Functions

subroutine slagge (M, N, KL, KU, D, A, LDA, ISEED, WORK, INFO)
SLAGGE
subroutine slagsy (N, K, D, A, LDA, ISEED, WORK, INFO)
SLAGSY
subroutine slahilb (N, NRHS, A, LDA, X, LDX, B, LDB, WORK, INFO)
SLAHILB
subroutine slakf2 (M, N, A, LDA, B, D, E, Z, LDZ)
SLAKF2
real function slaran (ISEED)
SLARAN
subroutine slarge (N, A, LDA, ISEED, WORK, INFO)
SLARGE
real function slarnd (IDIST, ISEED)
SLARND
subroutine slaror (SIDE, INIT, M, N, A, LDA, ISEED, X, INFO)
SLAROR
subroutine slarot (LROWS, LLEFT, LRIGHT, NL, C, S, A, LDA, XLEFT, XRIGHT)
SLAROT
subroutine slatm1 (MODE, COND, IRSIGN, IDIST, ISEED, D, N, INFO)
SLATM1
real function slatm2 (M, N, I, J, KL, KU, IDIST, ISEED, D, IGRADE, DL, DR, IPVTNG, IWORK, SPARSE)
SLATM2
real function slatm3 (M, N, I, J, ISUB, JSUB, KL, KU, IDIST, ISEED, D, IGRADE, DL, DR, IPVTNG, IWORK, SPARSE)
SLATM3
subroutine slatm5 (PRTYPE, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, R, LDR, L, LDL, ALPHA, QBLCKA, QBLCKB)
SLATM5
subroutine slatm6 (TYPE, N, A, LDA, B, X, LDX, Y, LDY, ALPHA, BETA, WX, WY, S, DIF)
SLATM6
subroutine slatm7 (MODE, COND, IRSIGN, IDIST, ISEED, D, N, RANK, INFO)
SLATM7
subroutine slatme (N, DIST, ISEED, D, MODE, COND, DMAX, EI, RSIGN, UPPER, SIM, DS, MODES, CONDS, KL, KU, ANORM, A, LDA, WORK, INFO)
SLATME
subroutine slatmr (M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX, RSIGN, GRADE, DL, MODEL, CONDL, DR, MODER, CONDR, PIVTNG, IPIVOT, KL, KU, SPARSE, ANORM, PACK, A, LDA, IWORK, INFO)
SLATMR
subroutine slatms (M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX, KL, KU, PACK, A, LDA, WORK, INFO)
SLATMS
subroutine slatmt (M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX, RANK, KL, KU, PACK, A, LDA, WORK, INFO)
SLATMT

Detailed Description

This is the group of real LAPACK TESTING MATGEN routines.

Function Documentation

subroutine slagge (integer M, integer N, integer KL, integer KU, real, dimension( * ) D, real, dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, real, dimension( * ) WORK, integer INFO)

SLAGGE

Purpose:

 SLAGGE generates a real general m by n matrix A, by pre- and post-
 multiplying a real diagonal matrix D with random orthogonal matrices:
 A = U*D*V. The lower and upper bandwidths may then be reduced to
 kl and ku by additional orthogonal transformations.

 

Parameters:

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= 0.

KL

          KL is INTEGER
          The number of nonzero subdiagonals within the band of A.
          0 <= KL <= M-1.

KU

          KU is INTEGER
          The number of nonzero superdiagonals within the band of A.
          0 <= KU <= N-1.

D

          D is REAL array, dimension (min(M,N))
          The diagonal elements of the diagonal matrix D.

A

          A is REAL array, dimension (LDA,N)
          The generated m by n matrix A.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= M.

ISEED

          ISEED is INTEGER array, dimension (4)
          On entry, the seed of the random number generator; the array
          elements must be between 0 and 4095, and ISEED(4) must be
          odd.
          On exit, the seed is updated.

WORK

          WORK is REAL array, dimension (M+N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2015

subroutine slagsy (integer N, integer K, real, dimension( * ) D, real, dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, real, dimension( * ) WORK, integer INFO)

SLAGSY

Purpose:

 SLAGSY generates a real symmetric matrix A, by pre- and post-
 multiplying a real diagonal matrix D with a random orthogonal matrix:
 A = U*D*U'. The semi-bandwidth may then be reduced to k by additional
 orthogonal transformations.

 

Parameters:

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

K

          K is INTEGER
          The number of nonzero subdiagonals within the band of A.
          0 <= K <= N-1.

D

          D is REAL array, dimension (N)
          The diagonal elements of the diagonal matrix D.

A

          A is REAL array, dimension (LDA,N)
          The generated n by n symmetric matrix A (the full matrix is
          stored).

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= N.

ISEED

          ISEED is INTEGER array, dimension (4)
          On entry, the seed of the random number generator; the array
          elements must be between 0 and 4095, and ISEED(4) must be
          odd.
          On exit, the seed is updated.

WORK

          WORK is REAL array, dimension (2*N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine slahilb (integer N, integer NRHS, real, dimension(lda, n) A, integer LDA, real, dimension(ldx, nrhs) X, integer LDX, real, dimension(ldb, nrhs) B, integer LDB, real, dimension(n) WORK, integer INFO)

SLAHILB

Purpose:

 SLAHILB generates an N by N scaled Hilbert matrix in A along with
 NRHS right-hand sides in B and solutions in X such that A*X=B.
 The Hilbert matrix is scaled by M = LCM(1, 2, ..., 2*N-1) so that all
 entries are integers.  The right-hand sides are the first NRHS 
 columns of M * the identity matrix, and the solutions are the 
 first NRHS columns of the inverse Hilbert matrix.
 The condition number of the Hilbert matrix grows exponentially with
 its size, roughly as O(e ** (3.5*N)).  Additionally, the inverse
 Hilbert matrices beyond a relatively small dimension cannot be
 generated exactly without extra precision.  Precision is exhausted
 when the largest entry in the inverse Hilbert matrix is greater than
 2 to the power of the number of bits in the fraction of the data type
 used plus one, which is 24 for single precision.  
 In single, the generated solution is exact for N <= 6 and has
 small componentwise error for 7 <= N <= 11.

 

Parameters:

N

          N is INTEGER
          The dimension of the matrix A.

NRHS

          NRHS is INTEGER
          The requested number of right-hand sides.

A

          A is REAL array, dimension (LDA, N)
          The generated scaled Hilbert matrix.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= N.

X

          X is REAL array, dimension (LDX, NRHS)
          The generated exact solutions.  Currently, the first NRHS
          columns of the inverse Hilbert matrix.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= N.

B

          B is REAL array, dimension (LDB, NRHS)
          The generated right-hand sides.  Currently, the first NRHS
          columns of LCM(1, 2, ..., 2*N-1) * the identity matrix.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= N.

WORK

          WORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0: successful exit
          = 1: N is too large; the data is still generated but may not
               be not exact.
          < 0: if INFO = -i, the i-th argument had an illegal value

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2015

subroutine slakf2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( lda, * ) B, real, dimension( lda, * ) D, real, dimension( lda, * ) E, real, dimension( ldz, * ) Z, integer LDZ)

SLAKF2

Purpose:

 Form the 2*M*N by 2*M*N matrix
        Z = [ kron(In, A)  -kron(B', Im) ]
            [ kron(In, D)  -kron(E', Im) ],
 where In is the identity matrix of size n and X' is the transpose
 of X. kron(X, Y) is the Kronecker product between the matrices X
 and Y.

 

Parameters:

M

          M is INTEGER
          Size of matrix, must be >= 1.

N

          N is INTEGER
          Size of matrix, must be >= 1.

A

          A is REAL, dimension ( LDA, M )
          The matrix A in the output matrix Z.

LDA

          LDA is INTEGER
          The leading dimension of A, B, D, and E. ( LDA >= M+N )

B

          B is REAL, dimension ( LDA, N )

D

          D is REAL, dimension ( LDA, M )

E

          E is REAL, dimension ( LDA, N )
          The matrices used in forming the output matrix Z.

Z

          Z is REAL, dimension ( LDZ, 2*M*N )
          The resultant Kronecker M*N*2 by M*N*2 matrix (see above.)

LDZ

          LDZ is INTEGER
          The leading dimension of Z. ( LDZ >= 2*M*N )

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

real function slaran (integer, dimension( 4 ) ISEED)

SLARAN

Purpose:

 SLARAN returns a random real number from a uniform (0,1)
 distribution.

 

Parameters:

ISEED

          ISEED is INTEGER array, dimension (4)
          On entry, the seed of the random number generator; the array
          elements must be between 0 and 4095, and ISEED(4) must be
          odd.
          On exit, the seed is updated.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Further Details:

  This routine uses a multiplicative congruential method with modulus
  2**48 and multiplier 33952834046453 (see G.S.Fishman,
  'Multiplicative congruential random number generators with modulus
  2**b: an exhaustive analysis for b = 32 and a partial analysis for
  b = 48', Math. Comp. 189, pp 331-344, 1990).
  48-bit integers are stored in 4 integer array elements with 12 bits
  per element. Hence the routine is portable across machines with
  integers of 32 bits or more.

 

subroutine slarge (integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, real, dimension( * ) WORK, integer INFO)

SLARGE

Purpose:

 SLARGE pre- and post-multiplies a real general n by n matrix A
 with a random orthogonal matrix: A = U*D*U'.

 

Parameters:

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is REAL array, dimension (LDA,N)
          On entry, the original n by n matrix A.
          On exit, A is overwritten by U*A*U' for some random
          orthogonal matrix U.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= N.

ISEED

          ISEED is INTEGER array, dimension (4)
          On entry, the seed of the random number generator; the array
          elements must be between 0 and 4095, and ISEED(4) must be
          odd.
          On exit, the seed is updated.

WORK

          WORK is REAL array, dimension (2*N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

real function slarnd (integer IDIST, integer, dimension( 4 ) ISEED)

SLARND

Purpose:

 SLARND returns a random real number from a uniform or normal
 distribution.

 

Parameters:

IDIST

          IDIST is INTEGER
          Specifies the distribution of the random numbers:
          = 1:  uniform (0,1)
          = 2:  uniform (-1,1)
          = 3:  normal (0,1)

ISEED

          ISEED is INTEGER array, dimension (4)
          On entry, the seed of the random number generator; the array
          elements must be between 0 and 4095, and ISEED(4) must be
          odd.
          On exit, the seed is updated.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Further Details:

  This routine calls the auxiliary routine SLARAN to generate a random
  real number from a uniform (0,1) distribution. The Box-Muller method
  is used to transform numbers from a uniform to a normal distribution.

