/ulp otherwise (with sorting of eigenvalues) (11) 0 if T(with VS) = T(without VS), 1/ulp otherwise (with sorting of eigenvalues) (12) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise (with sorting of eigenvalues) (13) if sorting worked and SDIM is the number of eigenvalues which were SELECTed The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A (transposed) Jordan block, with 1's on the diagonal. (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random complex angles. (ULP = (first number larger than 1)  1 ) (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random complex angles. (6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random complex angles. (7) Same as (4), but multiplied by a constant near the overflow threshold (8) Same as (4), but multiplied by a constant near the underflow threshold (9) A matrix of the form U' T U, where U is unitary and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (10) A matrix of the form U' T U, where U is unitary and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (11) A matrix of the form U' T U, where U is orthogonal and T has "clustered" entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (12) A matrix of the form U' T U, where U is unitary and T has complex eigenvalues randomly chosen from ULP < z < 1 and random O(1) entries in the upper triangle. (13) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (14) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (15) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (16) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has complex eigenvalues randomly chosen from ULP < z < 1 and random O(1) entries in the upper triangle. (17) Same as (16), but multiplied by a constant near the overflow threshold (18) Same as (16), but multiplied by a constant near the underflow threshold (19) Nonsymmetric matrix with random entries chosen from (1,1). If N is at least 4, all entries in first two rows and last row, and first column and last two columns are zero. (20) Same as (19), but multiplied by a constant near the overflow threshold (21) Same as (19), but multiplied by a constant near the underflow threshold
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, ZDRVES does nothing. It must be at least zero.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, ZDRVES does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZDRVES to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)
AA is COMPLEX*16 array, dimension (LDA, max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used.
LDALDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least max( NN ).
HH is COMPLEX*16 array, dimension (LDA, max(NN)) Another copy of the test matrix A, modified by ZGEES.
HTHT is COMPLEX*16 array, dimension (LDA, max(NN)) Yet another copy of the test matrix A, modified by ZGEES.
WW is COMPLEX*16 array, dimension (max(NN)) The computed eigenvalues of A.
WTWT is COMPLEX*16 array, dimension (max(NN)) Like W, this array contains the eigenvalues of A, but those computed when ZGEES only computes a partial eigendecomposition, i.e. not Schur vectors
VSVS is COMPLEX*16 array, dimension (LDVS, max(NN)) VS holds the computed Schur vectors.
LDVSLDVS is INTEGER Leading dimension of VS. Must be at least max(1,max(NN)).
RESULTRESULT is DOUBLE PRECISION array, dimension (13) The values computed by the 13 tests described above. The values are currently limited to 1/ulp, to avoid overflow.
WORKWORK is COMPLEX*16 array, dimension (NWORK)
NWORKNWORK is INTEGER The number of entries in WORK. This must be at least 5*NN(j)+2*NN(j)**2 for all j.
RWORKRWORK is DOUBLE PRECISION array, dimension (max(NN))
IWORKIWORK is INTEGER array, dimension (max(NN))
BWORKBWORK is LOGICAL array, dimension (max(NN))
INFOINFO is INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 6: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 15: LDVS < 1 or LDVS < NMAX, where NMAX is max( NN(j) ). 18: NWORK too small. If ZLATMR, CLATMS, CLATME or ZGEES returns an error code, the absolute value of it is returned.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NMAX Largest value in NN. NERRS The number of tests which have exceeded THRESH COND, CONDS, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTULP, RTULPI Square roots of the previous 4 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) KCONDS(j) Select whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.)
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zdrvev (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) H, complex*16, dimension( * ) W, complex*16, dimension( * ) W1, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, complex*16, dimension( ldlre, * ) LRE, integer LDLRE, double precision, dimension( 7 ) RESULT, complex*16, dimension( * ) WORK, integer NWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO)
ZDRVEV
Purpose:

ZDRVEV checks the nonsymmetric eigenvalue problem driver ZGEEV. When ZDRVEV is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 7 tests will be performed: (1)  A * VR  VR * W  / ( n A ulp ) Here VR is the matrix of unit right eigenvectors. W is a diagonal matrix with diagonal entries W(j). (2)  A**H * VL  VL * W**H  / ( n A ulp ) Here VL is the matrix of unit left eigenvectors, A**H is the conjugatetranspose of A, and W is as above. (3)  VR(i)  1  / ulp and whether largest component real VR(i) denotes the ith column of VR. (4)  VL(i)  1  / ulp and whether largest component real VL(i) denotes the ith column of VL. (5) W(full) = W(partial) W(full) denotes the eigenvalues computed when both VR and VL are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and VR, or only W and VL are computed. (6) VR(full) = VR(partial) VR(full) denotes the right eigenvectors computed when both VR and VL are computed, and VR(partial) denotes the result when only VR is computed. (7) VL(full) = VL(partial) VL(full) denotes the left eigenvectors computed when both VR and VL are also computed, and VL(partial) denotes the result when only VL is computed. The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A (transposed) Jordan block, with 1's on the diagonal. (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random complex angles. (ULP = (first number larger than 1)  1 ) (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random complex angles. (6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random complex angles. (7) Same as (4), but multiplied by a constant near the overflow threshold (8) Same as (4), but multiplied by a constant near the underflow threshold (9) A matrix of the form U' T U, where U is unitary and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (10) A matrix of the form U' T U, where U is unitary and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (11) A matrix of the form U' T U, where U is unitary and T has "clustered" entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (12) A matrix of the form U' T U, where U is unitary and T has complex eigenvalues randomly chosen from ULP < z < 1 and random O(1) entries in the upper triangle. (13) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (14) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (15) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (16) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has complex eigenvalues randomly chosen from ULP < z < 1 and random O(1) entries in the upper triangle. (17) Same as (16), but multiplied by a constant near the overflow threshold (18) Same as (16), but multiplied by a constant near the underflow threshold (19) Nonsymmetric matrix with random entries chosen from z < 1 If N is at least 4, all entries in first two rows and last row, and first column and last two columns are zero. (20) Same as (19), but multiplied by a constant near the overflow threshold (21) Same as (19), but multiplied by a constant near the underflow threshold
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, ZDRVEV does nothing. It must be at least zero.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. If it is zero, ZDRVEV does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZDRVEV to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)
AA is COMPLEX*16 array, dimension (LDA, max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used.
LDALDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least max(NN).
HH is COMPLEX*16 array, dimension (LDA, max(NN)) Another copy of the test matrix A, modified by ZGEEV.
WW is COMPLEX*16 array, dimension (max(NN)) The eigenvalues of A. On exit, W are the eigenvalues of the matrix in A.
W1W1 is COMPLEX*16 array, dimension (max(NN)) Like W, this array contains the eigenvalues of A, but those computed when ZGEEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors.
VLVL is COMPLEX*16 array, dimension (LDVL, max(NN)) VL holds the computed left eigenvectors.
LDVLLDVL is INTEGER Leading dimension of VL. Must be at least max(1,max(NN)).
VRVR is COMPLEX*16 array, dimension (LDVR, max(NN)) VR holds the computed right eigenvectors.
LDVRLDVR is INTEGER Leading dimension of VR. Must be at least max(1,max(NN)).
LRELRE is COMPLEX*16 array, dimension (LDLRE, max(NN)) LRE holds the computed right or left eigenvectors.
LDLRELDLRE is INTEGER Leading dimension of LRE. Must be at least max(1,max(NN)).
RESULTRESULT is DOUBLE PRECISION array, dimension (7) The values computed by the seven tests described above. The values are currently limited to 1/ulp, to avoid overflow.
WORKWORK is COMPLEX*16 array, dimension (NWORK)
NWORKNWORK is INTEGER The number of entries in WORK. This must be at least 5*NN(j)+2*NN(j)**2 for all j.
RWORKRWORK is DOUBLE PRECISION array, dimension (2*max(NN))
IWORKIWORK is INTEGER array, dimension (max(NN))
INFOINFO is INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 6: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 14: LDVL < 1 or LDVL < NMAX, where NMAX is max( NN(j) ). 16: LDVR < 1 or LDVR < NMAX, where NMAX is max( NN(j) ). 18: LDLRE < 1 or LDLRE < NMAX, where NMAX is max( NN(j) ). 21: NWORK too small. If ZLATMR, CLATMS, CLATME or ZGEEV returns an error code, the absolute value of it is returned.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NMAX Largest value in NN. NERRS The number of tests which have exceeded THRESH COND, CONDS, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTULP, RTULPI Square roots of the previous 4 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) KCONDS(j) Selectw whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.)
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zdrvsg (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) D, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( lda, * ) AB, complex*16, dimension( ldb, * ) BB, complex*16, dimension( * ) AP, complex*16, dimension( * ) BP, complex*16, dimension( * ) WORK, integer NWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, double precision, dimension( * ) RESULT, integer INFO)
ZDRVSG
Purpose:

ZDRVSG checks the complex Hermitian generalized eigenproblem drivers. ZHEGV computes all eigenvalues and, optionally, eigenvectors of a complex Hermitiandefinite generalized eigenproblem. ZHEGVD computes all eigenvalues and, optionally, eigenvectors of a complex Hermitiandefinite generalized eigenproblem using a divide and conquer algorithm. ZHEGVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitiandefinite generalized eigenproblem. ZHPGV computes all eigenvalues and, optionally, eigenvectors of a complex Hermitiandefinite generalized eigenproblem in packed storage. ZHPGVD computes all eigenvalues and, optionally, eigenvectors of a complex Hermitiandefinite generalized eigenproblem in packed storage using a divide and conquer algorithm. ZHPGVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitiandefinite generalized eigenproblem in packed storage. ZHBGV computes all eigenvalues and, optionally, eigenvectors of a complex Hermitiandefinite banded generalized eigenproblem. ZHBGVD computes all eigenvalues and, optionally, eigenvectors of a complex Hermitiandefinite banded generalized eigenproblem using a divide and conquer algorithm. ZHBGVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitiandefinite banded generalized eigenproblem. When ZDRVSG is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix A of the given type will be generated; a random wellconditioned matrix B is also generated and the pair (A,B) is used to test the drivers. For each pair (A,B), the following tests are performed: (1) ZHEGV with ITYPE = 1 and UPLO ='U':  A Z  B Z D  / ( A Z n ulp ) (2) as (1) but calling ZHPGV (3) as (1) but calling ZHBGV (4) as (1) but with UPLO = 'L' (5) as (4) but calling ZHPGV (6) as (4) but calling ZHBGV (7) ZHEGV with ITYPE = 2 and UPLO ='U':  A B Z  Z D  / ( A Z n ulp ) (8) as (7) but calling ZHPGV (9) as (7) but with UPLO = 'L' (10) as (9) but calling ZHPGV (11) ZHEGV with ITYPE = 3 and UPLO ='U':  B A Z  Z D  / ( A Z n ulp ) (12) as (11) but calling ZHPGV (13) as (11) but with UPLO = 'L' (14) as (13) but calling ZHPGV ZHEGVD, ZHPGVD and ZHBGVD performed the same 14 tests. ZHEGVX, ZHPGVX and ZHBGVX performed the above 14 tests with the parameter RANGE = 'A', 'N' and 'I', respectively. The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. This type is used for the matrix A which has halfbandwidth KA. B is generated as a wellconditioned positive definite matrix with halfbandwidth KB (<= KA). Currently, the list of possible types for A is: (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (4), but multiplied by SQRT( overflow threshold ) (7) Same as (4), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U* D U, where U is unitary and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U* D U, where U is unitary and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U* D U, where U is unitary and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Hermitian matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold ) (16) Same as (8), but with KA = 1 and KB = 1 (17) Same as (8), but with KA = 2 and KB = 1 (18) Same as (8), but with KA = 2 and KB = 2 (19) Same as (8), but with KA = 3 and KB = 1 (20) Same as (8), but with KA = 3 and KB = 2 (21) Same as (8), but with KA = 3 and KB = 3
NSIZES INTEGER The number of sizes of matrices to use. If it is zero, ZDRVSG does nothing. It must be at least zero. Not modified. NN INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. Not modified. NTYPES INTEGER The number of elements in DOTYPE. If it is zero, ZDRVSG does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . Not modified. DOTYPE LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. Not modified. ISEED INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZDRVSG to continue the same random number sequence. Modified. THRESH DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. Not modified. NOUNIT INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) Not modified. A COMPLEX*16 array, dimension (LDA , max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. Modified. LDA INTEGER The leading dimension of A. It must be at least 1 and at least max( NN ). Not modified. B COMPLEX*16 array, dimension (LDB , max(NN)) Used to hold the Hermitian positive definite matrix for the generailzed problem. On exit, B contains the last matrix actually used. Modified. LDB INTEGER The leading dimension of B. It must be at least 1 and at least max( NN ). Not modified. D DOUBLE PRECISION array, dimension (max(NN)) The eigenvalues of A. On exit, the eigenvalues in D correspond with the matrix in A. Modified. Z COMPLEX*16 array, dimension (LDZ, max(NN)) The matrix of eigenvectors. Modified. LDZ INTEGER The leading dimension of ZZ. It must be at least 1 and at least max( NN ). Not modified. AB COMPLEX*16 array, dimension (LDA, max(NN)) Workspace. Modified. BB COMPLEX*16 array, dimension (LDB, max(NN)) Workspace. Modified. AP COMPLEX*16 array, dimension (max(NN)**2) Workspace. Modified. BP COMPLEX*16 array, dimension (max(NN)**2) Workspace. Modified. WORK COMPLEX*16 array, dimension (NWORK) Workspace. Modified. NWORK INTEGER The number of entries in WORK. This must be at least 2*N + N**2 where N = max( NN(j), 2 ). Not modified. RWORK DOUBLE PRECISION array, dimension (LRWORK) Workspace. Modified. LRWORK INTEGER The number of entries in RWORK. This must be at least max( 7*N, 1 + 4*N + 2*N*lg(N) + 3*N**2 ) where N = max( NN(j) ) and lg( N ) = smallest integer k such that 2**k >= N . Not modified. IWORK INTEGER array, dimension (LIWORK)) Workspace. Modified. LIWORK INTEGER The number of entries in IWORK. This must be at least 2 + 5*max( NN(j) ). Not modified. RESULT DOUBLE PRECISION array, dimension (70) The values computed by the 70 tests described above. Modified. INFO INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 5: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 16: LDZ < 1 or LDZ < NMAX. 21: NWORK too small. 23: LRWORK too small. 25: LIWORK too small. If ZLATMR, CLATMS, ZHEGV, ZHPGV, ZHBGV, CHEGVD, CHPGVD, ZHPGVD, ZHEGVX, CHPGVX, ZHBGVX returns an error code, the absolute value of it is returned. Modified.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests that have been run on this matrix. NTESTT The total number of tests for this call. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far (computed by DLAFTS). COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL Square roots of the previous 2 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zdrvst (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D1, double precision, dimension( * ) D2, double precision, dimension( * ) D3, double precision, dimension( * ) WA1, double precision, dimension( * ) WA2, double precision, dimension( * ) WA3, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldu, * ) V, complex*16, dimension( * ) TAU, complex*16, dimension( ldu, * ) Z, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, double precision, dimension( * ) RESULT, integer INFO)
ZDRVST
Purpose:

ZDRVST checks the Hermitian eigenvalue problem drivers. ZHEEVD computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, using a divideandconquer algorithm. ZHEEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix. ZHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix using the Relatively Robust Representation where it can. ZHPEVD computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix in packed storage, using a divideandconquer algorithm. ZHPEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix in packed storage. ZHBEVD computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix, using a divideandconquer algorithm. ZHBEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix. ZHEEV computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix. ZHPEV computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix in packed storage. ZHBEV computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix. When ZDRVST is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the appropriate drivers. For each matrix and each driver routine called, the following tests will be performed: (1)  A  Z D Z'  / ( A n ulp ) (2)  I  Z Z'  / ( n ulp ) (3)  D1  D2  / ( D1 ulp ) where Z is the matrix of eigenvectors returned when the eigenvector option is given and D1 and D2 are the eigenvalues returned with and without the eigenvector option. The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (4), but multiplied by SQRT( overflow threshold ) (7) Same as (4), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U* D U, where U is unitary and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U* D U, where U is unitary and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U* D U, where U is unitary and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Symmetric matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold ) (16) A band matrix with half bandwidth randomly chosen between 0 and N1, with evenly spaced eigenvalues 1, ..., ULP with random signs. (17) Same as (16), but multiplied by SQRT( overflow threshold ) (18) Same as (16), but multiplied by SQRT( underflow threshold )
NSIZES INTEGER The number of sizes of matrices to use. If it is zero, ZDRVST does nothing. It must be at least zero. Not modified. NN INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. Not modified. NTYPES INTEGER The number of elements in DOTYPE. If it is zero, ZDRVST does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . Not modified. DOTYPE LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. Not modified. ISEED INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZDRVST to continue the same random number sequence. Modified. THRESH DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. Not modified. NOUNIT INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) Not modified. A COMPLEX*16 array, dimension (LDA , max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. Modified. LDA INTEGER The leading dimension of A. It must be at least 1 and at least max( NN ). Not modified. D1 DOUBLE PRECISION array, dimension (max(NN)) The eigenvalues of A, as computed by ZSTEQR simlutaneously with Z. On exit, the eigenvalues in D1 correspond with the matrix in A. Modified. D2 DOUBLE PRECISION array, dimension (max(NN)) The eigenvalues of A, as computed by ZSTEQR if Z is not computed. On exit, the eigenvalues in D2 correspond with the matrix in A. Modified. D3 DOUBLE PRECISION array, dimension (max(NN)) The eigenvalues of A, as computed by DSTERF. On exit, the eigenvalues in D3 correspond with the matrix in A. Modified. WA1 DOUBLE PRECISION array, dimension WA2 DOUBLE PRECISION array, dimension WA3 DOUBLE PRECISION array, dimension U COMPLEX*16 array, dimension (LDU, max(NN)) The unitary matrix computed by ZHETRD + ZUNGC3. Modified. LDU INTEGER The leading dimension of U, Z, and V. It must be at least 1 and at least max( NN ). Not modified. V COMPLEX*16 array, dimension (LDU, max(NN)) The Housholder vectors computed by ZHETRD in reducing A to tridiagonal form. Modified. TAU COMPLEX*16 array, dimension (max(NN)) The Householder factors computed by ZHETRD in reducing A to tridiagonal form. Modified. Z COMPLEX*16 array, dimension (LDU, max(NN)) The unitary matrix of eigenvectors computed by ZHEEVD, ZHEEVX, ZHPEVD, CHPEVX, ZHBEVD, and CHBEVX. Modified. WORK  COMPLEX*16 array of dimension ( LWORK ) Workspace. Modified. LWORK  INTEGER The number of entries in WORK. This must be at least 2*max( NN(j), 2 )**2. Not modified. RWORK DOUBLE PRECISION array, dimension (3*max(NN)) Workspace. Modified. LRWORK  INTEGER The number of entries in RWORK. IWORK INTEGER array, dimension (6*max(NN)) Workspace. Modified. LIWORK  INTEGER The number of entries in IWORK. RESULT DOUBLE PRECISION array, dimension (??) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. Modified. INFO INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 5: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 16: LDU < 1 or LDU < NMAX. 21: LWORK too small. If DLATMR, SLATMS, ZHETRD, DORGC3, ZSTEQR, DSTERF, or DORMC2 returns an error code, the absolute value of it is returned. Modified.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. NTESTT The total number of tests performed so far. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far (computed by DLAFTS). COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL Square roots of the previous 2 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zdrvsx (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NIUNIT, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) H, complex*16, dimension( lda, * ) HT, complex*16, dimension( * ) W, complex*16, dimension( * ) WT, complex*16, dimension( * ) WTMP, complex*16, dimension( ldvs, * ) VS, integer LDVS, complex*16, dimension( ldvs, * ) VS1, double precision, dimension( 17 ) RESULT, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)
ZDRVSX
Purpose:

ZDRVSX checks the nonsymmetric eigenvalue (Schur form) problem expert driver ZGEESX. ZDRVSX uses both test matrices generated randomly depending on data supplied in the calling sequence, as well as on data read from an input file and including precomputed condition numbers to which it compares the ones it computes. When ZDRVSX is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 15 tests will be performed: (1) 0 if T is in Schur form, 1/ulp otherwise (no sorting of eigenvalues) (2)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (no sorting of eigenvalues). (3)  I  VS VS'  / ( n ulp ) (no sorting of eigenvalues). (4) 0 if W are eigenvalues of T 1/ulp otherwise (no sorting of eigenvalues) (5) 0 if T(with VS) = T(without VS), 1/ulp otherwise (no sorting of eigenvalues) (6) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise (no sorting of eigenvalues) (7) 0 if T is in Schur form, 1/ulp otherwise (with sorting of eigenvalues) (8)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (with sorting of eigenvalues). (9)  I  VS VS'  / ( n ulp ) (with sorting of eigenvalues). (10) 0 if W are eigenvalues of T 1/ulp otherwise If workspace sufficient, also compare W with and without reciprocal condition numbers (with sorting of eigenvalues) (11) 0 if T(with VS) = T(without VS), 1/ulp otherwise If workspace sufficient, also compare T with and without reciprocal condition numbers (with sorting of eigenvalues) (12) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise If workspace sufficient, also compare VS with and without reciprocal condition numbers (with sorting of eigenvalues) (13) if sorting worked and SDIM is the number of eigenvalues which were SELECTed If workspace sufficient, also compare SDIM with and without reciprocal condition numbers (14) if RCONDE the same no matter if VS and/or RCONDV computed (15) if RCONDV the same no matter if VS and/or RCONDE computed The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A (transposed) Jordan block, with 1's on the diagonal. (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random complex angles. (ULP = (first number larger than 1)  1 ) (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random complex angles. (6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random complex angles. (7) Same as (4), but multiplied by a constant near the overflow threshold (8) Same as (4), but multiplied by a constant near the underflow threshold (9) A matrix of the form U' T U, where U is unitary and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (10) A matrix of the form U' T U, where U is unitary and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (11) A matrix of the form U' T U, where U is orthogonal and T has "clustered" entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (12) A matrix of the form U' T U, where U is unitary and T has complex eigenvalues randomly chosen from ULP < z < 1 and random O(1) entries in the upper triangle. (13) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (14) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (15) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (16) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has complex eigenvalues randomly chosen from ULP < z < 1 and random O(1) entries in the upper triangle. (17) Same as (16), but multiplied by a constant near the overflow threshold (18) Same as (16), but multiplied by a constant near the underflow threshold (19) Nonsymmetric matrix with random entries chosen from (1,1). If N is at least 4, all entries in first two rows and last row, and first column and last two columns are zero. (20) Same as (19), but multiplied by a constant near the overflow threshold (21) Same as (19), but multiplied by a constant near the underflow threshold In addition, an input file will be read from logical unit number NIUNIT. The file contains matrices along with precomputed eigenvalues and reciprocal condition numbers for the eigenvalue average and right invariant subspace. For these matrices, in addition to tests (1) to (15) we will compute the following two tests: (16) RCONDE  RCDEIN / cond(RCONDE) RCONDE is the reciprocal average eigenvalue condition number computed by ZGEESX and RCDEIN (the precomputed true value) is supplied as input. cond(RCONDE) is the condition number of RCONDE, and takes errors in computing RCONDE into account, so that the resulting quantity should be O(ULP). cond(RCONDE) is essentially given by norm(A)/RCONDV. (17) RCONDV  RCDVIN / cond(RCONDV) RCONDV is the reciprocal right invariant subspace condition number computed by ZGEESX and RCDVIN (the precomputed true value) is supplied as input. cond(RCONDV) is the condition number of RCONDV, and takes errors in computing RCONDV into account, so that the resulting quantity should be O(ULP). cond(RCONDV) is essentially given by norm(A)/RCONDE.
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. NSIZES must be at least zero. If it is zero, no randomly generated matrices are tested, but any test matrices read from NIUNIT will be tested.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. NTYPES must be at least zero. If it is zero, no randomly generated test matrices are tested, but and test matrices read from NIUNIT will be tested. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZDRVSX to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NIUNITNIUNIT is INTEGER The FORTRAN unit number for reading in the data file of problems to solve.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)
AA is COMPLEX*16 array, dimension (LDA, max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used.
LDALDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least max( NN ).
HH is COMPLEX*16 array, dimension (LDA, max(NN)) Another copy of the test matrix A, modified by ZGEESX.
HTHT is COMPLEX*16 array, dimension (LDA, max(NN)) Yet another copy of the test matrix A, modified by ZGEESX.
WW is COMPLEX*16 array, dimension (max(NN)) The computed eigenvalues of A.
WTWT is COMPLEX*16 array, dimension (max(NN)) Like W, this array contains the eigenvalues of A, but those computed when ZGEESX only computes a partial eigendecomposition, i.e. not Schur vectors
WTMPWTMP is COMPLEX*16 array, dimension (max(NN)) More temporary storage for eigenvalues.
VSVS is COMPLEX*16 array, dimension (LDVS, max(NN)) VS holds the computed Schur vectors.
LDVSLDVS is INTEGER Leading dimension of VS. Must be at least max(1,max(NN)).
VS1VS1 is COMPLEX*16 array, dimension (LDVS, max(NN)) VS1 holds another copy of the computed Schur vectors.
RESULTRESULT is DOUBLE PRECISION array, dimension (17) The values computed by the 17 tests described above. The values are currently limited to 1/ulp, to avoid overflow.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The number of entries in WORK. This must be at least max(1,2*NN(j)**2) for all j.
RWORKRWORK is DOUBLE PRECISION array, dimension (max(NN))
BWORKBWORK is LOGICAL array, dimension (max(NN))
INFOINFO is INTEGER If 0, successful exit. <0, input parameter INFO is incorrect >0, ZLATMR, CLATMS, CLATME or ZGET24 returned an error code and INFO is its absolute value  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NMAX Largest value in NN. NERRS The number of tests which have exceeded THRESH COND, CONDS, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTULP, RTULPI Square roots of the previous 4 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) KCONDS(j) Selectw whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.)
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zdrvvx (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, double precision THRESH, integer NIUNIT, integer NOUNIT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) H, complex*16, dimension( * ) W, complex*16, dimension( * ) W1, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, complex*16, dimension( ldlre, * ) LRE, integer LDLRE, double precision, dimension( * ) RCONDV, double precision, dimension( * ) RCNDV1, double precision, dimension( * ) RCDVIN, double precision, dimension( * ) RCONDE, double precision, dimension( * ) RCNDE1, double precision, dimension( * ) RCDEIN, double precision, dimension( * ) SCALE, double precision, dimension( * ) SCALE1, double precision, dimension( 11 ) RESULT, complex*16, dimension( * ) WORK, integer NWORK, double precision, dimension( * ) RWORK, integer INFO)
ZDRVVX
Purpose:

ZDRVVX checks the nonsymmetric eigenvalue problem expert driver ZGEEVX. ZDRVVX uses both test matrices generated randomly depending on data supplied in the calling sequence, as well as on data read from an input file and including precomputed condition numbers to which it compares the ones it computes. When ZDRVVX is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified in the calling sequence. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 9 tests will be performed: (1)  A * VR  VR * W  / ( n A ulp ) Here VR is the matrix of unit right eigenvectors. W is a diagonal matrix with diagonal entries W(j). (2)  A**H * VL  VL * W**H  / ( n A ulp ) Here VL is the matrix of unit left eigenvectors, A**H is the conjugate transpose of A, and W is as above. (3)  VR(i)  1  / ulp and largest component real VR(i) denotes the ith column of VR. (4)  VL(i)  1  / ulp and largest component real VL(i) denotes the ith column of VL. (5) W(full) = W(partial) W(full) denotes the eigenvalues computed when VR, VL, RCONDV and RCONDE are also computed, and W(partial) denotes the eigenvalues computed when only some of VR, VL, RCONDV, and RCONDE are computed. (6) VR(full) = VR(partial) VR(full) denotes the right eigenvectors computed when VL, RCONDV and RCONDE are computed, and VR(partial) denotes the result when only some of VL and RCONDV are computed. (7) VL(full) = VL(partial) VL(full) denotes the left eigenvectors computed when VR, RCONDV and RCONDE are computed, and VL(partial) denotes the result when only some of VR and RCONDV are computed. (8) 0 if SCALE, ILO, IHI, ABNRM (full) = SCALE, ILO, IHI, ABNRM (partial) 1/ulp otherwise SCALE, ILO, IHI and ABNRM describe how the matrix is balanced. (full) is when VR, VL, RCONDE and RCONDV are also computed, and (partial) is when some are not computed. (9) RCONDV(full) = RCONDV(partial) RCONDV(full) denotes the reciprocal condition numbers of the right eigenvectors computed when VR, VL and RCONDE are also computed. RCONDV(partial) denotes the reciprocal condition numbers when only some of VR, VL and RCONDE are computed. The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A (transposed) Jordan block, with 1's on the diagonal. (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random complex angles. (ULP = (first number larger than 1)  1 ) (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random complex angles. (6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random complex angles. (7) Same as (4), but multiplied by a constant near the overflow threshold (8) Same as (4), but multiplied by a constant near the underflow threshold (9) A matrix of the form U' T U, where U is unitary and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (10) A matrix of the form U' T U, where U is unitary and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (11) A matrix of the form U' T U, where U is unitary and T has "clustered" entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (12) A matrix of the form U' T U, where U is unitary and T has complex eigenvalues randomly chosen from ULP < z < 1 and random O(1) entries in the upper triangle. (13) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (14) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (15) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle. (16) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has complex eigenvalues randomly chosen from ULP < z < 1 and random O(1) entries in the upper triangle. (17) Same as (16), but multiplied by a constant near the overflow threshold (18) Same as (16), but multiplied by a constant near the underflow threshold (19) Nonsymmetric matrix with random entries chosen from z < 1 If N is at least 4, all entries in first two rows and last row, and first column and last two columns are zero. (20) Same as (19), but multiplied by a constant near the overflow threshold (21) Same as (19), but multiplied by a constant near the underflow threshold In addition, an input file will be read from logical unit number NIUNIT. The file contains matrices along with precomputed eigenvalues and reciprocal condition numbers for the eigenvalues and right eigenvectors. For these matrices, in addition to tests (1) to (9) we will compute the following two tests: (10) RCONDV  RCDVIN / cond(RCONDV) RCONDV is the reciprocal right eigenvector condition number computed by ZGEEVX and RCDVIN (the precomputed true value) is supplied as input. cond(RCONDV) is the condition number of RCONDV, and takes errors in computing RCONDV into account, so that the resulting quantity should be O(ULP). cond(RCONDV) is essentially given by norm(A)/RCONDE. (11) RCONDE  RCDEIN / cond(RCONDE) RCONDE is the reciprocal eigenvalue condition number computed by ZGEEVX and RCDEIN (the precomputed true value) is supplied as input. cond(RCONDE) is the condition number of RCONDE, and takes errors in computing RCONDE into account, so that the resulting quantity should be O(ULP). cond(RCONDE) is essentially given by norm(A)/RCONDV.
Parameters:

NSIZES
NSIZES is INTEGER The number of sizes of matrices to use. NSIZES must be at least zero. If it is zero, no randomly generated matrices are tested, but any test matrices read from NIUNIT will be tested.
NNNN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero.
NTYPESNTYPES is INTEGER The number of elements in DOTYPE. NTYPES must be at least zero. If it is zero, no randomly generated test matrices are tested, but and test matrices read from NIUNIT will be tested. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPEDOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to ZDRVVX to continue the same random number sequence.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NIUNITNIUNIT is INTEGER The FORTRAN unit number for reading in the data file of problems to solve.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)
AA is COMPLEX*16 array, dimension (LDA, max(NN,12)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used.
LDALDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least max( NN, 12 ). (12 is the dimension of the largest matrix on the precomputed input file.)
HH is COMPLEX*16 array, dimension (LDA, max(NN,12)) Another copy of the test matrix A, modified by ZGEEVX.
WW is COMPLEX*16 array, dimension (max(NN,12)) Contains the eigenvalues of A.
W1W1 is COMPLEX*16 array, dimension (max(NN,12)) Like W, this array contains the eigenvalues of A, but those computed when ZGEEVX only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors.
VLVL is COMPLEX*16 array, dimension (LDVL, max(NN,12)) VL holds the computed left eigenvectors.
LDVLLDVL is INTEGER Leading dimension of VL. Must be at least max(1,max(NN,12)).
VRVR is COMPLEX*16 array, dimension (LDVR, max(NN,12)) VR holds the computed right eigenvectors.
LDVRLDVR is INTEGER Leading dimension of VR. Must be at least max(1,max(NN,12)).
LRELRE is COMPLEX*16 array, dimension (LDLRE, max(NN,12)) LRE holds the computed right or left eigenvectors.
LDLRELDLRE is INTEGER Leading dimension of LRE. Must be at least max(1,max(NN,12))
RCONDVRCONDV is DOUBLE PRECISION array, dimension (N) RCONDV holds the computed reciprocal condition numbers for eigenvectors.
RCNDV1RCNDV1 is DOUBLE PRECISION array, dimension (N) RCNDV1 holds more computed reciprocal condition numbers for eigenvectors.
RCDVINRCDVIN is DOUBLE PRECISION array, dimension (N) When COMP = .TRUE. RCDVIN holds the precomputed reciprocal condition numbers for eigenvectors to be compared with RCONDV.
RCONDERCONDE is DOUBLE PRECISION array, dimension (N) RCONDE holds the computed reciprocal condition numbers for eigenvalues.
RCNDE1RCNDE1 is DOUBLE PRECISION array, dimension (N) RCNDE1 holds more computed reciprocal condition numbers for eigenvalues.
RCDEINRCDEIN is DOUBLE PRECISION array, dimension (N) When COMP = .TRUE. RCDEIN holds the precomputed reciprocal condition numbers for eigenvalues to be compared with RCONDE.
SCALESCALE is DOUBLE PRECISION array, dimension (N) Holds information describing balancing of matrix.
SCALE1SCALE1 is DOUBLE PRECISION array, dimension (N) Holds information describing balancing of matrix.
WORKWORK is COMPLEX*16 array, dimension (NWORK)
RESULTRESULT is DOUBLE PRECISION array, dimension (11) The values computed by the seven tests described above. The values are currently limited to 1/ulp, to avoid overflow.
NWORKNWORK is INTEGER The number of entries in WORK. This must be at least max(6*12+2*12**2,6*NN(j)+2*NN(j)**2) = max( 360 ,6*NN(j)+2*NN(j)**2) for all j.
RWORKRWORK is DOUBLE PRECISION array, dimension (2*max(NN,12))
INFOINFO is INTEGER If 0, then successful exit. If <0, then input parameter INFO is incorrect. If >0, ZLATMR, CLATMS, CLATME or ZGET23 returned an error code, and INFO is its absolute value.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NMAX Largest value in NN or 12. NERRS The number of tests which have exceeded THRESH COND, CONDS, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTULP, RTULPI Square roots of the previous 4 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) KCONDS(j) Selectw whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.)
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zerrbd (character*3 PATH, integer NUNIT)
ZERRBD
Purpose:

ZERRBD tests the error exits for ZGEBRD, ZUNGBR, ZUNMBR, and ZBDSQR.
Parameters:

PATH
PATH is CHARACTER*3 The LAPACK path name for the routines to be tested.
NUNITNUNIT is INTEGER The unit number for output.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zerrec (character*3 PATH, integer NUNIT)
ZERREC
Purpose:

ZERREC tests the error exits for the routines for eigen condition estimation for DOUBLE PRECISION matrices: ZTRSYL, CTREXC, CTRSNA and CTRSEN.
Parameters:

PATH
PATH is CHARACTER*3 The LAPACK path name for the routines to be tested.
NUNITNUNIT is INTEGER The unit number for output.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zerred (character*3 PATH, integer NUNIT)
ZERRED
Purpose:

ZERRED tests the error exits for the eigenvalue driver routines for DOUBLE COMPLEX PRECISION matrices: PATH driver description    ZEV ZGEEV find eigenvalues/eigenvectors for nonsymmetric A ZES ZGEES find eigenvalues/Schur form for nonsymmetric A ZVX ZGEEVX ZGEEV + balancing and condition estimation ZSX ZGEESX ZGEES + balancing and condition estimation ZBD ZGESVD compute SVD of an MbyN matrix A ZGESDD compute SVD of an MbyN matrix A(by divide and conquer) ZGEJSV compute SVD of an MbyN matrix A where M >= N ZGESVDX compute SVD of an MbyN matrix A(by bisection and inverse iteration)
Parameters:

PATH
PATH is CHARACTER*3 The LAPACK path name for the routines to be tested.
NUNITNUNIT is INTEGER The unit number for output.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zerrgg (character*3 PATH, integer NUNIT)
ZERRGG
Purpose:

ZERRGG tests the error exits for ZGGES, ZGGESX, ZGGEV, ZGGEVX, ZGGES3, ZGGEV3, ZGGGLM, ZGGHRD, ZGGLSE, ZGGQRF, ZGGRQF, ZGGSVD3, ZGGSVP3, ZHGEQZ, ZTGEVC, ZTGEXC, ZTGSEN, ZTGSJA, ZTGSNA, ZTGSYL, and ZUNCSD.
Parameters:

PATH
PATH is CHARACTER*3 The LAPACK path name for the routines to be tested.
NUNITNUNIT is INTEGER The unit number for output.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zerrhs (character*3 PATH, integer NUNIT)
ZERRHS
Purpose:

ZERRHS tests the error exits for ZGEBAK, CGEBAL, CGEHRD, ZUNGHR, ZUNMHR, ZHSEQR, CHSEIN, and ZTREVC.
Parameters:

PATH
PATH is CHARACTER*3 The LAPACK path name for the routines to be tested.
NUNITNUNIT is INTEGER The unit number for output.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zerrst (character*3 PATH, integer NUNIT)
ZERRST
Purpose:

ZERRST tests the error exits for ZHETRD, ZUNGTR, CUNMTR, ZHPTRD, ZUNGTR, ZUPMTR, ZSTEQR, CSTEIN, ZPTEQR, ZHBTRD, ZHEEV, CHEEVX, CHEEVD, ZHBEV, CHBEVX, CHBEVD, ZHPEV, CHPEVX, CHPEVD, and ZSTEDC.
Parameters:

PATH
PATH is CHARACTER*3 The LAPACK path name for the routines to be tested.
NUNITNUNIT is INTEGER The unit number for output.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget02 (character TRANS, integer M, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) RWORK, double precision RESID)
ZGET02
Purpose:

ZGET02 computes the residual for a solution of a system of linear equations A*x = b or A'*x = b: RESID = norm(B  A*X) / ( norm(A) * norm(X) * EPS ), where EPS is the machine epsilon.
Parameters:

TRANS
TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A *x = b = 'T': A^T*x = b, where A^T is the transpose of A = 'C': A^H*x = b, where A^H is the conjugate transpose of A
MM is INTEGER The number of rows of the matrix A. M >= 0.
NN is INTEGER The number of columns of the matrix A. N >= 0.
NRHSNRHS is INTEGER The number of columns of B, the matrix of right hand sides. NRHS >= 0.
AA is COMPLEX*16 array, dimension (LDA,N) The original M x N matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
XX is COMPLEX*16 array, dimension (LDX,NRHS) The computed solution vectors for the system of linear equations.
LDXLDX is INTEGER The leading dimension of the array X. If TRANS = 'N', LDX >= max(1,N); if TRANS = 'T' or 'C', LDX >= max(1,M).
BB is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the right hand side vectors for the system of linear equations. On exit, B is overwritten with the difference B  A*X.
LDBLDB is INTEGER The leading dimension of the array B. IF TRANS = 'N', LDB >= max(1,M); if TRANS = 'T' or 'C', LDB >= max(1,N).
RWORKRWORK is DOUBLE PRECISION array, dimension (M)
RESIDRESID is DOUBLE PRECISION The maximum over the number of right hand sides of norm(B  A*X) / ( norm(A) * norm(X) * EPS ).
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget10 (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision RESULT)
ZGET10
Purpose:

ZGET10 compares two matrices A and B and computes the ratio RESULT = norm( A  B ) / ( norm(A) * M * EPS )
Parameters:

M
M is INTEGER The number of rows of the matrices A and B.
NN is INTEGER The number of columns of the matrices A and B.
AA is COMPLEX*16 array, dimension (LDA,N) The m by n matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
BB is COMPLEX*16 array, dimension (LDB,N) The m by n matrix B.
LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,M).
WORKWORK is COMPLEX*16 array, dimension (M)
RWORKRWORK is COMPLEX*16 array, dimension (M)
RESULTRESULT is DOUBLE PRECISION RESULT = norm( A  B ) / ( norm(A) * M * EPS )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget22 (character TRANSA, character TRANSE, character TRANSW, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lde, * ) E, integer LDE, complex*16, dimension( * ) W, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision, dimension( 2 ) RESULT)
ZGET22
Purpose:

ZGET22 does an eigenvector check. The basic test is: RESULT(1) =  A E  E W  / ( A E ulp ) using the 1norm. It also tests the normalization of E: RESULT(2) = max  mnorm(E(j))  1  / ( n ulp ) j where E(j) is the jth eigenvector, and mnorm is the maxnorm of a vector. The maxnorm of a complex nvector x in this case is the maximum of re(x(i) + im(x(i) over i = 1, ..., n.
Parameters:

TRANSA
TRANSA is CHARACTER*1 Specifies whether or not A is transposed. = 'N': No transpose = 'T': Transpose = 'C': Conjugate transpose
TRANSETRANSE is CHARACTER*1 Specifies whether or not E is transposed. = 'N': No transpose, eigenvectors are in columns of E = 'T': Transpose, eigenvectors are in rows of E = 'C': Conjugate transpose, eigenvectors are in rows of E
TRANSWTRANSW is CHARACTER*1 Specifies whether or not W is transposed. = 'N': No transpose = 'T': Transpose, same as TRANSW = 'N' = 'C': Conjugate transpose, use WI(j) instead of WI(j)
NN is INTEGER The order of the matrix A. N >= 0.
AA is COMPLEX*16 array, dimension (LDA,N) The matrix whose eigenvectors are in E.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
EE is COMPLEX*16 array, dimension (LDE,N) The matrix of eigenvectors. If TRANSE = 'N', the eigenvectors are stored in the columns of E, if TRANSE = 'T' or 'C', the eigenvectors are stored in the rows of E.
LDELDE is INTEGER The leading dimension of the array E. LDE >= max(1,N).
WW is COMPLEX*16 array, dimension (N) The eigenvalues of A.
WORKWORK is COMPLEX*16 array, dimension (N*N)
RWORKRWORK is DOUBLE PRECISION array, dimension (N)
RESULTRESULT is DOUBLE PRECISION array, dimension (2) RESULT(1) =  A E  E W  / ( A E ulp ) RESULT(2) = max  mnorm(E(j))  1  / ( n ulp )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget23 (logical COMP, integer ISRT, character BALANC, integer JTYPE, double precision THRESH, integer, dimension( 4 ) ISEED, integer NOUNIT, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) H, complex*16, dimension( * ) W, complex*16, dimension( * ) W1, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, complex*16, dimension( ldlre, * ) LRE, integer LDLRE, double precision, dimension( * ) RCONDV, double precision, dimension( * ) RCNDV1, double precision, dimension( * ) RCDVIN, double precision, dimension( * ) RCONDE, double precision, dimension( * ) RCNDE1, double precision, dimension( * ) RCDEIN, double precision, dimension( * ) SCALE, double precision, dimension( * ) SCALE1, double precision, dimension( 11 ) RESULT, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer INFO)
ZGET23
Purpose:

ZGET23 checks the nonsymmetric eigenvalue problem driver CGEEVX. If COMP = .FALSE., the first 8 of the following tests will be performed on the input matrix A, and also test 9 if LWORK is sufficiently large. if COMP is .TRUE. all 11 tests will be performed. (1)  A * VR  VR * W  / ( n A ulp ) Here VR is the matrix of unit right eigenvectors. W is a diagonal matrix with diagonal entries W(j). (2)  A**H * VL  VL * W**H  / ( n A ulp ) Here VL is the matrix of unit left eigenvectors, A**H is the conjugate transpose of A, and W is as above. (3)  VR(i)  1  / ulp and largest component real VR(i) denotes the ith column of VR. (4)  VL(i)  1  / ulp and largest component real VL(i) denotes the ith column of VL. (5) 0 if W(full) = W(partial), 1/ulp otherwise W(full) denotes the eigenvalues computed when VR, VL, RCONDV and RCONDE are also computed, and W(partial) denotes the eigenvalues computed when only some of VR, VL, RCONDV, and RCONDE are computed. (6) 0 if VR(full) = VR(partial), 1/ulp otherwise VR(full) denotes the right eigenvectors computed when VL, RCONDV and RCONDE are computed, and VR(partial) denotes the result when only some of VL and RCONDV are computed. (7) 0 if VL(full) = VL(partial), 1/ulp otherwise VL(full) denotes the left eigenvectors computed when VR, RCONDV and RCONDE are computed, and VL(partial) denotes the result when only some of VR and RCONDV are computed. (8) 0 if SCALE, ILO, IHI, ABNRM (full) = SCALE, ILO, IHI, ABNRM (partial) 1/ulp otherwise SCALE, ILO, IHI and ABNRM describe how the matrix is balanced. (full) is when VR, VL, RCONDE and RCONDV are also computed, and (partial) is when some are not computed. (9) 0 if RCONDV(full) = RCONDV(partial), 1/ulp otherwise RCONDV(full) denotes the reciprocal condition numbers of the right eigenvectors computed when VR, VL and RCONDE are also computed. RCONDV(partial) denotes the reciprocal condition numbers when only some of VR, VL and RCONDE are computed. (10) RCONDV  RCDVIN / cond(RCONDV) RCONDV is the reciprocal right eigenvector condition number computed by ZGEEVX and RCDVIN (the precomputed true value) is supplied as input. cond(RCONDV) is the condition number of RCONDV, and takes errors in computing RCONDV into account, so that the resulting quantity should be O(ULP). cond(RCONDV) is essentially given by norm(A)/RCONDE. (11) RCONDE  RCDEIN / cond(RCONDE) RCONDE is the reciprocal eigenvalue condition number computed by ZGEEVX and RCDEIN (the precomputed true value) is supplied as input. cond(RCONDE) is the condition number of RCONDE, and takes errors in computing RCONDE into account, so that the resulting quantity should be O(ULP). cond(RCONDE) is essentially given by norm(A)/RCONDV.
Parameters:

COMP
COMP is LOGICAL COMP describes which input tests to perform: = .FALSE. if the computed condition numbers are not to be tested against RCDVIN and RCDEIN = .TRUE. if they are to be compared
ISRTISRT is INTEGER If COMP = .TRUE., ISRT indicates in how the eigenvalues corresponding to values in RCDVIN and RCDEIN are ordered: = 0 means the eigenvalues are sorted by increasing real part = 1 means the eigenvalues are sorted by increasing imaginary part If COMP = .FALSE., ISRT is not referenced.
BALANCBALANC is CHARACTER Describes the balancing option to be tested. = 'N' for no permuting or diagonal scaling = 'P' for permuting but no diagonal scaling = 'S' for no permuting but diagonal scaling = 'B' for permuting and diagonal scaling
JTYPEJTYPE is INTEGER Type of input matrix. Used to label output if error occurs.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
ISEEDISEED is INTEGER array, dimension (4) If COMP = .FALSE., the random number generator seed used to produce matrix. If COMP = .TRUE., ISEED(1) = the number of the example. Used to label output if error occurs.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)
NN is INTEGER The dimension of A. N must be at least 0.
AA is COMPLEX*16 array, dimension (LDA,N) Used to hold the matrix whose eigenvalues are to be computed.
LDALDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least N.
HH is COMPLEX*16 array, dimension (LDA,N) Another copy of the test matrix A, modified by ZGEEVX.
WW is COMPLEX*16 array, dimension (N) Contains the eigenvalues of A.
W1W1 is COMPLEX*16 array, dimension (N) Like W, this array contains the eigenvalues of A, but those computed when ZGEEVX only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors.
VLVL is COMPLEX*16 array, dimension (LDVL,N) VL holds the computed left eigenvectors.
LDVLLDVL is INTEGER Leading dimension of VL. Must be at least max(1,N).
VRVR is COMPLEX*16 array, dimension (LDVR,N) VR holds the computed right eigenvectors.
LDVRLDVR is INTEGER Leading dimension of VR. Must be at least max(1,N).
LRELRE is COMPLEX*16 array, dimension (LDLRE,N) LRE holds the computed right or left eigenvectors.
LDLRELDLRE is INTEGER Leading dimension of LRE. Must be at least max(1,N).
RCONDVRCONDV is DOUBLE PRECISION array, dimension (N) RCONDV holds the computed reciprocal condition numbers for eigenvectors.
RCNDV1RCNDV1 is DOUBLE PRECISION array, dimension (N) RCNDV1 holds more computed reciprocal condition numbers for eigenvectors.
RCDVINRCDVIN is DOUBLE PRECISION array, dimension (N) When COMP = .TRUE. RCDVIN holds the precomputed reciprocal condition numbers for eigenvectors to be compared with RCONDV.
RCONDERCONDE is DOUBLE PRECISION array, dimension (N) RCONDE holds the computed reciprocal condition numbers for eigenvalues.
RCNDE1RCNDE1 is DOUBLE PRECISION array, dimension (N) RCNDE1 holds more computed reciprocal condition numbers for eigenvalues.
RCDEINRCDEIN is DOUBLE PRECISION array, dimension (N) When COMP = .TRUE. RCDEIN holds the precomputed reciprocal condition numbers for eigenvalues to be compared with RCONDE.
SCALESCALE is DOUBLE PRECISION array, dimension (N) Holds information describing balancing of matrix.
SCALE1SCALE1 is DOUBLE PRECISION array, dimension (N) Holds information describing balancing of matrix.
RESULTRESULT is DOUBLE PRECISION array, dimension (11) The values computed by the 11 tests described above. The values are currently limited to 1/ulp, to avoid overflow.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The number of entries in WORK. This must be at least 2*N, and 2*N+N**2 if tests 9, 10 or 11 are to be performed.
RWORKRWORK is DOUBLE PRECISION array, dimension (2*N)
INFOINFO is INTEGER If 0, successful exit. If <0, input parameter INFO had an incorrect value. If >0, ZGEEVX returned an error code, the absolute value of which is returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget24 (logical COMP, integer JTYPE, double precision THRESH, integer, dimension( 4 ) ISEED, integer NOUNIT, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) H, complex*16, dimension( lda, * ) HT, complex*16, dimension( * ) W, complex*16, dimension( * ) WT, complex*16, dimension( * ) WTMP, complex*16, dimension( ldvs, * ) VS, integer LDVS, complex*16, dimension( ldvs, * ) VS1, double precision RCDEIN, double precision RCDVIN, integer NSLCT, integer, dimension( * ) ISLCT, integer ISRT, double precision, dimension( 17 ) RESULT, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)
ZGET24
Purpose:

ZGET24 checks the nonsymmetric eigenvalue (Schur form) problem expert driver ZGEESX. If COMP = .FALSE., the first 13 of the following tests will be be performed on the input matrix A, and also tests 14 and 15 if LWORK is sufficiently large. If COMP = .TRUE., all 17 test will be performed. (1) 0 if T is in Schur form, 1/ulp otherwise (no sorting of eigenvalues) (2)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (no sorting of eigenvalues). (3)  I  VS VS'  / ( n ulp ) (no sorting of eigenvalues). (4) 0 if W are eigenvalues of T 1/ulp otherwise (no sorting of eigenvalues) (5) 0 if T(with VS) = T(without VS), 1/ulp otherwise (no sorting of eigenvalues) (6) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise (no sorting of eigenvalues) (7) 0 if T is in Schur form, 1/ulp otherwise (with sorting of eigenvalues) (8)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (with sorting of eigenvalues). (9)  I  VS VS'  / ( n ulp ) (with sorting of eigenvalues). (10) 0 if W are eigenvalues of T 1/ulp otherwise If workspace sufficient, also compare W with and without reciprocal condition numbers (with sorting of eigenvalues) (11) 0 if T(with VS) = T(without VS), 1/ulp otherwise If workspace sufficient, also compare T with and without reciprocal condition numbers (with sorting of eigenvalues) (12) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise If workspace sufficient, also compare VS with and without reciprocal condition numbers (with sorting of eigenvalues) (13) if sorting worked and SDIM is the number of eigenvalues which were SELECTed If workspace sufficient, also compare SDIM with and without reciprocal condition numbers (14) if RCONDE the same no matter if VS and/or RCONDV computed (15) if RCONDV the same no matter if VS and/or RCONDE computed (16) RCONDE  RCDEIN / cond(RCONDE) RCONDE is the reciprocal average eigenvalue condition number computed by ZGEESX and RCDEIN (the precomputed true value) is supplied as input. cond(RCONDE) is the condition number of RCONDE, and takes errors in computing RCONDE into account, so that the resulting quantity should be O(ULP). cond(RCONDE) is essentially given by norm(A)/RCONDV. (17) RCONDV  RCDVIN / cond(RCONDV) RCONDV is the reciprocal right invariant subspace condition number computed by ZGEESX and RCDVIN (the precomputed true value) is supplied as input. cond(RCONDV) is the condition number of RCONDV, and takes errors in computing RCONDV into account, so that the resulting quantity should be O(ULP). cond(RCONDV) is essentially given by norm(A)/RCONDE.
Parameters:

COMP
COMP is LOGICAL COMP describes which input tests to perform: = .FALSE. if the computed condition numbers are not to be tested against RCDVIN and RCDEIN = .TRUE. if they are to be compared
JTYPEJTYPE is INTEGER Type of input matrix. Used to label output if error occurs.
ISEEDISEED is INTEGER array, dimension (4) If COMP = .FALSE., the random number generator seed used to produce matrix. If COMP = .TRUE., ISEED(1) = the number of the example. Used to label output if error occurs.
THRESHTHRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
NOUNITNOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)
NN is INTEGER The dimension of A. N must be at least 0.
AA is COMPLEX*16 array, dimension (LDA, N) Used to hold the matrix whose eigenvalues are to be computed.
LDALDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least N.
HH is COMPLEX*16 array, dimension (LDA, N) Another copy of the test matrix A, modified by ZGEESX.
HTHT is COMPLEX*16 array, dimension (LDA, N) Yet another copy of the test matrix A, modified by ZGEESX.
WW is COMPLEX*16 array, dimension (N) The computed eigenvalues of A.
WTWT is COMPLEX*16 array, dimension (N) Like W, this array contains the eigenvalues of A, but those computed when ZGEESX only computes a partial eigendecomposition, i.e. not Schur vectors
WTMPWTMP is COMPLEX*16 array, dimension (N) Like W, this array contains the eigenvalues of A, but sorted by increasing real or imaginary part.
VSVS is COMPLEX*16 array, dimension (LDVS, N) VS holds the computed Schur vectors.
LDVSLDVS is INTEGER Leading dimension of VS. Must be at least max(1, N).
VS1VS1 is COMPLEX*16 array, dimension (LDVS, N) VS1 holds another copy of the computed Schur vectors.
RCDEINRCDEIN is DOUBLE PRECISION When COMP = .TRUE. RCDEIN holds the precomputed reciprocal condition number for the average of selected eigenvalues.
RCDVINRCDVIN is DOUBLE PRECISION When COMP = .TRUE. RCDVIN holds the precomputed reciprocal condition number for the selected right invariant subspace.
NSLCTNSLCT is INTEGER When COMP = .TRUE. the number of selected eigenvalues corresponding to the precomputed values RCDEIN and RCDVIN.
ISLCTISLCT is INTEGER array, dimension (NSLCT) When COMP = .TRUE. ISLCT selects the eigenvalues of the input matrix corresponding to the precomputed values RCDEIN and RCDVIN. For I=1, ... ,NSLCT, if ISLCT(I) = J, then the eigenvalue with the Jth largest real or imaginary part is selected. The real part is used if ISRT = 0, and the imaginary part if ISRT = 1. Not referenced if COMP = .FALSE.
ISRTISRT is INTEGER When COMP = .TRUE., ISRT describes how ISLCT is used to choose a subset of the spectrum. Not referenced if COMP = .FALSE.
RESULTRESULT is DOUBLE PRECISION array, dimension (17) The values computed by the 17 tests described above. The values are currently limited to 1/ulp, to avoid overflow.
WORKWORK is COMPLEX*16 array, dimension (2*N*N)
LWORKLWORK is INTEGER The number of entries in WORK to be passed to ZGEESX. This must be at least 2*N, and N*(N+1)/2 if tests 1416 are to be performed.
RWORKRWORK is DOUBLE PRECISION array, dimension (N)
BWORKBWORK is LOGICAL array, dimension (N)
INFOINFO is INTEGER If 0, successful exit. If <0, input parameter INFO had an incorrect value. If >0, ZGEESX returned an error code, the absolute value of which is returned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget35 (double precision RMAX, integer LMAX, integer NINFO, integer KNT, integer NIN)
ZGET35
Purpose:

ZGET35 tests ZTRSYL, a routine for solving the Sylvester matrix equation op(A)*X + ISGN*X*op(B) = scale*C, A and B are assumed to be in Schur canonical form, op() represents an optional transpose, and ISGN can be 1 or +1. Scale is an output less than or equal to 1, chosen to avoid overflow in X. The test code verifies that the following residual is order 1: norm(op(A)*X + ISGN*X*op(B)  scale*C) / (EPS*max(norm(A),norm(B))*norm(X))
Parameters:

RMAX
RMAX is DOUBLE PRECISION Value of the largest test ratio.
LMAXLMAX is INTEGER Example number where largest test ratio achieved.
NINFONINFO is INTEGER Number of examples where INFO is nonzero.
KNTKNT is INTEGER Total number of examples tested.
NINNIN is INTEGER Input logical unit number.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget36 (double precision RMAX, integer LMAX, integer NINFO, integer KNT, integer NIN)
ZGET36
Purpose:

ZGET36 tests ZTREXC, a routine for reordering diagonal entries of a matrix in complex Schur form. Thus, ZLAEXC computes a unitary matrix Q such that Q' * T1 * Q = T2 and where one of the diagonal blocks of T1 (the one at row IFST) has been moved to position ILST. The test code verifies that the residual Q'*T1*QT2 is small, that T2 is in Schur form, and that the final position of the IFST block is ILST. The test matrices are read from a file with logical unit number NIN.
Parameters:

RMAX
RMAX is DOUBLE PRECISION Value of the largest test ratio.
LMAXLMAX is INTEGER Example number where largest test ratio achieved.
NINFONINFO is INTEGER Number of examples where INFO is nonzero.
KNTKNT is INTEGER Total number of examples tested.
NINNIN is INTEGER Input logical unit number.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget37 (double precision, dimension( 3 ) RMAX, integer, dimension( 3 ) LMAX, integer, dimension( 3 ) NINFO, integer KNT, integer NIN)
ZGET37
Purpose:

ZGET37 tests ZTRSNA, a routine for estimating condition numbers of eigenvalues and/or right eigenvectors of a matrix. The test matrices are read from a file with logical unit number NIN.
Parameters:

RMAX
RMAX is DOUBLE PRECISION array, dimension (3) Value of the largest test ratio. RMAX(1) = largest ratio comparing different calls to ZTRSNA RMAX(2) = largest error in reciprocal condition numbers taking their conditioning into account RMAX(3) = largest error in reciprocal condition numbers not taking their conditioning into account (may be larger than RMAX(2))
LMAXLMAX is INTEGER array, dimension (3) LMAX(i) is example number where largest test ratio RMAX(i) is achieved. Also: If ZGEHRD returns INFO nonzero on example i, LMAX(1)=i If ZHSEQR returns INFO nonzero on example i, LMAX(2)=i If ZTRSNA returns INFO nonzero on example i, LMAX(3)=i
NINFONINFO is INTEGER array, dimension (3) NINFO(1) = No. of times ZGEHRD returned INFO nonzero NINFO(2) = No. of times ZHSEQR returned INFO nonzero NINFO(3) = No. of times ZTRSNA returned INFO nonzero
KNTKNT is INTEGER Total number of examples tested.
NINNIN is INTEGER Input logical unit number
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget38 (double precision, dimension( 3 ) RMAX, integer, dimension( 3 ) LMAX, integer, dimension( 3 ) NINFO, integer KNT, integer NIN)
ZGET38
Purpose:

ZGET38 tests ZTRSEN, a routine for estimating condition numbers of a cluster of eigenvalues and/or its associated right invariant subspace The test matrices are read from a file with logical unit number NIN.
Parameters:

RMAX
RMAX is DOUBLE PRECISION array, dimension (3) Values of the largest test ratios. RMAX(1) = largest residuals from ZHST01 or comparing different calls to ZTRSEN RMAX(2) = largest error in reciprocal condition numbers taking their conditioning into account RMAX(3) = largest error in reciprocal condition numbers not taking their conditioning into account (may be larger than RMAX(2))
LMAXLMAX is INTEGER array, dimension (3) LMAX(i) is example number where largest test ratio RMAX(i) is achieved. Also: If ZGEHRD returns INFO nonzero on example i, LMAX(1)=i If ZHSEQR returns INFO nonzero on example i, LMAX(2)=i If ZTRSEN returns INFO nonzero on example i, LMAX(3)=i
NINFONINFO is INTEGER array, dimension (3) NINFO(1) = No. of times ZGEHRD returned INFO nonzero NINFO(2) = No. of times ZHSEQR returned INFO nonzero NINFO(3) = No. of times ZTRSEN returned INFO nonzero
KNTKNT is INTEGER Total number of examples tested.
NINNIN is INTEGER Input logical unit number.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget51 (integer ITYPE, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision RESULT)
ZGET51
Purpose:

ZGET51 generally checks a decomposition of the form A = U B VC> where * means conjugate transpose and U and V are unitary. Specifically, if ITYPE=1 RESULT =  A  U B V*  / ( A n ulp ) If ITYPE=2, then: RESULT =  A  B  / ( A n ulp ) If ITYPE=3, then: RESULT =  I  UU*  / ( n ulp )
Parameters:

ITYPE
ITYPE is INTEGER Specifies the type of tests to be performed. =1: RESULT =  A  U B V*  / ( A n ulp ) =2: RESULT =  A  B  / ( A n ulp ) =3: RESULT =  I  UU*  / ( n ulp )
NN is INTEGER The size of the matrix. If it is zero, ZGET51 does nothing. It must be at least zero.
AA is COMPLEX*16 array, dimension (LDA, N) The original (unfactored) matrix.
LDALDA is INTEGER The leading dimension of A. It must be at least 1 and at least N.
BB is COMPLEX*16 array, dimension (LDB, N) The factored matrix.
LDBLDB is INTEGER The leading dimension of B. It must be at least 1 and at least N.
UU is COMPLEX*16 array, dimension (LDU, N) The unitary matrix on the lefthand side in the decomposition. Not referenced if ITYPE=2
LDULDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1.
VV is COMPLEX*16 array, dimension (LDV, N) The unitary matrix on the lefthand side in the decomposition. Not referenced if ITYPE=2
LDVLDV is INTEGER The leading dimension of V. LDV must be at least N and at least 1.
WORKWORK is COMPLEX*16 array, dimension (2*N**2)
RWORKRWORK is DOUBLE PRECISION array, dimension (N)
RESULTRESULT is DOUBLE PRECISION The values computed by the test specified by ITYPE. The value is currently limited to 1/ulp, to avoid overflow. Errors are flagged by RESULT=10/ulp.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget52 (logical LEFT, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( lde, * ) E, integer LDE, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision, dimension( 2 ) RESULT)
ZGET52
Purpose:

ZGET52 does an eigenvector check for the generalized eigenvalue problem. The basic test for right eigenvectors is:  b(i) A E(i)  a(i) B E(i)  RESULT(1) = max  i n ulp max( b(i) A, a(i) B ) using the 1norm. Here, a(i)/b(i) = w is the ith generalized eigenvalue of A  w B, or, equivalently, b(i)/a(i) = m is the ith generalized eigenvalue of m A  B. H H _ _ For left eigenvectors, A , B , a, and b are used. ZGET52 also tests the normalization of E. Each eigenvector is supposed to be normalized so that the maximum "absolute value" of its elements is 1, where in this case, "absolute value" of a complex value x is Re(x) + Im(x) ; let us call this maximum "absolute value" norm of a vector v M(v). If a(i)=b(i)=0, then the eigenvector is set to be the jth coordinate vector. The normalization test is: RESULT(2) = max  M(v(i))  1  / ( n ulp ) eigenvectors v(i)
Parameters:

LEFT
LEFT is LOGICAL =.TRUE.: The eigenvectors in the columns of E are assumed to be *left* eigenvectors. =.FALSE.: The eigenvectors in the columns of E are assumed to be *right* eigenvectors.
NN is INTEGER The size of the matrices. If it is zero, ZGET52 does nothing. It must be at least zero.
AA is COMPLEX*16 array, dimension (LDA, N) The matrix A.
LDALDA is INTEGER The leading dimension of A. It must be at least 1 and at least N.
BB is COMPLEX*16 array, dimension (LDB, N) The matrix B.
LDBLDB is INTEGER The leading dimension of B. It must be at least 1 and at least N.
EE is COMPLEX*16 array, dimension (LDE, N) The matrix of eigenvectors. It must be O( 1 ).
LDELDE is INTEGER The leading dimension of E. It must be at least 1 and at least N.
ALPHAALPHA is COMPLEX*16 array, dimension (N) The values a(i) as described above, which, along with b(i), define the generalized eigenvalues.
BETABETA is COMPLEX*16 array, dimension (N) The values b(i) as described above, which, along with a(i), define the generalized eigenvalues.
WORKWORK is COMPLEX*16 array, dimension (N**2)
RWORKRWORK is DOUBLE PRECISION array, dimension (N)
RESULTRESULT is DOUBLE PRECISION array, dimension (2) The values computed by the test described above. If A E or B E is likely to overflow, then RESULT(1:2) is set to 10 / ulp.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zget54 (integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( lds, * ) S, integer LDS, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( * ) WORK, double precision RESULT)
ZGET54
Purpose:

ZGET54 checks a generalized decomposition of the form A = U*S*V' and B = U*T* V' where ' means conjugate transpose and U and V are unitary. Specifically, RESULT = ( A  U*S*V', B  U*T*V' ) / (( A, B )*n*ulp )
Parameters:

N
N is INTEGER The size of the matrix. If it is zero, DGET54 does nothing. It must be at least zero.
AA is COMPLEX*16 array, dimension (LDA, N) The original (unfactored) matrix A.
LDALDA is INTEGER The leading dimension of A. It must be at least 1 and at least N.
BB is COMPLEX*16 array, dimension (LDB, N) The original (unfactored) matrix B.
LDBLDB is INTEGER The leading dimension of B. It must be at least 1 and at least N.
SS is COMPLEX*16 array, dimension (LDS, N) The factored matrix S.
LDSLDS is INTEGER The leading dimension of S. It must be at least 1 and at least N.
TT is COMPLEX*16 array, dimension (LDT, N) The factored matrix T.
LDTLDT is INTEGER The leading dimension of T. It must be at least 1 and at least N.
UU is COMPLEX*16 array, dimension (LDU, N) The orthogonal matrix on the lefthand side in the decomposition.
LDULDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1.
VV is COMPLEX*16 array, dimension (LDV, N) The orthogonal matrix on the lefthand side in the decomposition.
LDVLDV is INTEGER The leading dimension of V. LDV must be at least N and at least 1.
WORKWORK is COMPLEX*16 array, dimension (3*N**2)
RESULTRESULT is DOUBLE PRECISION The value RESULT, It is currently limited to 1/ulp, to avoid overflow. Errors are flagged by RESULT=10/ulp.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zglmts (integer N, integer M, integer P, complex*16, dimension( lda, * ) A, complex*16, dimension( lda, * ) AF, integer LDA, complex*16, dimension( ldb, * ) B, complex*16, dimension( ldb, * ) BF, integer LDB, complex*16, dimension( * ) D, complex*16, dimension( * ) DF, complex*16, dimension( * ) X, complex*16, dimension( * ) U, complex*16, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision RESULT)
ZGLMTS
Purpose:

ZGLMTS tests ZGGGLM  a subroutine for solving the generalized linear model problem.
Parameters:

N
N is INTEGER The number of rows of the matrices A and B. N >= 0.
MM is INTEGER The number of columns of the matrix A. M >= 0.
PP is INTEGER The number of columns of the matrix B. P >= 0.
AA is COMPLEX*16 array, dimension (LDA,M) The NbyM matrix A.
AFAF is COMPLEX*16 array, dimension (LDA,M)
LDALDA is INTEGER The leading dimension of the arrays A, AF. LDA >= max(M,N).
BB is COMPLEX*16 array, dimension (LDB,P) The NbyP matrix A.
BFBF is COMPLEX*16 array, dimension (LDB,P)
LDBLDB is INTEGER The leading dimension of the arrays B, BF. LDB >= max(P,N).
DD is COMPLEX*16 array, dimension( N ) On input, the left hand side of the GLM.
DFDF is COMPLEX*16 array, dimension( N )
XX is COMPLEX*16 array, dimension( M ) solution vector X in the GLM problem.
UU is COMPLEX*16 array, dimension( P ) solution vector U in the GLM problem.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The dimension of the array WORK.
RWORKRWORK is DOUBLE PRECISION array, dimension (M)
RESULTRESULT is DOUBLE PRECISION The test ratio: norm( d  A*x  B*u ) RESULT =  (norm(A)+norm(B))*(norm(x)+norm(u))*EPS
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zgqrts (integer N, integer M, integer P, complex*16, dimension( lda, * ) A, complex*16, dimension( lda, * ) AF, complex*16, dimension( lda, * ) Q, complex*16, dimension( lda, * ) R, integer LDA, complex*16, dimension( * ) TAUA, complex*16, dimension( ldb, * ) B, complex*16, dimension( ldb, * ) BF, complex*16, dimension( ldb, * ) Z, complex*16, dimension( ldb, * ) T, complex*16, dimension( ldb, * ) BWK, integer LDB, complex*16, dimension( * ) TAUB, complex*16, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 4 ) RESULT)
ZGQRTS
Purpose:

ZGQRTS tests ZGGQRF, which computes the GQR factorization of an NbyM matrix A and a NbyP matrix B: A = Q*R and B = Q*T*Z.
Parameters:

N
N is INTEGER The number of rows of the matrices A and B. N >= 0.
MM is INTEGER The number of columns of the matrix A. M >= 0.
PP is INTEGER The number of columns of the matrix B. P >= 0.
AA is COMPLEX*16 array, dimension (LDA,M) The NbyM matrix A.
AFAF is COMPLEX*16 array, dimension (LDA,N) Details of the GQR factorization of A and B, as returned by ZGGQRF, see CGGQRF for further details.
QQ is COMPLEX*16 array, dimension (LDA,N) The MbyM unitary matrix Q.
RR is COMPLEX*16 array, dimension (LDA,MAX(M,N))
LDALDA is INTEGER The leading dimension of the arrays A, AF, R and Q. LDA >= max(M,N).
TAUATAUA is COMPLEX*16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors, as returned by ZGGQRF.
BB is COMPLEX*16 array, dimension (LDB,P) On entry, the NbyP matrix A.
BFBF is COMPLEX*16 array, dimension (LDB,N) Details of the GQR factorization of A and B, as returned by ZGGQRF, see CGGQRF for further details.
ZZ is COMPLEX*16 array, dimension (LDB,P) The PbyP unitary matrix Z.
TT is COMPLEX*16 array, dimension (LDB,max(P,N))
BWKBWK is COMPLEX*16 array, dimension (LDB,N)
LDBLDB is INTEGER The leading dimension of the arrays B, BF, Z and T. LDB >= max(P,N).
TAUBTAUB is COMPLEX*16 array, dimension (min(P,N)) The scalar factors of the elementary reflectors, as returned by DGGRQF.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The dimension of the array WORK, LWORK >= max(N,M,P)**2.
RWORKRWORK is DOUBLE PRECISION array, dimension (max(N,M,P))
RESULTRESULT is DOUBLE PRECISION array, dimension (4) The test ratios: RESULT(1) = norm( R  Q'*A ) / ( MAX(M,N)*norm(A)*ULP) RESULT(2) = norm( T*Z  Q'*B ) / (MAX(P,N)*norm(B)*ULP) RESULT(3) = norm( I  Q'*Q ) / ( M*ULP ) RESULT(4) = norm( I  Z'*Z ) / ( P*ULP )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zgrqts (integer M, integer P, integer N, complex*16, dimension( lda, * ) A, complex*16, dimension( lda, * ) AF, complex*16, dimension( lda, * ) Q, complex*16, dimension( lda, * ) R, integer LDA, complex*16, dimension( * ) TAUA, complex*16, dimension( ldb, * ) B, complex*16, dimension( ldb, * ) BF, complex*16, dimension( ldb, * ) Z, complex*16, dimension( ldb, * ) T, complex*16, dimension( ldb, * ) BWK, integer LDB, complex*16, dimension( * ) TAUB, complex*16, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 4 ) RESULT)
ZGRQTS
Purpose:

ZGRQTS tests ZGGRQF, which computes the GRQ factorization of an MbyN matrix A and a PbyN matrix B: A = R*Q and B = Z*T*Q.
Parameters:

M
M is INTEGER The number of rows of the matrix A. M >= 0.
PP is INTEGER The number of rows of the matrix B. P >= 0.
NN is INTEGER The number of columns of the matrices A and B. N >= 0.
AA is COMPLEX*16 array, dimension (LDA,N) The MbyN matrix A.
AFAF is COMPLEX*16 array, dimension (LDA,N) Details of the GRQ factorization of A and B, as returned by ZGGRQF, see CGGRQF for further details.
QQ is COMPLEX*16 array, dimension (LDA,N) The NbyN unitary matrix Q.
RR is COMPLEX*16 array, dimension (LDA,MAX(M,N))
LDALDA is INTEGER The leading dimension of the arrays A, AF, R and Q. LDA >= max(M,N).
TAUATAUA is COMPLEX*16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors, as returned by DGGQRC.
BB is COMPLEX*16 array, dimension (LDB,N) On entry, the PbyN matrix A.
BFBF is COMPLEX*16 array, dimension (LDB,N) Details of the GQR factorization of A and B, as returned by ZGGRQF, see CGGRQF for further details.
ZZ is DOUBLE PRECISION array, dimension (LDB,P) The PbyP unitary matrix Z.
TT is COMPLEX*16 array, dimension (LDB,max(P,N))
BWKBWK is COMPLEX*16 array, dimension (LDB,N)
LDBLDB is INTEGER The leading dimension of the arrays B, BF, Z and T. LDB >= max(P,N).
TAUBTAUB is COMPLEX*16 array, dimension (min(P,N)) The scalar factors of the elementary reflectors, as returned by DGGRQF.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The dimension of the array WORK, LWORK >= max(M,P,N)**2.
RWORKRWORK is DOUBLE PRECISION array, dimension (M)
RESULTRESULT is DOUBLE PRECISION array, dimension (4) The test ratios: RESULT(1) = norm( R  A*Q' ) / ( MAX(M,N)*norm(A)*ULP) RESULT(2) = norm( T*Q  Z'*B ) / (MAX(P,N)*norm(B)*ULP) RESULT(3) = norm( I  Q'*Q ) / ( N*ULP ) RESULT(4) = norm( I  Z'*Z ) / ( P*ULP )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zgsvts3 (integer M, integer P, integer N, complex*16, dimension( lda, * ) A, complex*16, dimension( lda, * ) AF, integer LDA, complex*16, dimension( ldb, * ) B, complex*16, dimension( ldb, * ) BF, integer LDB, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( * ) ALPHA, double precision, dimension( * ) BETA, complex*16, dimension( ldr, * ) R, integer LDR, integer, dimension( * ) IWORK, complex*16, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 6 ) RESULT)
ZGSVTS3
Purpose:

ZGSVTS3 tests ZGGSVD3, which computes the GSVD of an MbyN matrix A and a PbyN matrix B: U'*A*Q = D1*R and V'*B*Q = D2*R.
Parameters:

M
M is INTEGER The number of rows of the matrix A. M >= 0.
PP is INTEGER The number of rows of the matrix B. P >= 0.
NN is INTEGER The number of columns of the matrices A and B. N >= 0.
AA is COMPLEX*16 array, dimension (LDA,M) The MbyN matrix A.
AFAF is COMPLEX*16 array, dimension (LDA,N) Details of the GSVD of A and B, as returned by ZGGSVD3, see ZGGSVD3 for further details.
LDALDA is INTEGER The leading dimension of the arrays A and AF. LDA >= max( 1,M ).
BB is COMPLEX*16 array, dimension (LDB,P) On entry, the PbyN matrix B.
BFBF is COMPLEX*16 array, dimension (LDB,N) Details of the GSVD of A and B, as returned by ZGGSVD3, see ZGGSVD3 for further details.
LDBLDB is INTEGER The leading dimension of the arrays B and BF. LDB >= max(1,P).
UU is COMPLEX*16 array, dimension(LDU,M) The M by M unitary matrix U.
LDULDU is INTEGER The leading dimension of the array U. LDU >= max(1,M).
VV is COMPLEX*16 array, dimension(LDV,M) The P by P unitary matrix V.
LDVLDV is INTEGER The leading dimension of the array V. LDV >= max(1,P).
QQ is COMPLEX*16 array, dimension(LDQ,N) The N by N unitary matrix Q.
LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).
ALPHAALPHA is DOUBLE PRECISION array, dimension (N)
BETABETA is DOUBLE PRECISION array, dimension (N) The generalized singular value pairs of A and B, the ``diagonal'' matrices D1 and D2 are constructed from ALPHA and BETA, see subroutine ZGGSVD3 for details.
RR is COMPLEX*16 array, dimension(LDQ,N) The upper triangular matrix R.
LDRLDR is INTEGER The leading dimension of the array R. LDR >= max(1,N).
IWORKIWORK is INTEGER array, dimension (N)
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The dimension of the array WORK, LWORK >= max(M,P,N)*max(M,P,N).
RWORKRWORK is DOUBLE PRECISION array, dimension (max(M,P,N))
RESULTRESULT is DOUBLE PRECISION array, dimension (6) The test ratios: RESULT(1) = norm( U'*A*Q  D1*R ) / ( MAX(M,N)*norm(A)*ULP) RESULT(2) = norm( V'*B*Q  D2*R ) / ( MAX(P,N)*norm(B)*ULP) RESULT(3) = norm( I  U'*U ) / ( M*ULP ) RESULT(4) = norm( I  V'*V ) / ( P*ULP ) RESULT(5) = norm( I  Q'*Q ) / ( N*ULP ) RESULT(6) = 0 if ALPHA is in decreasing order; = ULPINV otherwise.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 August 2015
subroutine zhbt21 (character UPLO, integer N, integer KA, integer KS, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision, dimension( 2 ) RESULT)
ZHBT21
Purpose:

ZHBT21 generally checks a decomposition of the form A = U S UC> where * means conjugate transpose, A is hermitian banded, U is unitary, and S is diagonal (if KS=0) or symmetric tridiagonal (if KS=1). Specifically: RESULT(1) =  A  U S U*  / ( A n ulp ) *andC> RESULT(2) =  I  UU*  / ( n ulp )
Parameters:

UPLO
UPLO is CHARACTER If UPLO='U', the upper triangle of A and V will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A and V will be used and the (strictly) upper triangle will not be referenced.
NN is INTEGER The size of the matrix. If it is zero, ZHBT21 does nothing. It must be at least zero.
KAKA is INTEGER The bandwidth of the matrix A. It must be at least zero. If it is larger than N1, then max( 0, N1 ) will be used.
KSKS is INTEGER The bandwidth of the matrix S. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tridiagonal.
AA is COMPLEX*16 array, dimension (LDA, N) The original (unfactored) matrix. It is assumed to be hermitian, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced.
LDALDA is INTEGER The leading dimension of A. It must be at least 1 and at least min( KA, N1 ).
DD is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri) diagonal matrix S.
EE is DOUBLE PRECISION array, dimension (N1) The offdiagonal of the (symmetric tri) diagonal matrix S. E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and (3,2) element, etc. Not referenced if KS=0.
UU is COMPLEX*16 array, dimension (LDU, N) The unitary matrix in the decomposition, expressed as a dense matrix (i.e., not as a product of Householder transformations, Givens transformations, etc.)
LDULDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1.
WORKWORK is COMPLEX*16 array, dimension (N**2)
RWORKRWORK is DOUBLE PRECISION array, dimension (N)
RESULTRESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zhet21 (integer ITYPE, character UPLO, integer N, integer KBAND, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision, dimension( 2 ) RESULT)
ZHET21
Purpose:

ZHET21 generally checks a decomposition of the form A = U S UC> where * means conjugate transpose, A is hermitian, U is unitary, and S is diagonal (if KBAND=0) or (real) symmetric tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as a dense matrix; otherwise U is expressed as a product of Householder transformations, whose vectors are stored in the array "V" and whose scaling constants are in "TAU". We shall use the letter "V" to refer to the product of Householder transformations (which should be equal to U). Specifically, if ITYPE=1, then: RESULT(1) =  A  U S U*  / ( A n ulp ) *andC> RESULT(2) =  I  UU*  / ( n ulp ) If ITYPE=2, then: RESULT(1) =  A  V S V*  / ( A n ulp ) If ITYPE=3, then: RESULT(1) =  I  UV*  / ( n ulp ) For ITYPE > 1, the transformation U is expressed as a product V = H(1)...H(n2), where H(j) = I  tau(j) v(j) v(j)C> and each vector v(j) has its first j elements 0 and the remaining nj elements stored in V(j+1:n,j).
Parameters:

ITYPE
ITYPE is INTEGER Specifies the type of tests to be performed. 1: U expressed as a dense unitary matrix: RESULT(1) =  A  U S U*  / ( A n ulp ) *andC> RESULT(2) =  I  UU*  / ( n ulp ) 2: U expressed as a product V of Housholder transformations: RESULT(1) =  A  V S V*  / ( A n ulp ) 3: U expressed both as a dense unitary matrix and as a product of Housholder transformations: RESULT(1) =  I  UV*  / ( n ulp )
UPLOUPLO is CHARACTER If UPLO='U', the upper triangle of A and V will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A and V will be used and the (strictly) upper triangle will not be referenced.
NN is INTEGER The size of the matrix. If it is zero, ZHET21 does nothing. It must be at least zero.
KBANDKBAND is INTEGER The bandwidth of the matrix. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tridiagonal.
AA is COMPLEX*16 array, dimension (LDA, N) The original (unfactored) matrix. It is assumed to be hermitian, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced.
LDALDA is INTEGER The leading dimension of A. It must be at least 1 and at least N.
DD is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri) diagonal matrix.
EE is DOUBLE PRECISION array, dimension (N1) The offdiagonal of the (symmetric tri) diagonal matrix. E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and (3,2) element, etc. Not referenced if KBAND=0.
UU is COMPLEX*16 array, dimension (LDU, N) If ITYPE=1 or 3, this contains the unitary matrix in the decomposition, expressed as a dense matrix. If ITYPE=2, then it is not referenced.
LDULDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1.
VV is COMPLEX*16 array, dimension (LDV, N) If ITYPE=2 or 3, the columns of this array contain the Householder vectors used to describe the unitary matrix in the decomposition. If UPLO='L', then the vectors are in the lower triangle, if UPLO='U', then in the upper triangle. *NOTE* If ITYPE=2 or 3, V is modified and restored. The subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U') is set to one, and later reset to its original value, during the course of the calculation. If ITYPE=1, then it is neither referenced nor modified.
LDVLDV is INTEGER The leading dimension of V. LDV must be at least N and at least 1.
TAUTAU is COMPLEX*16 array, dimension (N) If ITYPE >= 2, then TAU(j) is the scalar factor of v(j) v(j)* in the Householder transformation H(j) of the product U = H(1)...H(n2) If ITYPE < 2, then TAU is not referenced.
WORKWORK is COMPLEX*16 array, dimension (2*N**2)
RWORKRWORK is DOUBLE PRECISION array, dimension (N)
RESULTRESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. RESULT(1) is always modified. RESULT(2) is modified only if ITYPE=1.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zhet22 (integer ITYPE, character UPLO, integer N, integer M, integer KBAND, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision, dimension( 2 ) RESULT)
ZHET22
Purpose:

ZHET22 generally checks a decomposition of the form A U = U S where A is complex Hermitian, the columns of U are orthonormal, and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as a dense matrix, otherwise the U is expressed as a product of Householder transformations, whose vectors are stored in the array "V" and whose scaling constants are in "TAU"; we shall use the letter "V" to refer to the product of Householder transformations (which should be equal to U). Specifically, if ITYPE=1, then: RESULT(1) =  U' A U  S  / ( A m ulp ) *andC> RESULT(2) =  I  U'U  / ( m ulp )
ITYPE INTEGER Specifies the type of tests to be performed. 1: U expressed as a dense orthogonal matrix: RESULT(1) =  A  U S U'  / ( A n ulp ) *andC> RESULT(2) =  I  UU'  / ( n ulp ) UPLO CHARACTER If UPLO='U', the upper triangle of A will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A will be used and the (strictly) upper triangle will not be referenced. Not modified. N INTEGER The size of the matrix. If it is zero, ZHET22 does nothing. It must be at least zero. Not modified. M INTEGER The number of columns of U. If it is zero, ZHET22 does nothing. It must be at least zero. Not modified. KBAND INTEGER The bandwidth of the matrix. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tridiagonal. Not modified. A COMPLEX*16 array, dimension (LDA , N) The original (unfactored) matrix. It is assumed to be symmetric, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced. Not modified. LDA INTEGER The leading dimension of A. It must be at least 1 and at least N. Not modified. D DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri) diagonal matrix. Not modified. E DOUBLE PRECISION array, dimension (N) The offdiagonal of the (symmetric tri) diagonal matrix. E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc. Not referenced if KBAND=0. Not modified. U COMPLEX*16 array, dimension (LDU, N) If ITYPE=1, this contains the orthogonal matrix in the decomposition, expressed as a dense matrix. Not modified. LDU INTEGER The leading dimension of U. LDU must be at least N and at least 1. Not modified. V COMPLEX*16 array, dimension (LDV, N) If ITYPE=2 or 3, the lower triangle of this array contains the Householder vectors used to describe the orthogonal matrix in the decomposition. If ITYPE=1, then it is not referenced. Not modified. LDV INTEGER The leading dimension of V. LDV must be at least N and at least 1. Not modified. TAU COMPLEX*16 array, dimension (N) If ITYPE >= 2, then TAU(j) is the scalar factor of v(j) v(j)' in the Householder transformation H(j) of the product U = H(1)...H(n2) If ITYPE < 2, then TAU is not referenced. Not modified. WORK COMPLEX*16 array, dimension (2*N**2) Workspace. Modified. RWORK DOUBLE PRECISION array, dimension (N) Workspace. Modified. RESULT DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. RESULT(1) is always modified. RESULT(2) is modified only if LDU is at least N. Modified.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zhpt21 (integer ITYPE, character UPLO, integer N, integer KBAND, complex*16, dimension( * ) AP, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( * ) VP, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision, dimension( 2 ) RESULT)
ZHPT21
Purpose:

ZHPT21 generally checks a decomposition of the form A = U S UC> where * means conjugate transpose, A is hermitian, U is unitary, and S is diagonal (if KBAND=0) or (real) symmetric tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as a dense matrix, otherwise the U is expressed as a product of Householder transformations, whose vectors are stored in the array "V" and whose scaling constants are in "TAU"; we shall use the letter "V" to refer to the product of Householder transformations (which should be equal to U). Specifically, if ITYPE=1, then: RESULT(1) =  A  U S U*  / ( A n ulp ) *andC> RESULT(2) =  I  UU*  / ( n ulp ) If ITYPE=2, then: RESULT(1) =  A  V S V*  / ( A n ulp ) If ITYPE=3, then: RESULT(1) =  I  UV*  / ( n ulp ) Packed storage means that, for example, if UPLO='U', then the columns of the upper triangle of A are stored one after another, so that A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if UPLO='L', then the columns of the lower triangle of A are stored one after another in AP, so that A(j+1,j+1) immediately follows A(n,j) in the array AP. This means that A(i,j) is stored in: AP( i + j*(j1)/2 ) if UPLO='U' AP( i + (2*nj)*(j1)/2 ) if UPLO='L' The array VP bears the same relation to the matrix V that A does to AP. For ITYPE > 1, the transformation U is expressed as a product of Householder transformations: If UPLO='U', then V = H(n1)...H(1), where H(j) = I  tau(j) v(j) v(j)C> and the first j1 elements of v(j) are stored in V(1:j1,j+1), (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j1 ) ), the jth element is 1, and the last nj elements are 0. If UPLO='L', then V = H(1)...H(n1), where H(j) = I  tau(j) v(j) v(j)C> and the first j elements of v(j) are 0, the (j+1)st is 1, and the (j+2)nd through nth elements are stored in V(j+2:n,j) (i.e., in VP( (2*nj)*(j1)/2 + j+2 : (2*nj)*(j1)/2 + n ) .)
Parameters:

ITYPE
ITYPE is INTEGER Specifies the type of tests to be performed. 1: U expressed as a dense unitary matrix: RESULT(1) =  A  U S U*  / ( A n ulp ) *andC> RESULT(2) =  I  UU*  / ( n ulp ) 2: U expressed as a product V of Housholder transformations: RESULT(1) =  A  V S V*  / ( A n ulp ) 3: U expressed both as a dense unitary matrix and as a product of Housholder transformations: RESULT(1) =  I  UV*  / ( n ulp )
UPLOUPLO is CHARACTER If UPLO='U', the upper triangle of A and V will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A and V will be used and the (strictly) upper triangle will not be referenced.
NN is INTEGER The size of the matrix. If it is zero, ZHPT21 does nothing. It must be at least zero.
KBANDKBAND is INTEGER The bandwidth of the matrix. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tridiagonal.
APAP is COMPLEX*16 array, dimension (N*(N+1)/2) The original (unfactored) matrix. It is assumed to be hermitian, and contains the columns of just the upper triangle (UPLO='U') or only the lower triangle (UPLO='L'), packed one after another.
DD is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri) diagonal matrix.
EE is DOUBLE PRECISION array, dimension (N) The offdiagonal of the (symmetric tri) diagonal matrix. E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and (3,2) element, etc. Not referenced if KBAND=0.
UU is COMPLEX*16 array, dimension (LDU, N) If ITYPE=1 or 3, this contains the unitary matrix in the decomposition, expressed as a dense matrix. If ITYPE=2, then it is not referenced.
LDULDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1.
VPVP is DOUBLE PRECISION array, dimension (N*(N+1)/2) If ITYPE=2 or 3, the columns of this array contain the Householder vectors used to describe the unitary matrix in the decomposition, as described in purpose. *NOTE* If ITYPE=2 or 3, V is modified and restored. The subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U') is set to one, and later reset to its original value, during the course of the calculation. If ITYPE=1, then it is neither referenced nor modified.
TAUTAU is COMPLEX*16 array, dimension (N) If ITYPE >= 2, then TAU(j) is the scalar factor of v(j) v(j)* in the Householder transformation H(j) of the product U = H(1)...H(n2) If ITYPE < 2, then TAU is not referenced.
WORKWORK is COMPLEX*16 array, dimension (N**2) Workspace.
RWORKRWORK is DOUBLE PRECISION array, dimension (N) Workspace.
RESULTRESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. RESULT(1) is always modified. RESULT(2) is modified only if ITYPE=1.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zhst01 (integer N, integer ILO, integer IHI, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 2 ) RESULT)
ZHST01
Purpose:

ZHST01 tests the reduction of a general matrix A to upper Hessenberg form: A = Q*H*Q'. Two test ratios are computed; RESULT(1) = norm( A  Q*H*Q' ) / ( norm(A) * N * EPS ) RESULT(2) = norm( I  Q'*Q ) / ( N * EPS ) The matrix Q is assumed to be given explicitly as it would be following ZGEHRD + ZUNGHR. In this version, ILO and IHI are not used, but they could be used to save some work if this is desired.
Parameters:

N
N is INTEGER The order of the matrix A. N >= 0.
ILOILO is INTEGER
IHIIHI is INTEGER A is assumed to be upper triangular in rows and columns 1:ILO1 and IHI+1:N, so Q differs from the identity only in rows and columns ILO+1:IHI.
AA is COMPLEX*16 array, dimension (LDA,N) The original n by n matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
HH is COMPLEX*16 array, dimension (LDH,N) The upper Hessenberg matrix H from the reduction A = Q*H*Q' as computed by ZGEHRD. H is assumed to be zero below the first subdiagonal.
LDHLDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).
QQ is COMPLEX*16 array, dimension (LDQ,N) The orthogonal matrix Q from the reduction A = Q*H*Q' as computed by ZGEHRD + ZUNGHR.
LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The length of the array WORK. LWORK >= 2*N*N.
RWORKRWORK is DOUBLE PRECISION array, dimension (N)
RESULTRESULT is DOUBLE PRECISION array, dimension (2) RESULT(1) = norm( A  Q*H*Q' ) / ( norm(A) * N * EPS ) RESULT(2) = norm( I  Q'*Q ) / ( N * EPS )
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zlarfy (character UPLO, integer N, complex*16, dimension( * ) V, integer INCV, complex*16 TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK)
ZLARFY
Purpose:

ZLARFY applies an elementary reflector, or Householder matrix, H, to an n x n Hermitian matrix C, from both the left and the right. H is represented in the form H = I  tau * v * v' where tau is a scalar and v is a vector. If tau is zero, then H is taken to be the unit matrix.
Parameters:

UPLO
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix C is stored. = 'U': Upper triangle = 'L': Lower triangle
NN is INTEGER The number of rows and columns of the matrix C. N >= 0.
VV is COMPLEX*16 array, dimension (1 + (N1)*abs(INCV)) The vector v as described above.
INCVINCV is INTEGER The increment between successive elements of v. INCV must not be zero.
TAUTAU is COMPLEX*16 The value tau as described above.
CC is COMPLEX*16 array, dimension (LDC, N) On entry, the matrix C. On exit, C is overwritten by H * C * H'.
LDCLDC is INTEGER The leading dimension of the array C. LDC >= max( 1, N ).
WORKWORK is COMPLEX*16 array, dimension (N)
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zlarhs (character*3 PATH, character XTYPE, character UPLO, character TRANS, integer M, integer N, integer KL, integer KU, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( ldb, * ) B, integer LDB, integer, dimension( 4 ) ISEED, integer INFO)
ZLARHS
Purpose:

ZLARHS chooses a set of NRHS random solution vectors and sets up the right hand sides for the linear system op( A ) * X = B, where op( A ) may be A, A**T (transpose of A), or A**H (conjugate transpose of A).
Parameters:

PATH
PATH is CHARACTER*3 The type of the complex matrix A. PATH may be given in any combination of upper and lower case. Valid paths include xGE: General m x n matrix xGB: General banded matrix xPO: Hermitian positive definite, 2D storage xPP: Hermitian positive definite packed xPB: Hermitian positive definite banded xHE: Hermitian indefinite, 2D storage xHP: Hermitian indefinite packed xHB: Hermitian indefinite banded xSY: Symmetric indefinite, 2D storage xSP: Symmetric indefinite packed xSB: Symmetric indefinite banded xTR: Triangular xTP: Triangular packed xTB: Triangular banded xQR: General m x n matrix xLQ: General m x n matrix xQL: General m x n matrix xRQ: General m x n matrix where the leading character indicates the precision.
XTYPEXTYPE is CHARACTER*1 Specifies how the exact solution X will be determined: = 'N': New solution; generate a random X. = 'C': Computed; use value of X on entry.
UPLOUPLO is CHARACTER*1 Used only if A is symmetric or triangular; specifies whether the upper or lower triangular part of the matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular
TRANSTRANS is CHARACTER*1 Used only if A is nonsymmetric; specifies the operation applied to the matrix A. = 'N': B := A * X = 'T': B := A**T * X = 'C': B := A**H * X
MM is INTEGER The number of rows of the matrix A. M >= 0.
NN is INTEGER The number of columns of the matrix A. N >= 0.
KLKL is INTEGER Used only if A is a band matrix; specifies the number of subdiagonals of A if A is a general band matrix or if A is symmetric or triangular and UPLO = 'L'; specifies the number of superdiagonals of A if A is symmetric or triangular and UPLO = 'U'. 0 <= KL <= M1.
KUKU is INTEGER Used only if A is a general band matrix or if A is triangular. If PATH = xGB, specifies the number of superdiagonals of A, and 0 <= KU <= N1. If PATH = xTR, xTP, or xTB, specifies whether or not the matrix has unit diagonal: = 1: matrix has nonunit diagonal (default) = 2: matrix has unit diagonal
NRHSNRHS is INTEGER The number of right hand side vectors in the system A*X = B.
AA is COMPLEX*16 array, dimension (LDA,N) The test matrix whose type is given by PATH.
LDALDA is INTEGER The leading dimension of the array A. If PATH = xGB, LDA >= KL+KU+1. If PATH = xPB, xSB, xHB, or xTB, LDA >= KL+1. Otherwise, LDA >= max(1,M).
XX is or output) COMPLEX*16 array, dimension (LDX,NRHS) On entry, if XTYPE = 'C' (for 'Computed'), then X contains the exact solution to the system of linear equations. On exit, if XTYPE = 'N' (for 'New'), then X is initialized with random values.
LDXLDX is INTEGER The leading dimension of the array X. If TRANS = 'N', LDX >= max(1,N); if TRANS = 'T', LDX >= max(1,M).
BB is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side vector(s) for the system of equations, computed from B = op(A) * X, where op(A) is determined by TRANS.
LDBLDB is INTEGER The leading dimension of the array B. If TRANS = 'N', LDB >= max(1,M); if TRANS = 'T', LDB >= max(1,N).
ISEEDISEED is INTEGER array, dimension (4) The seed vector for the random number generator (used in ZLATMS). Modified on exit.
INFOINFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zlatm4 (integer ITYPE, integer N, integer NZ1, integer NZ2, logical RSIGN, double precision AMAGN, double precision RCOND, double precision TRIANG, integer IDIST, integer, dimension( 4 ) ISEED, complex*16, dimension( lda, * ) A, integer LDA)
ZLATM4
Purpose:

ZLATM4 generates basic square matrices, which may later be multiplied by others in order to produce test matrices. It is intended mainly to be used to test the generalized eigenvalue routines. It first generates the diagonal and (possibly) subdiagonal, according to the value of ITYPE, NZ1, NZ2, RSIGN, AMAGN, and RCOND. It then fills in the upper triangle with random numbers, if TRIANG is nonzero.
Parameters:

ITYPE
ITYPE is INTEGER The "type" of matrix on the diagonal and subdiagonal. If ITYPE < 0, then type abs(ITYPE) is generated and then swapped end for end (A(I,J) := A'(NJ,NI).) See also the description of AMAGN and RSIGN. Special types: = 0: the zero matrix. = 1: the identity. = 2: a transposed Jordan block. = 3: If N is odd, then a k+1 x k+1 transposed Jordan block followed by a k x k identity block, where k=(N1)/2. If N is even, then k=(N2)/2, and a zero diagonal entry is tacked onto the end. Diagonal types. The diagonal consists of NZ1 zeros, then k=NNZ1NZ2 nonzeros. The subdiagonal is zero. ITYPE specifies the nonzero diagonal entries as follows: = 4: 1, ..., k = 5: 1, RCOND, ..., RCOND = 6: 1, ..., 1, RCOND = 7: 1, a, a^2, ..., a^(k1)=RCOND = 8: 1, 1d, 12*d, ..., 1(k1)*d=RCOND = 9: random numbers chosen from (RCOND,1) = 10: random numbers with distribution IDIST (see ZLARND.)
NN is INTEGER The order of the matrix.
NZ1NZ1 is INTEGER If abs(ITYPE) > 3, then the first NZ1 diagonal entries will be zero.
NZ2NZ2 is INTEGER If abs(ITYPE) > 3, then the last NZ2 diagonal entries will be zero.
RSIGNRSIGN is LOGICAL = .TRUE.: The diagonal and subdiagonal entries will be multiplied by random numbers of magnitude 1. = .FALSE.: The diagonal and subdiagonal entries will be left as they are (usually nonnegative real.)
AMAGNAMAGN is DOUBLE PRECISION The diagonal and subdiagonal entries will be multiplied by AMAGN.
RCONDRCOND is DOUBLE PRECISION If abs(ITYPE) > 4, then the smallest diagonal entry will be RCOND. RCOND must be between 0 and 1.
TRIANGTRIANG is DOUBLE PRECISION The entries above the diagonal will be random numbers with magnitude bounded by TRIANG (i.e., random numbers multiplied by TRIANG.)
IDISTIDIST is INTEGER On entry, DIST specifies the type of distribution to be used to generate a random matrix . = 1: real and imaginary parts each UNIFORM( 0, 1 ) = 2: real and imaginary parts each UNIFORM( 1, 1 ) = 3: real and imaginary parts each NORMAL( 0, 1 ) = 4: complex number uniform in DISK( 0, 1 )
ISEEDISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The values of ISEED are changed on exit, and can be used in the next call to ZLATM4 to continue the same random number sequence. Note: ISEED(4) should be odd, for the random number generator used at present.
AA is COMPLEX*16 array, dimension (LDA, N) Array to be computed.
LDALDA is INTEGER Leading dimension of A. Must be at least 1 and at least N.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
logical function zlctes (complex*16 Z, complex*16 D)
ZLCTES
Purpose:

ZLCTES returns .TRUE. if the eigenvalue Z/D is to be selected (specifically, in this subroutine, if the real part of the eigenvalue is negative), and otherwise it returns .FALSE.. It is used by the test routine ZDRGES to test whether the driver routine ZGGES successfully sorts eigenvalues.
Parameters:

Z
Z is COMPLEX*16 The numerator part of a complex eigenvalue Z/D.
DD is COMPLEX*16 The denominator part of a complex eigenvalue Z/D.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
logical function zlctsx (complex*16 ALPHA, complex*16 BETA)
ZLCTSX
Purpose:

This function is used to determine what eigenvalues will be selected. If this is part of the test driver ZDRGSX, do not change the code UNLESS you are testing input examples and not using the builtin examples.
Parameters:

ALPHA
ALPHA is COMPLEX*16
BETABETA is COMPLEX*16 parameters to decide whether the pair (ALPHA, BETA) is selected.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zlsets (integer M, integer P, integer N, complex*16, dimension( lda, * ) A, complex*16, dimension( lda, * ) AF, integer LDA, complex*16, dimension( ldb, * ) B, complex*16, dimension( ldb, * ) BF, integer LDB, complex*16, dimension( * ) C, complex*16, dimension( * ) CF, complex*16, dimension( * ) D, complex*16, dimension( * ) DF, complex*16, dimension( * ) X, complex*16, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( 2 ) RESULT)
ZLSETS
Purpose:

ZLSETS tests ZGGLSE  a subroutine for solving linear equality constrained least square problem (LSE).
Parameters:

M
M is INTEGER The number of rows of the matrix A. M >= 0.
PP is INTEGER The number of rows of the matrix B. P >= 0.
NN is INTEGER The number of columns of the matrices A and B. N >= 0.
AA is COMPLEX*16 array, dimension (LDA,N) The MbyN matrix A.
AFAF is COMPLEX*16 array, dimension (LDA,N)
LDALDA is INTEGER The leading dimension of the arrays A, AF, Q and R. LDA >= max(M,N).
BB is COMPLEX*16 array, dimension (LDB,N) The PbyN matrix A.
BFBF is COMPLEX*16 array, dimension (LDB,N)
LDBLDB is INTEGER The leading dimension of the arrays B, BF, V and S. LDB >= max(P,N).
CC is COMPLEX*16 array, dimension( M ) the vector C in the LSE problem.
CFCF is COMPLEX*16 array, dimension( M )
DD is COMPLEX*16 array, dimension( P ) the vector D in the LSE problem.
DFDF is COMPLEX*16 array, dimension( P )
XX is COMPLEX*16 array, dimension( N ) solution vector X in the LSE problem.
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The dimension of the array WORK.
RWORKRWORK is DOUBLE PRECISION array, dimension (M)
RESULTRESULT is DOUBLE PRECISION array, dimension (2) The test ratios: RESULT(1) = norm( A*x  c )/ norm(A)*norm(X)*EPS RESULT(2) = norm( B*x  d )/ norm(B)*norm(X)*EPS
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zsbmv (character UPLO, integer N, integer K, complex*16 ALPHA, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) X, integer INCX, complex*16 BETA, complex*16, dimension( * ) Y, integer INCY)
ZSBMV
Purpose:

ZSBMV performs the matrixvector operation y := alpha*A*x + beta*y, where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric band matrix, with k superdiagonals.
UPLO  CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the band matrix A is being supplied as follows: UPLO = 'U' or 'u' The upper triangular part of A is being supplied. UPLO = 'L' or 'l' The lower triangular part of A is being supplied. Unchanged on exit. N  INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. Unchanged on exit. K  INTEGER On entry, K specifies the number of superdiagonals of the matrix A. K must satisfy 0 .le. K. Unchanged on exit. ALPHA  COMPLEX*16 On entry, ALPHA specifies the scalar alpha. Unchanged on exit. A  COMPLEX*16 array, dimension( LDA, N ) Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) by n part of the array A must contain the upper triangular band part of the symmetric matrix, supplied column by column, with the leading diagonal of the matrix in row ( k + 1 ) of the array, the first superdiagonal starting at position 2 in row k, and so on. The top left k by k triangle of the array A is not referenced. The following program segment will transfer the upper triangular part of a symmetric band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N M = K + 1  J DO 10, I = MAX( 1, J  K ), J A( M + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) by n part of the array A must contain the lower triangular band part of the symmetric matrix, supplied column by column, with the leading diagonal of the matrix in row 1 of the array, the first subdiagonal starting at position 1 in row 2, and so on. The bottom right k by k triangle of the array A is not referenced. The following program segment will transfer the lower triangular part of a symmetric band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N M = 1  J DO 10, I = J, MIN( N, J + K ) A( M + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE Unchanged on exit. LDA  INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( k + 1 ). Unchanged on exit. X  COMPLEX*16 array, dimension at least ( 1 + ( N  1 )*abs( INCX ) ). Before entry, the incremented array X must contain the vector x. Unchanged on exit. INCX  INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit. BETA  COMPLEX*16 On entry, BETA specifies the scalar beta. Unchanged on exit. Y  COMPLEX*16 array, dimension at least ( 1 + ( N  1 )*abs( INCY ) ). Before entry, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y. INCY  INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zsgt01 (integer ITYPE, character UPLO, integer N, integer M, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) D, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision, dimension( * ) RESULT)
ZSGT01
Purpose:

CDGT01 checks a decomposition of the form A Z = B Z D or A B Z = Z D or B A Z = Z D where A is a Hermitian matrix, B is Hermitian positive definite, Z is unitary, and D is diagonal. One of the following test ratios is computed: ITYPE = 1: RESULT(1) =  A Z  B Z D  / ( A Z n ulp ) ITYPE = 2: RESULT(1) =  A B Z  Z D  / ( A Z n ulp ) ITYPE = 3: RESULT(1) =  B A Z  Z D  / ( A Z n ulp )
Parameters:

ITYPE
ITYPE is INTEGER The form of the Hermitian generalized eigenproblem. = 1: A*z = (lambda)*B*z = 2: A*B*z = (lambda)*z = 3: B*A*z = (lambda)*z
UPLOUPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrices A and B is stored. = 'U': Upper triangular = 'L': Lower triangular
NN is INTEGER The order of the matrix A. N >= 0.
MM is INTEGER The number of eigenvalues found. M >= 0.
AA is COMPLEX*16 array, dimension (LDA, N) The original Hermitian matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
BB is COMPLEX*16 array, dimension (LDB, N) The original Hermitian positive definite matrix B.
LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
ZZ is COMPLEX*16 array, dimension (LDZ, M) The computed eigenvectors of the generalized eigenproblem.
LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,N).
DD is DOUBLE PRECISION array, dimension (M) The computed eigenvalues of the generalized eigenproblem.
WORKWORK is COMPLEX*16 array, dimension (N*N)
RWORKRWORK is DOUBLE PRECISION array, dimension (N)
RESULTRESULT is DOUBLE PRECISION array, dimension (1) The test ratio as described above.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
logical function zslect (complex*16 Z)
ZSLECT
Purpose:

ZSLECT returns .TRUE. if the eigenvalue Z is to be selected, otherwise it returns .FALSE. It is used by ZCHK41 to test if ZGEES successfully sorts eigenvalues, and by ZCHK43 to test if ZGEESX successfully sorts eigenvalues. The common block /SSLCT/ controls how eigenvalues are selected. If SELOPT = 0, then ZSLECT return .TRUE. when real(Z) is less than zero, and .FALSE. otherwise. If SELOPT is at least 1, ZSLECT returns SELVAL(SELOPT) and adds 1 to SELOPT, cycling back to 1 at SELMAX.
Parameters:

Z
Z is COMPLEX*16 The eigenvalue Z.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
subroutine zstt21 (integer N, integer KBAND, double precision, dimension( * ) AD, double precision, dimension( * ) AE, double precision, dimension( * ) SD, double precision, dimension( * ) SE, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, double precision, dimension( 2 ) RESULT)
ZSTT21
Purpose:

ZSTT21 checks a decomposition of the form A = U S UC> where * means conjugate transpose, A is real symmetric tridiagonal, U is unitary, and S is real and diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1). Two tests are performed: RESULT(1) =  A  U S U*  / ( A n ulp ) RESULT(2) =  I  UU*  / ( n ulp )
Parameters:

N
N is INTEGER The size of the matrix. If it is zero, ZSTT21 does nothing. It must be at least zero.
KBANDKBAND is INTEGER The bandwidth of the matrix S. It may only be zero or one. If zero, then S is diagonal, and SE is not referenced. If one, then S is symmetric tridiagonal.
ADAD is DOUBLE PRECISION array, dimension (N) The diagonal of the original (unfactored) matrix A. A is assumed to be real symmetric tridiagonal.
AEAE is DOUBLE PRECISION array, dimension (N1) The offdiagonal of the original (unfactored) matrix A. A is assumed to be symmetric tridiagonal. AE(1) is the (1,2) and (2,1) element, AE(2) is the (2,3) and (3,2) element, etc.
SDSD is DOUBLE PRECISION array, dimension (N) The diagonal of the real (symmetric tri) diagonal matrix S.
SESE is DOUBLE PRECISION array, dimension (N1) The offdiagonal of the (symmetric tri) diagonal matrix S. Not referenced if KBSND=0. If KBAND=1, then AE(1) is the (1,2) and (2,1) element, SE(2) is the (2,3) and (3,2) element, etc.
UU is COMPLEX*16 array, dimension (LDU, N) The unitary matrix in the decomposition.
LDULDU is INTEGER The leading dimension of U. LDU must be at least N.
WORKWORK is COMPLEX*16 array, dimension (N**2)
RWORKRWORK is DOUBLE PRECISION array, dimension (N)
RESULTRESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. RESULT(1) is always modified.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zstt22 (integer N, integer M, integer KBAND, double precision, dimension( * ) AD, double precision, dimension( * ) AE, double precision, dimension( * ) SD, double precision, dimension( * ) SE, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldwork, * ) WORK, integer LDWORK, double precision, dimension( * ) RWORK, double precision, dimension( 2 ) RESULT)
ZSTT22
Purpose:

ZSTT22 checks a set of M eigenvalues and eigenvectors, A U = U S where A is Hermitian tridiagonal, the columns of U are unitary, and S is diagonal (if KBAND=0) or Hermitian tridiagonal (if KBAND=1). Two tests are performed: RESULT(1) =  U* A U  S  / ( A m ulp ) RESULT(2) =  I  U*U  / ( m ulp )
Parameters:

N
N is INTEGER The size of the matrix. If it is zero, ZSTT22 does nothing. It must be at least zero.
MM is INTEGER The number of eigenpairs to check. If it is zero, ZSTT22 does nothing. It must be at least zero.
KBANDKBAND is INTEGER The bandwidth of the matrix S. It may only be zero or one. If zero, then S is diagonal, and SE is not referenced. If one, then S is Hermitian tridiagonal.
ADAD is DOUBLE PRECISION array, dimension (N) The diagonal of the original (unfactored) matrix A. A is assumed to be Hermitian tridiagonal.
AEAE is DOUBLE PRECISION array, dimension (N) The offdiagonal of the original (unfactored) matrix A. A is assumed to be Hermitian tridiagonal. AE(1) is ignored, AE(2) is the (1,2) and (2,1) element, etc.
SDSD is DOUBLE PRECISION array, dimension (N) The diagonal of the (Hermitian tri) diagonal matrix S.
SESE is DOUBLE PRECISION array, dimension (N) The offdiagonal of the (Hermitian tri) diagonal matrix S. Not referenced if KBSND=0. If KBAND=1, then AE(1) is ignored, SE(2) is the (1,2) and (2,1) element, etc.
UU is DOUBLE PRECISION array, dimension (LDU, N) The unitary matrix in the decomposition.
LDULDU is INTEGER The leading dimension of U. LDU must be at least N.
WORKWORK is COMPLEX*16 array, dimension (LDWORK, M+1)
LDWORKLDWORK is INTEGER The leading dimension of WORK. LDWORK must be at least max(1,M).
RWORKRWORK is DOUBLE PRECISION array, dimension (N)
RESULTRESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zunt01 (character ROWCOL, integer M, integer N, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision RESID)
ZUNT01
Purpose:

ZUNT01 checks that the matrix U is unitary by computing the ratio RESID = norm( I  U*U' ) / ( n * EPS ), if ROWCOL = 'R', or RESID = norm( I  U'*U ) / ( m * EPS ), if ROWCOL = 'C'. Alternatively, if there isn't sufficient workspace to form I  U*U' or I  U'*U, the ratio is computed as RESID = abs( I  U*U' ) / ( n * EPS ), if ROWCOL = 'R', or RESID = abs( I  U'*U ) / ( m * EPS ), if ROWCOL = 'C'. where EPS is the machine precision. ROWCOL is used only if m = n; if m > n, ROWCOL is assumed to be 'C', and if m < n, ROWCOL is assumed to be 'R'.
Parameters:

ROWCOL
ROWCOL is CHARACTER Specifies whether the rows or columns of U should be checked for orthogonality. Used only if M = N. = 'R': Check for orthogonal rows of U = 'C': Check for orthogonal columns of U
MM is INTEGER The number of rows of the matrix U.
NN is INTEGER The number of columns of the matrix U.
UU is COMPLEX*16 array, dimension (LDU,N) The unitary matrix U. U is checked for orthogonal columns if m > n or if m = n and ROWCOL = 'C'. U is checked for orthogonal rows if m < n or if m = n and ROWCOL = 'R'.
LDULDU is INTEGER The leading dimension of the array U. LDU >= max(1,M).
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The length of the array WORK. For best performance, LWORK should be at least N*N if ROWCOL = 'C' or M*M if ROWCOL = 'R', but the test will be done even if LWORK is 0.
RWORKRWORK is DOUBLE PRECISION array, dimension (min(M,N)) Used only if LWORK is large enough to use the Level 3 BLAS code.
RESIDRESID is DOUBLE PRECISION RESID = norm( I  U * U' ) / ( n * EPS ), if ROWCOL = 'R', or RESID = norm( I  U' * U ) / ( m * EPS ), if ROWCOL = 'C'.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
subroutine zunt03 (character*( * ) RC, integer MU, integer MV, integer N, integer K, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision RESULT, integer INFO)
ZUNT03
Purpose:

ZUNT03 compares two unitary matrices U and V to see if their corresponding rows or columns span the same spaces. The rows are checked if RC = 'R', and the columns are checked if RC = 'C'. RESULT is the maximum of  V*V'  I  / ( MV ulp ), if RC = 'R', or  V'*V  I  / ( MV ulp ), if RC = 'C', and the maximum over rows (or columns) 1 to K of  U(i)  S*V(i) / ( N ulp ) where abs(S) = 1 (chosen to minimize the expression), U(i) is the ith row (column) of U, and V(i) is the ith row (column) of V.
Parameters:

RC
RC is CHARACTER*1 If RC = 'R' the rows of U and V are to be compared. If RC = 'C' the columns of U and V are to be compared.
MUMU is INTEGER The number of rows of U if RC = 'R', and the number of columns if RC = 'C'. If MU = 0 ZUNT03 does nothing. MU must be at least zero.
MVMV is INTEGER The number of rows of V if RC = 'R', and the number of columns if RC = 'C'. If MV = 0 ZUNT03 does nothing. MV must be at least zero.
NN is INTEGER If RC = 'R', the number of columns in the matrices U and V, and if RC = 'C', the number of rows in U and V. If N = 0 ZUNT03 does nothing. N must be at least zero.
KK is INTEGER The number of rows or columns of U and V to compare. 0 <= K <= max(MU,MV).
UU is COMPLEX*16 array, dimension (LDU,N) The first matrix to compare. If RC = 'R', U is MU by N, and if RC = 'C', U is N by MU.
LDULDU is INTEGER The leading dimension of U. If RC = 'R', LDU >= max(1,MU), and if RC = 'C', LDU >= max(1,N).
VV is COMPLEX*16 array, dimension (LDV,N) The second matrix to compare. If RC = 'R', V is MV by N, and if RC = 'C', V is N by MV.
LDVLDV is INTEGER The leading dimension of V. If RC = 'R', LDV >= max(1,MV), and if RC = 'C', LDV >= max(1,N).
WORKWORK is COMPLEX*16 array, dimension (LWORK)
LWORKLWORK is INTEGER The length of the array WORK. For best performance, LWORK should be at least N*N if RC = 'C' or M*M if RC = 'R', but the tests will be done even if LWORK is 0.
RWORKRWORK is DOUBLE PRECISION array, dimension (max(MV,N))
RESULTRESULT is DOUBLE PRECISION The value computed by the test described above. RESULT is limited to 1/ulp to avoid overflow.
INFOINFO is INTEGER 0 indicates a successful exit k indicates the kth parameter had an illegal value
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2011
Author
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