 

subroutine slaror (character SIDE, character INIT, integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, real, dimension( * ) X, integer INFO)

SLAROR

Purpose:

 SLAROR pre- or post-multiplies an M by N matrix A by a random
 orthogonal matrix U, overwriting A.  A may optionally be initialized
 to the identity matrix before multiplying by U.  U is generated using
 the method of G.W. Stewart (SIAM J. Numer. Anal. 17, 1980, 403-409).

 

Parameters:

SIDE

          SIDE is CHARACTER*1
          Specifies whether A is multiplied on the left or right by U.
          = 'L':         Multiply A on the left (premultiply) by U
          = 'R':         Multiply A on the right (postmultiply) by U'
          = 'C' or 'T':  Multiply A on the left by U and the right
                          by U' (Here, U' means U-transpose.)

INIT

          INIT is CHARACTER*1
          Specifies whether or not A should be initialized to the
          identity matrix.
          = 'I':  Initialize A to (a section of) the identity matrix
                   before applying U.
          = 'N':  No initialization.  Apply U to the input matrix A.
          INIT = 'I' may be used to generate square or rectangular
          orthogonal matrices:
          For M = N and SIDE = 'L' or 'R', the rows will be orthogonal
          to each other, as will the columns.
          If M < N, SIDE = 'R' produces a dense matrix whose rows are
          orthogonal and whose columns are not, while SIDE = 'L'
          produces a matrix whose rows are orthogonal, and whose first
          M columns are orthogonal, and whose remaining columns are
          zero.
          If M > N, SIDE = 'L' produces a dense matrix whose columns
          are orthogonal and whose rows are not, while SIDE = 'R'
          produces a matrix whose columns are orthogonal, and whose
          first M rows are orthogonal, and whose remaining rows are
          zero.

M

          M is INTEGER
          The number of rows of A.

N

          N is INTEGER
          The number of columns of A.

A

          A is REAL array, dimension (LDA, N)
          On entry, the array A.
          On exit, overwritten by U A ( if SIDE = 'L' ),
           or by A U ( if SIDE = 'R' ),
           or by U A U' ( if SIDE = 'C' or 'T').

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

ISEED

          ISEED 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 SLAROR to continue the same random number
          sequence.

X

          X is REAL array, dimension (3*MAX( M, N ))
          Workspace of length
              2*M + N if SIDE = 'L',
              2*N + M if SIDE = 'R',
              3*N     if SIDE = 'C' or 'T'.

INFO

          INFO is INTEGER
          An error flag.  It is set to:
          = 0:  normal return
          < 0:  if INFO = -k, the k-th argument had an illegal value
          = 1:  if the random numbers generated by SLARND are bad.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine slarot (logical LROWS, logical LLEFT, logical LRIGHT, integer NL, real C, real S, real, dimension( * ) A, integer LDA, real XLEFT, real XRIGHT)

SLAROT

Purpose:

    SLAROT applies a (Givens) rotation to two adjacent rows or
    columns, where one element of the first and/or last column/row
    for use on matrices stored in some format other than GE, so
    that elements of the matrix may be used or modified for which
    no array element is provided.
    One example is a symmetric matrix in SB format (bandwidth=4), for
    which UPLO='L':  Two adjacent rows will have the format:
    row j:     C> C> C> C> C> .  .  .  .
    row j+1:      C> C> C> C> C> .  .  .  .
    '*' indicates elements for which storage is provided,
    '.' indicates elements for which no storage is provided, but
    are not necessarily zero; their values are determined by
    symmetry.  ' ' indicates elements which are necessarily zero,
     and have no storage provided.
    Those columns which have two '*'s can be handled by SROT.
    Those columns which have no '*'s can be ignored, since as long
    as the Givens rotations are carefully applied to preserve
    symmetry, their values are determined.
    Those columns which have one '*' have to be handled separately,
    by using separate variables "p" and "q":
    row j:     C> C> C> C> C> p  .  .  .
    row j+1:   q  C> C> C> C> C> .  .  .  .
    The element p would have to be set correctly, then that column
    is rotated, setting p to its new value.  The next call to
    SLAROT would rotate columns j and j+1, using p, and restore
    symmetry.  The element q would start out being zero, and be
    made non-zero by the rotation.  Later, rotations would presumably
    be chosen to zero q out.
    Typical Calling Sequences: rotating the i-th and (i+1)-st rows.
    ------- ------- ---------
      General dense matrix:
              CALL SLAROT(.TRUE.,.FALSE.,.FALSE., N, C,S,
                      A(i,1),LDA, DUMMY, DUMMY)
      General banded matrix in GB format:
              j = MAX(1, i-KL )
              NL = MIN( N, i+KU+1 ) + 1-j
              CALL SLAROT( .TRUE., i-KL.GE.1, i+KU.LT.N, NL, C,S,
                      A(KU+i+1-j,j),LDA-1, XLEFT, XRIGHT )
              [ note that i+1-j is just MIN(i,KL+1) ]
      Symmetric banded matrix in SY format, bandwidth K,
      lower triangle only:
              j = MAX(1, i-K )
              NL = MIN( K+1, i ) + 1
              CALL SLAROT( .TRUE., i-K.GE.1, .TRUE., NL, C,S,
                      A(i,j), LDA, XLEFT, XRIGHT )
      Same, but upper triangle only:
              NL = MIN( K+1, N-i ) + 1
              CALL SLAROT( .TRUE., .TRUE., i+K.LT.N, NL, C,S,
                      A(i,i), LDA, XLEFT, XRIGHT )
      Symmetric banded matrix in SB format, bandwidth K,
      lower triangle only:
              [ same as for SY, except:]
                  . . . .
                      A(i+1-j,j), LDA-1, XLEFT, XRIGHT )
              [ note that i+1-j is just MIN(i,K+1) ]
      Same, but upper triangle only:
                   . . .
                      A(K+1,i), LDA-1, XLEFT, XRIGHT )
      Rotating columns is just the transpose of rotating rows, except
      for GB and SB: (rotating columns i and i+1)
      GB:
              j = MAX(1, i-KU )
              NL = MIN( N, i+KL+1 ) + 1-j
              CALL SLAROT( .TRUE., i-KU.GE.1, i+KL.LT.N, NL, C,S,
                      A(KU+j+1-i,i),LDA-1, XTOP, XBOTTM )
              [note that KU+j+1-i is just MAX(1,KU+2-i)]
      SB: (upper triangle)
                   . . . . . .
                      A(K+j+1-i,i),LDA-1, XTOP, XBOTTM )
      SB: (lower triangle)
                   . . . . . .
                      A(1,i),LDA-1, XTOP, XBOTTM )

 

  LROWS  - LOGICAL
           If .TRUE., then SLAROT will rotate two rows.  If .FALSE.,
           then it will rotate two columns.
           Not modified.
  LLEFT  - LOGICAL
           If .TRUE., then XLEFT will be used instead of the
           corresponding element of A for the first element in the
           second row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.)
           If .FALSE., then the corresponding element of A will be
           used.
           Not modified.
  LRIGHT - LOGICAL
           If .TRUE., then XRIGHT will be used instead of the
           corresponding element of A for the last element in the
           first row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.) If
           .FALSE., then the corresponding element of A will be used.
           Not modified.
  NL     - INTEGER
           The length of the rows (if LROWS=.TRUE.) or columns (if
           LROWS=.FALSE.) to be rotated.  If XLEFT and/or XRIGHT are
           used, the columns/rows they are in should be included in
           NL, e.g., if LLEFT = LRIGHT = .TRUE., then NL must be at
           least 2.  The number of rows/columns to be rotated
           exclusive of those involving XLEFT and/or XRIGHT may
           not be negative, i.e., NL minus how many of LLEFT and
           LRIGHT are .TRUE. must be at least zero; if not, XERBLA
           will be called.
           Not modified.
  C, S   - REAL
           Specify the Givens rotation to be applied.  If LROWS is
           true, then the matrix ( c  s )
                                 (-s  c )  is applied from the left;
           if false, then the transpose thereof is applied from the
           right.  For a Givens rotation, C**2 + S**2 should be 1,
           but this is not checked.
           Not modified.
  A      - REAL array.
           The array containing the rows/columns to be rotated.  The
           first element of A should be the upper left element to
           be rotated.
           Read and modified.
  LDA    - INTEGER
           The "effective" leading dimension of A.  If A contains
           a matrix stored in GE or SY format, then this is just
           the leading dimension of A as dimensioned in the calling
           routine.  If A contains a matrix stored in band (GB or SB)
           format, then this should be *one less* than the leading
           dimension used in the calling routine.  Thus, if
           A were dimensioned A(LDA,*) in SLAROT, then A(1,j) would
           be the j-th element in the first of the two rows
           to be rotated, and A(2,j) would be the j-th in the second,
           regardless of how the array may be stored in the calling
           routine.  [A cannot, however, actually be dimensioned thus,
           since for band format, the row number may exceed LDA, which
           is not legal FORTRAN.]
           If LROWS=.TRUE., then LDA must be at least 1, otherwise
           it must be at least NL minus the number of .TRUE. values
           in XLEFT and XRIGHT.
           Not modified.
  XLEFT  - REAL
           If LLEFT is .TRUE., then XLEFT will be used and modified
           instead of A(2,1) (if LROWS=.TRUE.) or A(1,2)
           (if LROWS=.FALSE.).
           Read and modified.
  XRIGHT - REAL
           If LRIGHT is .TRUE., then XRIGHT will be used and modified
           instead of A(1,NL) (if LROWS=.TRUE.) or A(NL,1)
           (if LROWS=.FALSE.).
           Read and modified.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine slatm1 (integer MODE, real COND, integer IRSIGN, integer IDIST, integer, dimension( 4 ) ISEED, real, dimension( * ) D, integer N, integer INFO)

SLATM1

Purpose:

    SLATM1 computes the entries of D(1..N) as specified by
    MODE, COND and IRSIGN. IDIST and ISEED determine the generation
    of random numbers. SLATM1 is called by SLATMR to generate
    random test matrices for LAPACK programs.

 

Parameters:

MODE

          MODE is INTEGER
           On entry describes how D is to be computed:
           MODE = 0 means do not change D.
           MODE = 1 sets D(1)=1 and D(2:N)=1.0/COND
           MODE = 2 sets D(1:N-1)=1 and D(N)=1.0/COND
           MODE = 3 sets D(I)=COND**(-(I-1)/(N-1))
           MODE = 4 sets D(i)=1 - (i-1)/(N-1)*(1 - 1/COND)
           MODE = 5 sets D to random numbers in the range
                    ( 1/COND , 1 ) such that their logarithms
                    are uniformly distributed.
           MODE = 6 set D to random numbers from same distribution
                    as the rest of the matrix.
           MODE < 0 has the same meaning as ABS(MODE), except that
              the order of the elements of D is reversed.
           Thus if MODE is positive, D has entries ranging from
              1 to 1/COND, if negative, from 1/COND to 1,
           Not modified.

COND

          COND is REAL
           On entry, used as described under MODE above.
           If used, it must be >= 1. Not modified.

IRSIGN

          IRSIGN is INTEGER
           On entry, if MODE neither -6, 0 nor 6, determines sign of
           entries of D
           0 => leave entries of D unchanged
           1 => multiply each entry of D by 1 or -1 with probability .5

IDIST

          IDIST is INTEGER
           On entry, IDIST specifies the type of distribution to be
           used to generate a random matrix .
           1 => UNIFORM( 0, 1 )
           2 => UNIFORM( -1, 1 )
           3 => NORMAL( 0, 1 )
           Not modified.

ISEED

          ISEED is INTEGER array, dimension ( 4 )
           On entry ISEED specifies the seed of the random number
           generator. 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 SLATM1
           to continue the same random number sequence.
           Changed on exit.

D

          D is REAL array, dimension ( N )
           Array to be computed according to MODE, COND and IRSIGN.
           May be changed on exit if MODE is nonzero.

N

          N is INTEGER
           Number of entries of D. Not modified.

INFO

          INFO is INTEGER
            0  => normal termination
           -1  => if MODE not in range -6 to 6
           -2  => if MODE neither -6, 0 nor 6, and
                  IRSIGN neither 0 nor 1
           -3  => if MODE neither -6, 0 nor 6 and COND less than 1
           -4  => if MODE equals 6 or -6 and IDIST not in range 1 to 3
           -7  => if N negative

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2015

real function slatm2 (integer M, integer N, integer I, integer J, integer KL, integer KU, integer IDIST, integer, dimension( 4 ) ISEED, real, dimension( * ) D, integer IGRADE, real, dimension( * ) DL, real, dimension( * ) DR, integer IPVTNG, integer, dimension( * ) IWORK, real SPARSE)

SLATM2

Purpose:

    SLATM2 returns the (I,J) entry of a random matrix of dimension
    (M, N) described by the other parameters. It is called by the
    SLATMR routine in order to build random test matrices. No error
    checking on parameters is done, because this routine is called in
    a tight loop by SLATMR which has already checked the parameters.
    Use of SLATM2 differs from SLATM3 in the order in which the random
    number generator is called to fill in random matrix entries.
    With SLATM2, the generator is called to fill in the pivoted matrix
    columnwise. With SLATM3, the generator is called to fill in the
    matrix columnwise, after which it is pivoted. Thus, SLATM3 can
    be used to construct random matrices which differ only in their
    order of rows and/or columns. SLATM2 is used to construct band
    matrices while avoiding calling the random number generator for
    entries outside the band (and therefore generating random numbers
    The matrix whose (I,J) entry is returned is constructed as
    follows (this routine only computes one entry):
      If I is outside (1..M) or J is outside (1..N), return zero
         (this is convenient for generating matrices in band format).
      Generate a matrix A with random entries of distribution IDIST.
      Set the diagonal to D.
      Grade the matrix, if desired, from the left (by DL) and/or
         from the right (by DR or DL) as specified by IGRADE.
      Permute, if desired, the rows and/or columns as specified by
         IPVTNG and IWORK.
      Band the matrix to have lower bandwidth KL and upper
         bandwidth KU.
      Set random entries to zero as specified by SPARSE.

 

Parameters:

M

          M is INTEGER
           Number of rows of matrix. Not modified.

N

          N is INTEGER
           Number of columns of matrix. Not modified.

I

          I is INTEGER
           Row of entry to be returned. Not modified.

J

          J is INTEGER
           Column of entry to be returned. Not modified.

KL

          KL is INTEGER
           Lower bandwidth. Not modified.

KU

          KU is INTEGER
           Upper bandwidth. Not modified.

IDIST

          IDIST is INTEGER
           On entry, IDIST specifies the type of distribution to be
           used to generate a random matrix .
           1 => UNIFORM( 0, 1 )
           2 => UNIFORM( -1, 1 )
           3 => NORMAL( 0, 1 )
           Not modified.

ISEED

          ISEED is INTEGER array of dimension ( 4 )
           Seed for random number generator.
           Changed on exit.

D

          D is REAL array of dimension ( MIN( I , J ) )
           Diagonal entries of matrix. Not modified.

IGRADE

          IGRADE is INTEGER
           Specifies grading of matrix as follows:
           0  => no grading
           1  => matrix premultiplied by diag( DL )
           2  => matrix postmultiplied by diag( DR )
           3  => matrix premultiplied by diag( DL ) and
                         postmultiplied by diag( DR )
           4  => matrix premultiplied by diag( DL ) and
                         postmultiplied by inv( diag( DL ) )
           5  => matrix premultiplied by diag( DL ) and
                         postmultiplied by diag( DL )
           Not modified.

DL

          DL is REAL array ( I or J, as appropriate )
           Left scale factors for grading matrix.  Not modified.

DR

          DR is REAL array ( I or J, as appropriate )
           Right scale factors for grading matrix.  Not modified.

IPVTNG

          IPVTNG is INTEGER
           On entry specifies pivoting permutations as follows:
           0 => none.
           1 => row pivoting.
           2 => column pivoting.
           3 => full pivoting, i.e., on both sides.
           Not modified.

IWORK

          IWORK is INTEGER array ( I or J, as appropriate )
           This array specifies the permutation used. The
           row (or column) in position K was originally in
           position IWORK( K ).
           This differs from IWORK for SLATM3. Not modified.

SPARSE

          SPARSE is REAL between 0. and 1.
           On entry specifies the sparsity of the matrix
           if sparse matix is to be generated.
           SPARSE should lie between 0 and 1.
           A uniform ( 0, 1 ) random number x is generated and
           compared to SPARSE; if x is larger the matrix entry
           is unchanged and if x is smaller the entry is set
           to zero. Thus on the average a fraction SPARSE of the
           entries will be set to zero.
           Not modified.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

real function slatm3 (integer M, integer N, integer I, integer J, integer ISUB, integer JSUB, integer KL, integer KU, integer IDIST, integer, dimension( 4 ) ISEED, real, dimension( * ) D, integer IGRADE, real, dimension( * ) DL, real, dimension( * ) DR, integer IPVTNG, integer, dimension( * ) IWORK, real SPARSE)

SLATM3

Purpose:

    SLATM3 returns the (ISUB,JSUB) entry of a random matrix of
    dimension (M, N) described by the other parameters. (ISUB,JSUB)
    is the final position of the (I,J) entry after pivoting
    according to IPVTNG and IWORK. SLATM3 is called by the
    SLATMR routine in order to build random test matrices. No error
    checking on parameters is done, because this routine is called in
    a tight loop by SLATMR which has already checked the parameters.
    Use of SLATM3 differs from SLATM2 in the order in which the random
    number generator is called to fill in random matrix entries.
    With SLATM2, the generator is called to fill in the pivoted matrix
    columnwise. With SLATM3, the generator is called to fill in the
    matrix columnwise, after which it is pivoted. Thus, SLATM3 can
    be used to construct random matrices which differ only in their
    order of rows and/or columns. SLATM2 is used to construct band
    matrices while avoiding calling the random number generator for
    entries outside the band (and therefore generating random numbers
    in different orders for different pivot orders).
    The matrix whose (ISUB,JSUB) entry is returned is constructed as
    follows (this routine only computes one entry):
      If ISUB is outside (1..M) or JSUB is outside (1..N), return zero
         (this is convenient for generating matrices in band format).
      Generate a matrix A with random entries of distribution IDIST.
      Set the diagonal to D.
      Grade the matrix, if desired, from the left (by DL) and/or
         from the right (by DR or DL) as specified by IGRADE.
      Permute, if desired, the rows and/or columns as specified by
         IPVTNG and IWORK.
      Band the matrix to have lower bandwidth KL and upper
         bandwidth KU.
      Set random entries to zero as specified by SPARSE.

 

Parameters:

M

          M is INTEGER
           Number of rows of matrix. Not modified.

N

          N is INTEGER
           Number of columns of matrix. Not modified.

I

          I is INTEGER
           Row of unpivoted entry to be returned. Not modified.

J

          J is INTEGER
           Column of unpivoted entry to be returned. Not modified.

ISUB

          ISUB is INTEGER
           Row of pivoted entry to be returned. Changed on exit.

JSUB

          JSUB is INTEGER
           Column of pivoted entry to be returned. Changed on exit.

KL

          KL is INTEGER
           Lower bandwidth. Not modified.

KU

          KU is INTEGER
           Upper bandwidth. Not modified.

IDIST

          IDIST is INTEGER
           On entry, IDIST specifies the type of distribution to be
           used to generate a random matrix .
           1 => UNIFORM( 0, 1 )
           2 => UNIFORM( -1, 1 )
           3 => NORMAL( 0, 1 )
           Not modified.

ISEED

          ISEED is INTEGER array of dimension ( 4 )
           Seed for random number generator.
           Changed on exit.

D

          D is REAL array of dimension ( MIN( I , J ) )
           Diagonal entries of matrix. Not modified.

IGRADE

          IGRADE is INTEGER
           Specifies grading of matrix as follows:
           0  => no grading
           1  => matrix premultiplied by diag( DL )
           2  => matrix postmultiplied by diag( DR )
           3  => matrix premultiplied by diag( DL ) and
                         postmultiplied by diag( DR )
           4  => matrix premultiplied by diag( DL ) and
                         postmultiplied by inv( diag( DL ) )
           5  => matrix premultiplied by diag( DL ) and
                         postmultiplied by diag( DL )
           Not modified.

DL

          DL is REAL array ( I or J, as appropriate )
           Left scale factors for grading matrix.  Not modified.

DR

          DR is REAL array ( I or J, as appropriate )
           Right scale factors for grading matrix.  Not modified.

IPVTNG

          IPVTNG is INTEGER
           On entry specifies pivoting permutations as follows:
           0 => none.
           1 => row pivoting.
           2 => column pivoting.
           3 => full pivoting, i.e., on both sides.
           Not modified.

IWORK

          IWORK is INTEGER array ( I or J, as appropriate )
           This array specifies the permutation used. The
           row (or column) originally in position K is in
           position IWORK( K ) after pivoting.
           This differs from IWORK for SLATM2. Not modified.

SPARSE

          SPARSE is REAL between 0. and 1.
           On entry specifies the sparsity of the matrix
           if sparse matix is to be generated.
           SPARSE should lie between 0 and 1.
           A uniform ( 0, 1 ) random number x is generated and
           compared to SPARSE; if x is larger the matrix entry
           is unchanged and if x is smaller the entry is set
           to zero. Thus on the average a fraction SPARSE of the
           entries will be set to zero.
           Not modified.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine slatm5 (integer PRTYPE, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldc, * ) C, integer LDC, real, dimension( ldd, * ) D, integer LDD, real, dimension( lde, * ) E, integer LDE, real, dimension( ldf, * ) F, integer LDF, real, dimension( ldr, * ) R, integer LDR, real, dimension( ldl, * ) L, integer LDL, real ALPHA, integer QBLCKA, integer QBLCKB)

SLATM5

Purpose:

 SLATM5 generates matrices involved in the Generalized Sylvester
 equation:
     A * R - L * B = C
     D * R - L * E = F
 They also satisfy (the diagonalization condition)
  [ I -L ] ( [ A  -C ], [ D -F ] ) [ I  R ] = ( [ A    ], [ D    ] )
  [    I ] ( [     B ]  [    E ] ) [    I ]   ( [    B ]  [    E ] )

 

Parameters:

PRTYPE

          PRTYPE is INTEGER
          "Points" to a certain type of the matrices to generate
          (see further details).

M

          M is INTEGER
          Specifies the order of A and D and the number of rows in
          C, F,  R and L.

N

          N is INTEGER
          Specifies the order of B and E and the number of columns in
          C, F, R and L.

A

          A is REAL array, dimension (LDA, M).
          On exit A M-by-M is initialized according to PRTYPE.

LDA

          LDA is INTEGER
          The leading dimension of A.

B

          B is REAL array, dimension (LDB, N).
          On exit B N-by-N is initialized according to PRTYPE.

LDB

          LDB is INTEGER
          The leading dimension of B.

C

          C is REAL array, dimension (LDC, N).
          On exit C M-by-N is initialized according to PRTYPE.

LDC

          LDC is INTEGER
          The leading dimension of C.

D

          D is REAL array, dimension (LDD, M).
          On exit D M-by-M is initialized according to PRTYPE.

LDD

          LDD is INTEGER
          The leading dimension of D.

E

          E is REAL array, dimension (LDE, N).
          On exit E N-by-N is initialized according to PRTYPE.

LDE

          LDE is INTEGER
          The leading dimension of E.

F

          F is REAL array, dimension (LDF, N).
          On exit F M-by-N is initialized according to PRTYPE.

LDF

          LDF is INTEGER
          The leading dimension of F.

R

          R is REAL array, dimension (LDR, N).
          On exit R M-by-N is initialized according to PRTYPE.

LDR

          LDR is INTEGER
          The leading dimension of R.

L

          L is REAL array, dimension (LDL, N).
          On exit L M-by-N is initialized according to PRTYPE.

LDL

          LDL is INTEGER
          The leading dimension of L.

ALPHA

          ALPHA is REAL
          Parameter used in generating PRTYPE = 1 and 5 matrices.

QBLCKA

          QBLCKA is INTEGER
          When PRTYPE = 3, specifies the distance between 2-by-2
          blocks on the diagonal in A. Otherwise, QBLCKA is not
          referenced. QBLCKA > 1.

QBLCKB

          QBLCKB is INTEGER
          When PRTYPE = 3, specifies the distance between 2-by-2
          blocks on the diagonal in B. Otherwise, QBLCKB is not
          referenced. QBLCKB > 1.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Further Details:

  PRTYPE = 1: A and B are Jordan blocks, D and E are identity matrices
             A : if (i == j) then A(i, j) = 1.0
                 if (j == i + 1) then A(i, j) = -1.0
                 else A(i, j) = 0.0,            i, j = 1...M
             B : if (i == j) then B(i, j) = 1.0 - ALPHA
                 if (j == i + 1) then B(i, j) = 1.0
                 else B(i, j) = 0.0,            i, j = 1...N
             D : if (i == j) then D(i, j) = 1.0
                 else D(i, j) = 0.0,            i, j = 1...M
             E : if (i == j) then E(i, j) = 1.0
                 else E(i, j) = 0.0,            i, j = 1...N
             L =  R are chosen from [-10...10],
                  which specifies the right hand sides (C, F).
  PRTYPE = 2 or 3: Triangular and/or quasi- triangular.
             A : if (i <= j) then A(i, j) = [-1...1]
                 else A(i, j) = 0.0,             i, j = 1...M
                 if (PRTYPE = 3) then
                    A(k + 1, k + 1) = A(k, k)
                    A(k + 1, k) = [-1...1]
                    sign(A(k, k + 1) = -(sin(A(k + 1, k))
                        k = 1, M - 1, QBLCKA
             B : if (i <= j) then B(i, j) = [-1...1]
                 else B(i, j) = 0.0,            i, j = 1...N
                 if (PRTYPE = 3) then
                    B(k + 1, k + 1) = B(k, k)
                    B(k + 1, k) = [-1...1]
                    sign(B(k, k + 1) = -(sign(B(k + 1, k))
                        k = 1, N - 1, QBLCKB
             D : if (i <= j) then D(i, j) = [-1...1].
                 else D(i, j) = 0.0,            i, j = 1...M
             E : if (i <= j) then D(i, j) = [-1...1]
                 else E(i, j) = 0.0,            i, j = 1...N
                 L, R are chosen from [-10...10],
                 which specifies the right hand sides (C, F).
  PRTYPE = 4 Full
             A(i, j) = [-10...10]
             D(i, j) = [-1...1]    i,j = 1...M
             B(i, j) = [-10...10]
             E(i, j) = [-1...1]    i,j = 1...N
             R(i, j) = [-10...10]
             L(i, j) = [-1...1]    i = 1..M ,j = 1...N
             L, R specifies the right hand sides (C, F).
  PRTYPE = 5 special case common and/or close eigs.

 

subroutine slatm6 (integer TYPE, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( lda, * ) B, real, dimension( ldx, * ) X, integer LDX, real, dimension( ldy, * ) Y, integer LDY, real ALPHA, real BETA, real WX, real WY, real, dimension( * ) S, real, dimension( * ) DIF)

SLATM6

Purpose:

 SLATM6 generates test matrices for the generalized eigenvalue
 problem, their corresponding right and left eigenvector matrices,
 and also reciprocal condition numbers for all eigenvalues and
 the reciprocal condition numbers of eigenvectors corresponding to
 the 1th and 5th eigenvalues.
 Test Matrices
 =============
 Two kinds of test matrix pairs
       (A, B) = inverse(YH) * (Da, Db) * inverse(X)
 are used in the tests:
 Type 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
 Type 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  -1-b  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.

 

Parameters:

TYPE

          TYPE is INTEGER
          Specifies the problem type (see further details).

N

          N is INTEGER
          Size of the matrices A and B.

A

          A is REAL array, dimension (LDA, N).
          On exit A N-by-N is initialized according to TYPE.

LDA

          LDA is INTEGER
          The leading dimension of A and of B.

B

          B is REAL array, dimension (LDA, N).
          On exit B N-by-N is initialized according to TYPE.

X

          X is REAL array, dimension (LDX, N).
          On exit X is the N-by-N matrix of right eigenvectors.

LDX

          LDX is INTEGER
          The leading dimension of X.

Y

          Y is REAL array, dimension (LDY, N).
          On exit Y is the N-by-N matrix of left eigenvectors.

LDY

          LDY is INTEGER
          The leading dimension of Y.

ALPHA

          ALPHA is REAL

BETA

          BETA is REAL
          Weighting constants for matrix A.

WX

          WX is REAL
          Constant for right eigenvector matrix.

WY

          WY is REAL
          Constant for left eigenvector matrix.

S

          S is REAL array, dimension (N)
          S(i) is the reciprocal condition number for eigenvalue i.

DIF

          DIF is REAL array, dimension (N)
          DIF(i) is the reciprocal condition number for eigenvector i.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine slatm7 (integer MODE, real COND, integer IRSIGN, integer IDIST, integer, dimension( 4 ) ISEED, real, dimension( * ) D, integer N, integer RANK, integer INFO)

SLATM7

Purpose:

    SLATM7 computes the entries of D as specified by MODE
    COND and IRSIGN. IDIST and ISEED determine the generation
    of random numbers. SLATM7 is called by SLATMT to generate
    random test matrices.

 

  MODE   - INTEGER
           On entry describes how D is to be computed:
           MODE = 0 means do not change D.
           MODE = 1 sets D(1)=1 and D(2:RANK)=1.0/COND
           MODE = 2 sets D(1:RANK-1)=1 and D(RANK)=1.0/COND
           MODE = 3 sets D(I)=COND**(-(I-1)/(RANK-1)) I=1:RANK
           MODE = 4 sets D(i)=1 - (i-1)/(N-1)*(1 - 1/COND)
           MODE = 5 sets D to random numbers in the range
                    ( 1/COND , 1 ) such that their logarithms
                    are uniformly distributed.
           MODE = 6 set D to random numbers from same distribution
                    as the rest of the matrix.
           MODE < 0 has the same meaning as ABS(MODE), except that
              the order of the elements of D is reversed.
           Thus if MODE is positive, D has entries ranging from
              1 to 1/COND, if negative, from 1/COND to 1,
           Not modified.
  COND   - REAL
           On entry, used as described under MODE above.
           If used, it must be >= 1. Not modified.
  IRSIGN - INTEGER
           On entry, if MODE neither -6, 0 nor 6, determines sign of
           entries of D
           0 => leave entries of D unchanged
           1 => multiply each entry of D by 1 or -1 with probability .5
  IDIST  - CHARACTER*1
           On entry, IDIST specifies the type of distribution to be
           used to generate a random matrix .
           1 => UNIFORM( 0, 1 )
           2 => UNIFORM( -1, 1 )
           3 => NORMAL( 0, 1 )
           Not modified.
  ISEED  - INTEGER array, dimension ( 4 )
           On entry ISEED specifies the seed of the random number
           generator. 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 SLATM7
           to continue the same random number sequence.
           Changed on exit.
  D      - REAL array, dimension ( MIN( M , N ) )
           Array to be computed according to MODE, COND and IRSIGN.
           May be changed on exit if MODE is nonzero.
  N      - INTEGER
           Number of entries of D. Not modified.
  RANK   - INTEGER
           The rank of matrix to be generated for modes 1,2,3 only.
           D( RANK+1:N ) = 0.
           Not modified.
  INFO   - INTEGER
            0  => normal termination
           -1  => if MODE not in range -6 to 6
           -2  => if MODE neither -6, 0 nor 6, and
                  IRSIGN neither 0 nor 1
           -3  => if MODE neither -6, 0 nor 6 and COND less than 1
           -4  => if MODE equals 6 or -6 and IDIST not in range 1 to 3
           -7  => if N negative

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

subroutine slatme (integer N, character DIST, integer, dimension( 4 ) ISEED, real, dimension( * ) D, integer MODE, real COND, real DMAX, character, dimension( * ) EI, character RSIGN, character UPPER, character SIM, real, dimension( * ) DS, integer MODES, real CONDS, integer KL, integer KU, real ANORM, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK, integer INFO)

SLATME

Purpose:

    SLATME generates random non-symmetric square matrices with
    specified eigenvalues for testing LAPACK programs.
    SLATME operates by applying the following sequence of
    operations:
    1. Set the diagonal to D, where D may be input or
         computed according to MODE, COND, DMAX, and RSIGN
         as described below.
    2. If complex conjugate pairs are desired (MODE=0 and EI(1)='R',
         or MODE=5), certain pairs of adjacent elements of D are
         interpreted as the real and complex parts of a complex
         conjugate pair; A thus becomes block diagonal, with 1x1
         and 2x2 blocks.
    3. If UPPER='T', the upper triangle of A is set to random values
         out of distribution DIST.
    4. If SIM='T', A is multiplied on the left by a random matrix
         X, whose singular values are specified by DS, MODES, and
         CONDS, and on the right by X inverse.
    5. If KL < N-1, the lower bandwidth is reduced to KL using
         Householder transformations.  If KU < N-1, the upper
         bandwidth is reduced to KU.
    6. If ANORM is not negative, the matrix is scaled to have
         maximum-element-norm ANORM.
    (Note: since the matrix cannot be reduced beyond Hessenberg form,
     no packing options are available.)

 

Parameters:

N

          N is INTEGER
           The number of columns (or rows) of A. Not modified.

DIST

          DIST is CHARACTER*1
           On entry, DIST specifies the type of distribution to be used
           to generate the random eigen-/singular values, and for the
           upper triangle (see UPPER).
           'U' => UNIFORM( 0, 1 )  ( 'U' for uniform )
           'S' => UNIFORM( -1, 1 ) ( 'S' for symmetric )
           'N' => NORMAL( 0, 1 )   ( 'N' for normal )
           Not modified.

ISEED

          ISEED is INTEGER array, dimension ( 4 )
           On entry ISEED specifies the seed of the random number
           generator. They should lie between 0 and 4095 inclusive,
           and ISEED(4) should 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 SLATME
           to continue the same random number sequence.
           Changed on exit.

D

          D is REAL array, dimension ( N )
           This array is used to specify the eigenvalues of A.  If
           MODE=0, then D is assumed to contain the eigenvalues (but
           see the description of EI), otherwise they will be
           computed according to MODE, COND, DMAX, and RSIGN and
           placed in D.
           Modified if MODE is nonzero.

MODE

          MODE is INTEGER
           On entry this describes how the eigenvalues are to
           be specified:
           MODE = 0 means use D (with EI) as input
           MODE = 1 sets D(1)=1 and D(2:N)=1.0/COND
           MODE = 2 sets D(1:N-1)=1 and D(N)=1.0/COND
           MODE = 3 sets D(I)=COND**(-(I-1)/(N-1))
           MODE = 4 sets D(i)=1 - (i-1)/(N-1)*(1 - 1/COND)
           MODE = 5 sets D to random numbers in the range
                    ( 1/COND , 1 ) such that their logarithms
                    are uniformly distributed.  Each odd-even pair
                    of elements will be either used as two real
                    eigenvalues or as the real and imaginary part
                    of a complex conjugate pair of eigenvalues;
                    the choice of which is done is random, with
                    50-50 probability, for each pair.
           MODE = 6 set D to random numbers from same distribution
                    as the rest of the matrix.
           MODE < 0 has the same meaning as ABS(MODE), except that
              the order of the elements of D is reversed.
           Thus if MODE is between 1 and 4, D has entries ranging
              from 1 to 1/COND, if between -1 and -4, D has entries
              ranging from 1/COND to 1,
           Not modified.

COND

          COND is REAL
           On entry, this is used as described under MODE above.
           If used, it must be >= 1. Not modified.

DMAX

          DMAX is REAL
           If MODE is neither -6, 0 nor 6, the contents of D, as
           computed according to MODE and COND, will be scaled by
           DMAX / max(abs(D(i))).  Note that DMAX need not be
           positive: if DMAX is negative (or zero), D will be
           scaled by a negative number (or zero).
           Not modified.

EI

          EI is CHARACTER*1 array, dimension ( N )
           If MODE is 0, and EI(1) is not ' ' (space character),
           this array specifies which elements of D (on input) are
           real eigenvalues and which are the real and imaginary parts
           of a complex conjugate pair of eigenvalues.  The elements
           of EI may then only have the values 'R' and 'I'.  If
           EI(j)='R' and EI(j+1)='I', then the j-th eigenvalue is
           CMPLX( D(j) , D(j+1) ), and the (j+1)-th is the complex
           conjugate thereof.  If EI(j)=EI(j+1)='R', then the j-th
           eigenvalue is D(j) (i.e., real).  EI(1) may not be 'I',
           nor may two adjacent elements of EI both have the value 'I'.
           If MODE is not 0, then EI is ignored.  If MODE is 0 and
           EI(1)=' ', then the eigenvalues will all be real.
           Not modified.

RSIGN

          RSIGN is CHARACTER*1
           If MODE is not 0, 6, or -6, and RSIGN='T', then the
           elements of D, as computed according to MODE and COND, will
           be multiplied by a random sign (+1 or -1).  If RSIGN='F',
           they will not be.  RSIGN may only have the values 'T' or
           'F'.
           Not modified.

UPPER

          UPPER is CHARACTER*1
           If UPPER='T', then the elements of A above the diagonal
           (and above the 2x2 diagonal blocks, if A has complex
           eigenvalues) will be set to random numbers out of DIST.
           If UPPER='F', they will not.  UPPER may only have the
           values 'T' or 'F'.
           Not modified.

SIM

          SIM is CHARACTER*1
           If SIM='T', then A will be operated on by a "similarity
           transform", i.e., multiplied on the left by a matrix X and
           on the right by X inverse.  X = U S V, where U and V are
           random unitary matrices and S is a (diagonal) matrix of
           singular values specified by DS, MODES, and CONDS.  If
           SIM='F', then A will not be transformed.
           Not modified.

DS

          DS is REAL array, dimension ( N )
           This array is used to specify the singular values of X,
           in the same way that D specifies the eigenvalues of A.
           If MODE=0, the DS contains the singular values, which
           may not be zero.
           Modified if MODE is nonzero.

MODES

          MODES is INTEGER

CONDS

          CONDS is REAL
           Same as MODE and COND, but for specifying the diagonal
           of S.  MODES=-6 and +6 are not allowed (since they would
           result in randomly ill-conditioned eigenvalues.)

KL

          KL is INTEGER
           This specifies the lower bandwidth of the  matrix.  KL=1
           specifies upper Hessenberg form.  If KL is at least N-1,
           then A will have full lower bandwidth.  KL must be at
           least 1.
           Not modified.

KU

          KU is INTEGER
           This specifies the upper bandwidth of the  matrix.  KU=1
           specifies lower Hessenberg form.  If KU is at least N-1,
           then A will have full upper bandwidth; if KU and KL
           are both at least N-1, then A will be dense.  Only one of
           KU and KL may be less than N-1.  KU must be at least 1.
           Not modified.

ANORM

          ANORM is REAL
           If ANORM is not negative, then A will be scaled by a non-
           negative real number to make the maximum-element-norm of A
           to be ANORM.
           Not modified.

A

          A is REAL array, dimension ( LDA, N )
           On exit A is the desired test matrix.
           Modified.

LDA

          LDA is INTEGER
           LDA specifies the first dimension of A as declared in the
           calling program.  LDA must be at least N.
           Not modified.

WORK

          WORK is REAL array, dimension ( 3*N )
           Workspace.
           Modified.

INFO

          INFO is INTEGER
           Error code.  On exit, INFO will be set to one of the
           following values:
             0 => normal return
            -1 => N negative
            -2 => DIST illegal string
            -5 => MODE not in range -6 to 6
            -6 => COND less than 1.0, and MODE neither -6, 0 nor 6
            -8 => EI(1) is not ' ' or 'R', EI(j) is not 'R' or 'I', or
                  two adjacent elements of EI are 'I'.
            -9 => RSIGN is not 'T' or 'F'
           -10 => UPPER is not 'T' or 'F'
           -11 => SIM   is not 'T' or 'F'
           -12 => MODES=0 and DS has a zero singular value.
           -13 => MODES is not in the range -5 to 5.
           -14 => MODES is nonzero and CONDS is less than 1.
           -15 => KL is less than 1.
           -16 => KU is less than 1, or KL and KU are both less than
                  N-1.
           -19 => LDA is less than N.
            1  => Error return from SLATM1 (computing D)
            2  => Cannot scale to DMAX (max. eigenvalue is 0)
            3  => Error return from SLATM1 (computing DS)
            4  => Error return from SLARGE
            5  => Zero singular value from SLATM1.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine slatmr (integer M, integer N, character DIST, integer, dimension( 4 ) ISEED, character SYM, real, dimension( * ) D, integer MODE, real COND, real DMAX, character RSIGN, character GRADE, real, dimension( * ) DL, integer MODEL, real CONDL, real, dimension( * ) DR, integer MODER, real CONDR, character PIVTNG, integer, dimension( * ) IPIVOT, integer KL, integer KU, real SPARSE, real ANORM, character PACK, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IWORK, integer INFO)

SLATMR

Purpose:

    SLATMR generates random matrices of various types for testing
    LAPACK programs.
    SLATMR operates by applying the following sequence of
    operations:
      Generate a matrix A with random entries of distribution DIST
         which is symmetric if SYM='S', and nonsymmetric
         if SYM='N'.
      Set the diagonal to D, where D may be input or
         computed according to MODE, COND, DMAX and RSIGN
         as described below.
      Grade the matrix, if desired, from the left and/or right
         as specified by GRADE. The inputs DL, MODEL, CONDL, DR,
         MODER and CONDR also determine the grading as described
         below.
      Permute, if desired, the rows and/or columns as specified by
         PIVTNG and IPIVOT.
      Set random entries to zero, if desired, to get a random sparse
         matrix as specified by SPARSE.
      Make A a band matrix, if desired, by zeroing out the matrix
         outside a band of lower bandwidth KL and upper bandwidth KU.
      Scale A, if desired, to have maximum entry ANORM.
      Pack the matrix if desired. Options specified by PACK are:
         no packing
         zero out upper half (if symmetric)
         zero out lower half (if symmetric)
         store the upper half columnwise (if symmetric or
             square upper triangular)
         store the lower half columnwise (if symmetric or
             square lower triangular)
             same as upper half rowwise if symmetric
         store the lower triangle in banded format (if symmetric)
         store the upper triangle in banded format (if symmetric)
         store the entire matrix in banded format
    Note: If two calls to SLATMR differ only in the PACK parameter,
          they will generate mathematically equivalent matrices.
          If two calls to SLATMR both have full bandwidth (KL = M-1
          and KU = N-1), and differ only in the PIVTNG and PACK
          parameters, then the matrices generated will differ only
          in the order of the rows and/or columns, and otherwise
          contain the same data. This consistency cannot be and
          is not maintained with less than full bandwidth.

 

Parameters:

M

          M is INTEGER
           Number of rows of A. Not modified.

N

          N is INTEGER
           Number of columns of A. Not modified.

DIST

          DIST is CHARACTER*1
           On entry, DIST specifies the type of distribution to be used
           to generate a random matrix .
           'U' => UNIFORM( 0, 1 )  ( 'U' for uniform )
           'S' => UNIFORM( -1, 1 ) ( 'S' for symmetric )
           'N' => NORMAL( 0, 1 )   ( 'N' for normal )
           Not modified.

ISEED

          ISEED is INTEGER array, dimension (4)
           On entry ISEED specifies the seed of the random number
           generator. They should lie between 0 and 4095 inclusive,
           and ISEED(4) should 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 SLATMR
           to continue the same random number sequence.
           Changed on exit.

SYM

          SYM is CHARACTER*1
           If SYM='S' or 'H', generated matrix is symmetric.
           If SYM='N', generated matrix is nonsymmetric.
           Not modified.

D

          D is REAL array, dimension (min(M,N))
           On entry this array specifies the diagonal entries
           of the diagonal of A.  D may either be specified
           on entry, or set according to MODE and COND as described
           below. May be changed on exit if MODE is nonzero.

MODE

          MODE is INTEGER
           On entry describes how D is to be used:
           MODE = 0 means use D as input
           MODE = 1 sets D(1)=1 and D(2:N)=1.0/COND
           MODE = 2 sets D(1:N-1)=1 and D(N)=1.0/COND
           MODE = 3 sets D(I)=COND**(-(I-1)/(N-1))
           MODE = 4 sets D(i)=1 - (i-1)/(N-1)*(1 - 1/COND)
           MODE = 5 sets D to random numbers in the range
                    ( 1/COND , 1 ) such that their logarithms
                    are uniformly distributed.
           MODE = 6 set D to random numbers from same distribution
                    as the rest of the matrix.
           MODE < 0 has the same meaning as ABS(MODE), except that
              the order of the elements of D is reversed.
           Thus if MODE is positive, D has entries ranging from
              1 to 1/COND, if negative, from 1/COND to 1,
           Not modified.

COND

          COND is REAL
           On entry, used as described under MODE above.
           If used, it must be >= 1. Not modified.

DMAX

          DMAX is REAL
           If MODE neither -6, 0 nor 6, the diagonal is scaled by
           DMAX / max(abs(D(i))), so that maximum absolute entry
           of diagonal is abs(DMAX). If DMAX is negative (or zero),
           diagonal will be scaled by a negative number (or zero).

RSIGN

          RSIGN is CHARACTER*1
           If MODE neither -6, 0 nor 6, specifies sign of diagonal
           as follows:
           'T' => diagonal entries are multiplied by 1 or -1
                  with probability .5
           'F' => diagonal unchanged
           Not modified.

GRADE

          GRADE is CHARACTER*1
           Specifies grading of matrix as follows:
           'N'  => no grading
           'L'  => matrix premultiplied by diag( DL )
                   (only if matrix nonsymmetric)
           'R'  => matrix postmultiplied by diag( DR )
                   (only if matrix nonsymmetric)
           'B'  => matrix premultiplied by diag( DL ) and
                         postmultiplied by diag( DR )
                   (only if matrix nonsymmetric)
           'S' or 'H'  => matrix premultiplied by diag( DL ) and
                          postmultiplied by diag( DL )
                          ('S' for symmetric, or 'H' for Hermitian)
           'E'  => matrix premultiplied by diag( DL ) and
                         postmultiplied by inv( diag( DL ) )
                         ( 'E' for eigenvalue invariance)
                   (only if matrix nonsymmetric)
                   Note: if GRADE='E', then M must equal N.
           Not modified.

DL

          DL is REAL array, dimension (M)
           If MODEL=0, then on entry this array specifies the diagonal
           entries of a diagonal matrix used as described under GRADE
           above. If MODEL is not zero, then DL will be set according
           to MODEL and CONDL, analogous to the way D is set according
           to MODE and COND (except there is no DMAX parameter for DL).
           If GRADE='E', then DL cannot have zero entries.
           Not referenced if GRADE = 'N' or 'R'. Changed on exit.

MODEL

          MODEL is INTEGER
           This specifies how the diagonal array DL is to be computed,
           just as MODE specifies how D is to be computed.
           Not modified.

CONDL

          CONDL is REAL
           When MODEL is not zero, this specifies the condition number
           of the computed DL.  Not modified.

DR

          DR is REAL array, dimension (N)
           If MODER=0, then on entry this array specifies the diagonal
           entries of a diagonal matrix used as described under GRADE
           above. If MODER is not zero, then DR will be set according
           to MODER and CONDR, analogous to the way D is set according
           to MODE and COND (except there is no DMAX parameter for DR).
           Not referenced if GRADE = 'N', 'L', 'H', 'S' or 'E'.
           Changed on exit.

MODER

          MODER is INTEGER
           This specifies how the diagonal array DR is to be computed,
           just as MODE specifies how D is to be computed.
           Not modified.

CONDR

          CONDR is REAL
           When MODER is not zero, this specifies the condition number
           of the computed DR.  Not modified.

PIVTNG

          PIVTNG is CHARACTER*1
           On entry specifies pivoting permutations as follows:
           'N' or ' ' => none.
           'L' => left or row pivoting (matrix must be nonsymmetric).
           'R' => right or column pivoting (matrix must be
                  nonsymmetric).
           'B' or 'F' => both or full pivoting, i.e., on both sides.
                         In this case, M must equal N
           If two calls to SLATMR both have full bandwidth (KL = M-1
           and KU = N-1), and differ only in the PIVTNG and PACK
           parameters, then the matrices generated will differ only
           in the order of the rows and/or columns, and otherwise
           contain the same data. This consistency cannot be
           maintained with less than full bandwidth.

IPIVOT

          IPIVOT is INTEGER array, dimension (N or M)
           This array specifies the permutation used.  After the
           basic matrix is generated, the rows, columns, or both
           are permuted.   If, say, row pivoting is selected, SLATMR
           starts with the *last* row and interchanges the M-th and
           IPIVOT(M)-th rows, then moves to the next-to-last row,
           interchanging the (M-1)-th and the IPIVOT(M-1)-th rows,
           and so on.  In terms of "2-cycles", the permutation is
           (1 IPIVOT(1)) (2 IPIVOT(2)) ... (M IPIVOT(M))
           where the rightmost cycle is applied first.  This is the
           *inverse* of the effect of pivoting in LINPACK.  The idea
           is that factoring (with pivoting) an identity matrix
           which has been inverse-pivoted in this way should
           result in a pivot vector identical to IPIVOT.
           Not referenced if PIVTNG = 'N'. Not modified.

SPARSE

          SPARSE is REAL
           On entry specifies the sparsity of the matrix if a sparse
           matrix is to be generated. SPARSE should lie between
           0 and 1. To generate a sparse matrix, for each matrix entry
           a uniform ( 0, 1 ) random number x is generated and
           compared to SPARSE; if x is larger the matrix entry
           is unchanged and if x is smaller the entry is set
           to zero. Thus on the average a fraction SPARSE of the
           entries will be set to zero.
           Not modified.

KL

          KL is INTEGER
           On entry specifies the lower bandwidth of the  matrix. For
           example, KL=0 implies upper triangular, KL=1 implies upper
           Hessenberg, and KL at least M-1 implies the matrix is not
           banded. Must equal KU if matrix is symmetric.
           Not modified.

KU

          KU is INTEGER
           On entry specifies the upper bandwidth of the  matrix. For
           example, KU=0 implies lower triangular, KU=1 implies lower
           Hessenberg, and KU at least N-1 implies the matrix is not
           banded. Must equal KL if matrix is symmetric.
           Not modified.

ANORM

          ANORM is REAL
           On entry specifies maximum entry of output matrix
           (output matrix will by multiplied by a constant so that
           its largest absolute entry equal ANORM)
           if ANORM is nonnegative. If ANORM is negative no scaling
           is done. Not modified.

PACK

          PACK is CHARACTER*1
           On entry specifies packing of matrix as follows:
           'N' => no packing
           'U' => zero out all subdiagonal entries (if symmetric)
           'L' => zero out all superdiagonal entries (if symmetric)
           'C' => store the upper triangle columnwise
                  (only if matrix symmetric or square upper triangular)
           'R' => store the lower triangle columnwise
                  (only if matrix symmetric or square lower triangular)
                  (same as upper half rowwise if symmetric)
           'B' => store the lower triangle in band storage scheme
                  (only if matrix symmetric)
           'Q' => store the upper triangle in band storage scheme
                  (only if matrix symmetric)
           'Z' => store the entire matrix in band storage scheme
                      (pivoting can be provided for by using this
                      option to store A in the trailing rows of
                      the allocated storage)
           Using these options, the various LAPACK packed and banded
           storage schemes can be obtained:
           GB               - use 'Z'
           PB, SB or TB     - use 'B' or 'Q'
           PP, SP or TP     - use 'C' or 'R'
           If two calls to SLATMR differ only in the PACK parameter,
           they will generate mathematically equivalent matrices.
           Not modified.

A

          A is REAL array, dimension (LDA,N)
           On exit A is the desired test matrix. Only those
           entries of A which are significant on output
           will be referenced (even if A is in packed or band
           storage format). The 'unoccupied corners' of A in
           band format will be zeroed out.

LDA

          LDA is INTEGER
           on entry LDA specifies the first dimension of A as
           declared in the calling program.
           If PACK='N', 'U' or 'L', LDA must be at least max ( 1, M ).
           If PACK='C' or 'R', LDA must be at least 1.
           If PACK='B', or 'Q', LDA must be MIN ( KU+1, N )
           If PACK='Z', LDA must be at least KUU+KLL+1, where
           KUU = MIN ( KU, N-1 ) and KLL = MIN ( KL, N-1 )
           Not modified.

IWORK

          IWORK is INTEGER array, dimension ( N or M)
           Workspace. Not referenced if PIVTNG = 'N'. Changed on exit.

INFO

          INFO is INTEGER
           Error parameter on exit:
             0 => normal return
            -1 => M negative or unequal to N and SYM='S' or 'H'
            -2 => N negative
            -3 => DIST illegal string
            -5 => SYM illegal string
            -7 => MODE not in range -6 to 6
            -8 => COND less than 1.0, and MODE neither -6, 0 nor 6
           -10 => MODE neither -6, 0 nor 6 and RSIGN illegal string
           -11 => GRADE illegal string, or GRADE='E' and
                  M not equal to N, or GRADE='L', 'R', 'B' or 'E' and
                  SYM = 'S' or 'H'
           -12 => GRADE = 'E' and DL contains zero
           -13 => MODEL not in range -6 to 6 and GRADE= 'L', 'B', 'H',
                  'S' or 'E'
           -14 => CONDL less than 1.0, GRADE='L', 'B', 'H', 'S' or 'E',
                  and MODEL neither -6, 0 nor 6
           -16 => MODER not in range -6 to 6 and GRADE= 'R' or 'B'
           -17 => CONDR less than 1.0, GRADE='R' or 'B', and
                  MODER neither -6, 0 nor 6
           -18 => PIVTNG illegal string, or PIVTNG='B' or 'F' and
                  M not equal to N, or PIVTNG='L' or 'R' and SYM='S'
                  or 'H'
           -19 => IPIVOT contains out of range number and
                  PIVTNG not equal to 'N'
           -20 => KL negative
           -21 => KU negative, or SYM='S' or 'H' and KU not equal to KL
           -22 => SPARSE not in range 0. to 1.
           -24 => PACK illegal string, or PACK='U', 'L', 'B' or 'Q'
                  and SYM='N', or PACK='C' and SYM='N' and either KL
                  not equal to 0 or N not equal to M, or PACK='R' and
                  SYM='N', and either KU not equal to 0 or N not equal
                  to M
           -26 => LDA too small
             1 => Error return from SLATM1 (computing D)
             2 => Cannot scale diagonal to DMAX (max. entry is 0)
             3 => Error return from SLATM1 (computing DL)
             4 => Error return from SLATM1 (computing DR)
             5 => ANORM is positive, but matrix constructed prior to
                  attempting to scale it to have norm ANORM, is zero

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine slatms (integer M, integer N, character DIST, integer, dimension( 4 ) ISEED, character SYM, real, dimension( * ) D, integer MODE, real COND, real DMAX, integer KL, integer KU, character PACK, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK, integer INFO)

SLATMS

Purpose:

    SLATMS generates random matrices with specified singular values
    (or symmetric/hermitian with specified eigenvalues)
    for testing LAPACK programs.
    SLATMS operates by applying the following sequence of
    operations:
      Set the diagonal to D, where D may be input or
         computed according to MODE, COND, DMAX, and SYM
         as described below.
      Generate a matrix with the appropriate band structure, by one
         of two methods:
      Method A:
          Generate a dense M x N matrix by multiplying D on the left
              and the right by random unitary matrices, then:
          Reduce the bandwidth according to KL and KU, using
          Householder transformations.
      Method B:
          Convert the bandwidth-0 (i.e., diagonal) matrix to a
              bandwidth-1 matrix using Givens rotations, "chasing"
              out-of-band elements back, much as in QR; then
              convert the bandwidth-1 to a bandwidth-2 matrix, etc.
              Note that for reasonably small bandwidths (relative to
              M and N) this requires less storage, as a dense matrix
              is not generated.  Also, for symmetric matrices, only
              one triangle is generated.
      Method A is chosen if the bandwidth is a large fraction of the
          order of the matrix, and LDA is at least M (so a dense
          matrix can be stored.)  Method B is chosen if the bandwidth
          is small (< 1/2 N for symmetric, < .3 N+M for
          non-symmetric), or LDA is less than M and not less than the
          bandwidth.
      Pack the matrix if desired. Options specified by PACK are:
         no packing
         zero out upper half (if symmetric)
         zero out lower half (if symmetric)
         store the upper half columnwise (if symmetric or upper
               triangular)
         store the lower half columnwise (if symmetric or lower
               triangular)
         store the lower triangle in banded format (if symmetric
               or lower triangular)
         store the upper triangle in banded format (if symmetric
               or upper triangular)
         store the entire matrix in banded format
      If Method B is chosen, and band format is specified, then the
         matrix will be generated in the band format, so no repacking
         will be necessary.

 

Parameters:

M

          M is INTEGER
           The number of rows of A. Not modified.

N

          N is INTEGER
           The number of columns of A. Not modified.

DIST

          DIST is CHARACTER*1
           On entry, DIST specifies the type of distribution to be used
           to generate the random eigen-/singular values.
           'U' => UNIFORM( 0, 1 )  ( 'U' for uniform )
           'S' => UNIFORM( -1, 1 ) ( 'S' for symmetric )
           'N' => NORMAL( 0, 1 )   ( 'N' for normal )
           Not modified.

ISEED

          ISEED is INTEGER array, dimension ( 4 )
           On entry ISEED specifies the seed of the random number
           generator. They should lie between 0 and 4095 inclusive,
           and ISEED(4) should 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 SLATMS
           to continue the same random number sequence.
           Changed on exit.

SYM

          SYM is CHARACTER*1
           If SYM='S' or 'H', the generated matrix is symmetric, with
             eigenvalues specified by D, COND, MODE, and DMAX; they
             may be positive, negative, or zero.
           If SYM='P', the generated matrix is symmetric, with
             eigenvalues (= singular values) specified by D, COND,
             MODE, and DMAX; they will not be negative.
           If SYM='N', the generated matrix is nonsymmetric, with
             singular values specified by D, COND, MODE, and DMAX;
             they will not be negative.
           Not modified.

D

          D is REAL array, dimension ( MIN( M , N ) )
           This array is used to specify the singular values or
           eigenvalues of A (see SYM, above.)  If MODE=0, then D is
           assumed to contain the singular/eigenvalues, otherwise
           they will be computed according to MODE, COND, and DMAX,
           and placed in D.
           Modified if MODE is nonzero.

MODE

          MODE is INTEGER
           On entry this describes how the singular/eigenvalues are to
           be specified:
           MODE = 0 means use D as input
           MODE = 1 sets D(1)=1 and D(2:N)=1.0/COND
           MODE = 2 sets D(1:N-1)=1 and D(N)=1.0/COND
           MODE = 3 sets D(I)=COND**(-(I-1)/(N-1))
           MODE = 4 sets D(i)=1 - (i-1)/(N-1)*(1 - 1/COND)
           MODE = 5 sets D to random numbers in the range
                    ( 1/COND , 1 ) such that their logarithms
                    are uniformly distributed.
           MODE = 6 set D to random numbers from same distribution
                    as the rest of the matrix.
           MODE < 0 has the same meaning as ABS(MODE), except that
              the order of the elements of D is reversed.
           Thus if MODE is positive, D has entries ranging from
              1 to 1/COND, if negative, from 1/COND to 1,
           If SYM='S' or 'H', and MODE is neither 0, 6, nor -6, then
              the elements of D will also be multiplied by a random
              sign (i.e., +1 or -1.)
           Not modified.

COND

          COND is REAL
           On entry, this is used as described under MODE above.
           If used, it must be >= 1. Not modified.

DMAX

          DMAX is REAL
           If MODE is neither -6, 0 nor 6, the contents of D, as
           computed according to MODE and COND, will be scaled by
           DMAX / max(abs(D(i))); thus, the maximum absolute eigen- or
           singular value (which is to say the norm) will be abs(DMAX).
           Note that DMAX need not be positive: if DMAX is negative
           (or zero), D will be scaled by a negative number (or zero).
           Not modified.

KL

          KL is INTEGER
           This specifies the lower bandwidth of the  matrix. For
           example, KL=0 implies upper triangular, KL=1 implies upper
           Hessenberg, and KL being at least M-1 means that the matrix
           has full lower bandwidth.  KL must equal KU if the matrix
           is symmetric.
           Not modified.

KU

          KU is INTEGER
           This specifies the upper bandwidth of the  matrix. For
           example, KU=0 implies lower triangular, KU=1 implies lower
           Hessenberg, and KU being at least N-1 means that the matrix
           has full upper bandwidth.  KL must equal KU if the matrix
           is symmetric.
           Not modified.

PACK

          PACK is CHARACTER*1
           This specifies packing of matrix as follows:
           'N' => no packing
           'U' => zero out all subdiagonal entries (if symmetric)
           'L' => zero out all superdiagonal entries (if symmetric)
           'C' => store the upper triangle columnwise
                  (only if the matrix is symmetric or upper triangular)
           'R' => store the lower triangle columnwise
                  (only if the matrix is symmetric or lower triangular)
           'B' => store the lower triangle in band storage scheme
                  (only if matrix symmetric or lower triangular)
           'Q' => store the upper triangle in band storage scheme
                  (only if matrix symmetric or upper triangular)
           'Z' => store the entire matrix in band storage scheme
                      (pivoting can be provided for by using this
                      option to store A in the trailing rows of
                      the allocated storage)
           Using these options, the various LAPACK packed and banded
           storage schemes can be obtained:
           GB               - use 'Z'
           PB, SB or TB     - use 'B' or 'Q'
           PP, SP or TP     - use 'C' or 'R'
           If two calls to SLATMS differ only in the PACK parameter,
           they will generate mathematically equivalent matrices.
           Not modified.

A

          A is REAL array, dimension ( LDA, N )
           On exit A is the desired test matrix.  A is first generated
           in full (unpacked) form, and then packed, if so specified
           by PACK.  Thus, the first M elements of the first N
           columns will always be modified.  If PACK specifies a
           packed or banded storage scheme, all LDA elements of the
           first N columns will be modified; the elements of the
           array which do not correspond to elements of the generated
           matrix are set to zero.
           Modified.

LDA

          LDA is INTEGER
           LDA specifies the first dimension of A as declared in the
           calling program.  If PACK='N', 'U', 'L', 'C', or 'R', then
           LDA must be at least M.  If PACK='B' or 'Q', then LDA must
           be at least MIN( KL, M-1) (which is equal to MIN(KU,N-1)).
           If PACK='Z', LDA must be large enough to hold the packed
           array: MIN( KU, N-1) + MIN( KL, M-1) + 1.
           Not modified.

WORK

          WORK is REAL array, dimension ( 3*MAX( N , M ) )
           Workspace.
           Modified.

INFO

          INFO is INTEGER
           Error code.  On exit, INFO will be set to one of the
           following values:
             0 => normal return
            -1 => M negative or unequal to N and SYM='S', 'H', or 'P'
            -2 => N negative
            -3 => DIST illegal string
            -5 => SYM illegal string
            -7 => MODE not in range -6 to 6
            -8 => COND less than 1.0, and MODE neither -6, 0 nor 6
           -10 => KL negative
           -11 => KU negative, or SYM='S' or 'H' and KU not equal to KL
           -12 => PACK illegal string, or PACK='U' or 'L', and SYM='N';
                  or PACK='C' or 'Q' and SYM='N' and KL is not zero;
                  or PACK='R' or 'B' and SYM='N' and KU is not zero;
                  or PACK='U', 'L', 'C', 'R', 'B', or 'Q', and M is not
                  N.
           -14 => LDA is less than M, or PACK='Z' and LDA is less than
                  MIN(KU,N-1) + MIN(KL,M-1) + 1.
            1  => Error return from SLATM1
            2  => Cannot scale to DMAX (max. sing. value is 0)
            3  => Error return from SLAGGE or SLAGSY

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine slatmt (integer M, integer N, character DIST, integer, dimension( 4 ) ISEED, character SYM, real, dimension( * ) D, integer MODE, real COND, real DMAX, integer RANK, integer KL, integer KU, character PACK, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK, integer INFO)

SLATMT

Purpose:

    SLATMT generates random matrices with specified singular values
    (or symmetric/hermitian with specified eigenvalues)
    for testing LAPACK programs.
    SLATMT operates by applying the following sequence of
    operations:
      Set the diagonal to D, where D may be input or
         computed according to MODE, COND, DMAX, and SYM
         as described below.
      Generate a matrix with the appropriate band structure, by one
         of two methods:
      Method A:
          Generate a dense M x N matrix by multiplying D on the left
              and the right by random unitary matrices, then:
          Reduce the bandwidth according to KL and KU, using
          Householder transformations.
      Method B:
          Convert the bandwidth-0 (i.e., diagonal) matrix to a
              bandwidth-1 matrix using Givens rotations, "chasing"
              out-of-band elements back, much as in QR; then
              convert the bandwidth-1 to a bandwidth-2 matrix, etc.
              Note that for reasonably small bandwidths (relative to
              M and N) this requires less storage, as a dense matrix
              is not generated.  Also, for symmetric matrices, only
              one triangle is generated.
      Method A is chosen if the bandwidth is a large fraction of the
          order of the matrix, and LDA is at least M (so a dense
          matrix can be stored.)  Method B is chosen if the bandwidth
          is small (< 1/2 N for symmetric, < .3 N+M for
          non-symmetric), or LDA is less than M and not less than the
          bandwidth.
      Pack the matrix if desired. Options specified by PACK are:
         no packing
         zero out upper half (if symmetric)
         zero out lower half (if symmetric)
         store the upper half columnwise (if symmetric or upper
               triangular)
         store the lower half columnwise (if symmetric or lower
               triangular)
         store the lower triangle in banded format (if symmetric
               or lower triangular)
         store the upper triangle in banded format (if symmetric
               or upper triangular)
         store the entire matrix in banded format
      If Method B is chosen, and band format is specified, then the
         matrix will be generated in the band format, so no repacking
         will be necessary.

 

Parameters:

M

          M is INTEGER
           The number of rows of A. Not modified.

N

          N is INTEGER
           The number of columns of A. Not modified.

DIST

          DIST is CHARACTER*1
           On entry, DIST specifies the type of distribution to be used
           to generate the random eigen-/singular values.
           'U' => UNIFORM( 0, 1 )  ( 'U' for uniform )
           'S' => UNIFORM( -1, 1 ) ( 'S' for symmetric )
           'N' => NORMAL( 0, 1 )   ( 'N' for normal )
           Not modified.

ISEED

          ISEED is INTEGER array, dimension ( 4 )
           On entry ISEED specifies the seed of the random number
           generator. They should lie between 0 and 4095 inclusive,
           and ISEED(4) should 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 SLATMT
           to continue the same random number sequence.
           Changed on exit.

SYM

          SYM is CHARACTER*1
           If SYM='S' or 'H', the generated matrix is symmetric, with
             eigenvalues specified by D, COND, MODE, and DMAX; they
             may be positive, negative, or zero.
           If SYM='P', the generated matrix is symmetric, with
             eigenvalues (= singular values) specified by D, COND,
             MODE, and DMAX; they will not be negative.
           If SYM='N', the generated matrix is nonsymmetric, with
             singular values specified by D, COND, MODE, and DMAX;
             they will not be negative.
           Not modified.

D

          D is REAL array, dimension ( MIN( M , N ) )
           This array is used to specify the singular values or
           eigenvalues of A (see SYM, above.)  If MODE=0, then D is
           assumed to contain the singular/eigenvalues, otherwise
           they will be computed according to MODE, COND, and DMAX,
           and placed in D.
           Modified if MODE is nonzero.

MODE

          MODE is INTEGER
           On entry this describes how the singular/eigenvalues are to
           be specified:
           MODE = 0 means use D as input
           MODE = 1 sets D(1)=1 and D(2:RANK)=1.0/COND
           MODE = 2 sets D(1:RANK-1)=1 and D(RANK)=1.0/COND
           MODE = 3 sets D(I)=COND**(-(I-1)/(RANK-1))
           MODE = 4 sets D(i)=1 - (i-1)/(N-1)*(1 - 1/COND)
           MODE = 5 sets D to random numbers in the range
                    ( 1/COND , 1 ) such that their logarithms
                    are uniformly distributed.
           MODE = 6 set D to random numbers from same distribution
                    as the rest of the matrix.
           MODE < 0 has the same meaning as ABS(MODE), except that
              the order of the elements of D is reversed.
           Thus if MODE is positive, D has entries ranging from
              1 to 1/COND, if negative, from 1/COND to 1,
           If SYM='S' or 'H', and MODE is neither 0, 6, nor -6, then
              the elements of D will also be multiplied by a random
              sign (i.e., +1 or -1.)
           Not modified.

COND

          COND is REAL
           On entry, this is used as described under MODE above.
           If used, it must be >= 1. Not modified.

DMAX

          DMAX is REAL
           If MODE is neither -6, 0 nor 6, the contents of D, as
           computed according to MODE and COND, will be scaled by
           DMAX / max(abs(D(i))); thus, the maximum absolute eigen- or
           singular value (which is to say the norm) will be abs(DMAX).
           Note that DMAX need not be positive: if DMAX is negative
           (or zero), D will be scaled by a negative number (or zero).
           Not modified.

RANK

          RANK is INTEGER
           The rank of matrix to be generated for modes 1,2,3 only.
           D( RANK+1:N ) = 0.
           Not modified.

KL

          KL is INTEGER
           This specifies the lower bandwidth of the  matrix. For
           example, KL=0 implies upper triangular, KL=1 implies upper
           Hessenberg, and KL being at least M-1 means that the matrix
           has full lower bandwidth.  KL must equal KU if the matrix
           is symmetric.
           Not modified.

KU

          KU is INTEGER
           This specifies the upper bandwidth of the  matrix. For
           example, KU=0 implies lower triangular, KU=1 implies lower
           Hessenberg, and KU being at least N-1 means that the matrix
           has full upper bandwidth.  KL must equal KU if the matrix
           is symmetric.
           Not modified.

PACK

          PACK is CHARACTER*1
           This specifies packing of matrix as follows:
           'N' => no packing
           'U' => zero out all subdiagonal entries (if symmetric)
           'L' => zero out all superdiagonal entries (if symmetric)
           'C' => store the upper triangle columnwise
                  (only if the matrix is symmetric or upper triangular)
           'R' => store the lower triangle columnwise
                  (only if the matrix is symmetric or lower triangular)
           'B' => store the lower triangle in band storage scheme
                  (only if matrix symmetric or lower triangular)
           'Q' => store the upper triangle in band storage scheme
                  (only if matrix symmetric or upper triangular)
           'Z' => store the entire matrix in band storage scheme
                      (pivoting can be provided for by using this
                      option to store A in the trailing rows of
                      the allocated storage)
           Using these options, the various LAPACK packed and banded
           storage schemes can be obtained:
           GB               - use 'Z'
           PB, SB or TB     - use 'B' or 'Q'
           PP, SP or TP     - use 'C' or 'R'
           If two calls to SLATMT differ only in the PACK parameter,
           they will generate mathematically equivalent matrices.
           Not modified.

A

          A is REAL array, dimension ( LDA, N )
           On exit A is the desired test matrix.  A is first generated
           in full (unpacked) form, and then packed, if so specified
           by PACK.  Thus, the first M elements of the first N
           columns will always be modified.  If PACK specifies a
           packed or banded storage scheme, all LDA elements of the
           first N columns will be modified; the elements of the
           array which do not correspond to elements of the generated
           matrix are set to zero.
           Modified.

LDA

          LDA is INTEGER
           LDA specifies the first dimension of A as declared in the
           calling program.  If PACK='N', 'U', 'L', 'C', or 'R', then
           LDA must be at least M.  If PACK='B' or 'Q', then LDA must
           be at least MIN( KL, M-1) (which is equal to MIN(KU,N-1)).
           If PACK='Z', LDA must be large enough to hold the packed
           array: MIN( KU, N-1) + MIN( KL, M-1) + 1.
           Not modified.

WORK

          WORK is REAL array, dimension ( 3*MAX( N , M ) )
           Workspace.
           Modified.

INFO

          INFO is INTEGER
           Error code.  On exit, INFO will be set to one of the
           following values:
             0 => normal return
            -1 => M negative or unequal to N and SYM='S', 'H', or 'P'
            -2 => N negative
            -3 => DIST illegal string
            -5 => SYM illegal string
            -7 => MODE not in range -6 to 6
            -8 => COND less than 1.0, and MODE neither -6, 0 nor 6
           -10 => KL negative
           -11 => KU negative, or SYM='S' or 'H' and KU not equal to KL
           -12 => PACK illegal string, or PACK='U' or 'L', and SYM='N';
                  or PACK='C' or 'Q' and SYM='N' and KL is not zero;
                  or PACK='R' or 'B' and SYM='N' and KU is not zero;
                  or PACK='U', 'L', 'C', 'R', 'B', or 'Q', and M is not
                  N.
           -14 => LDA is less than M, or PACK='Z' and LDA is less than
                  MIN(KU,N-1) + MIN(KL,M-1) + 1.
            1  => Error return from SLATM7
            2  => Cannot scale to DMAX (max. sing. value is 0)
            3  => Error return from SLAGGE or SLAGSY

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Author

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