Functions
subroutine zgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE)
ZGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
subroutine zgetc2 (N, A, LDA, IPIV, JPIV, INFO)
ZGETC2 computes the LU factorization with complete pivoting of the general nbyn matrix.
double precision function zlange (NORM, M, N, A, LDA, WORK)
ZLANGE returns the value of the 1norm, Frobenius norm, infinitynorm, or the largest absolute value of any element of a general rectangular matrix.
subroutine zlaqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
ZLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
subroutine ztgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, INFO)
ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.
Detailed Description
This is the group of complex16 auxiliary functions for GE matrices
Function Documentation
subroutine zgesc2 (integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) RHS, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, double precision SCALE)
ZGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
Purpose:

ZGESC2 solves a system of linear equations A * X = scale* RHS with a general NbyN matrix A using the LU factorization with complete pivoting computed by ZGETC2.
Parameters:

N
N is INTEGER The number of columns of the matrix A.
AA is COMPLEX*16 array, dimension (LDA, N) On entry, the LU part of the factorization of the nbyn matrix A computed by ZGETC2: A = P * L * U * Q
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1, N).
RHSRHS is COMPLEX*16 array, dimension N. On entry, the right hand side vector b. On exit, the solution vector X.
IPIVIPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).
JPIVJPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).
SCALESCALE is DOUBLE PRECISION On exit, SCALE contains the scale factor. SCALE is chosen 0 <= SCALE <= 1 to prevent owerflow in the solution.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 September 2012
Contributors:
 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S901 87 Umea, Sweden.
subroutine zgetc2 (integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, integer INFO)
ZGETC2 computes the LU factorization with complete pivoting of the general nbyn matrix.
Purpose:

ZGETC2 computes an LU factorization, using complete pivoting, of the nbyn matrix A. The factorization has the form A = P * L * U * Q, where P and Q are permutation matrices, L is lower triangular with unit diagonal elements and U is upper triangular. This is a level 1 BLAS version of the algorithm.
Parameters:

N
N is INTEGER The order of the matrix A. N >= 0.
AA is COMPLEX*16 array, dimension (LDA, N) On entry, the nbyn matrix to be factored. On exit, the factors L and U from the factorization A = P*L*U*Q; the unit diagonal elements of L are not stored. If U(k, k) appears to be less than SMIN, U(k, k) is given the value of SMIN, giving a nonsingular perturbed system.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1, N).
IPIVIPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).
JPIVJPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).
INFOINFO is INTEGER = 0: successful exit > 0: if INFO = k, U(k, k) is likely to produce overflow if one tries to solve for x in Ax = b. So U is perturbed to avoid the overflow.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
Contributors:
 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S901 87 Umea, Sweden.
double precision function zlange (character NORM, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) WORK)
ZLANGE returns the value of the 1norm, Frobenius norm, infinitynorm, or the largest absolute value of any element of a general rectangular matrix.
Purpose:

ZLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex matrix A.
Returns:

ZLANGE
ZLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters:

NORM
NORM is CHARACTER*1 Specifies the value to be returned in ZLANGE as described above.
MM is INTEGER The number of rows of the matrix A. M >= 0. When M = 0, ZLANGE is set to zero.
NN is INTEGER The number of columns of the matrix A. N >= 0. When N = 0, ZLANGE is set to zero.
AA is COMPLEX*16 array, dimension (LDA,N) The m by n matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(M,1).
WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= M when NORM = 'I'; otherwise, WORK is not referenced.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 September 2012
subroutine zlaqge (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, character EQUED)
ZLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
Purpose:

ZLAQGE equilibrates a general M by N matrix A using the row and column scaling factors in the vectors R and C.
Parameters:

M
M is INTEGER The number of rows of the matrix A. M >= 0.
NN is INTEGER The number of columns of the matrix A. N >= 0.
AA is COMPLEX*16 array, dimension (LDA,N) On entry, the M by N matrix A. On exit, the equilibrated matrix. See EQUED for the form of the equilibrated matrix.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(M,1).
RR is DOUBLE PRECISION array, dimension (M) The row scale factors for A.
CC is DOUBLE PRECISION array, dimension (N) The column scale factors for A.
ROWCNDROWCND is DOUBLE PRECISION Ratio of the smallest R(i) to the largest R(i).
COLCNDCOLCND is DOUBLE PRECISION Ratio of the smallest C(i) to the largest C(i).
AMAXAMAX is DOUBLE PRECISION Absolute value of largest matrix entry.
EQUEDEQUED is CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration = 'R': Row equilibration, i.e., A has been premultiplied by diag(R). = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C). = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).
Internal Parameters:

THRESH is a threshold value used to decide if row or column scaling should be done based on the ratio of the row or column scaling factors. If ROWCND < THRESH, row scaling is done, and if COLCND < THRESH, column scaling is done. LARGE and SMALL are threshold values used to decide if row scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, row scaling is done.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 September 2012
subroutine ztgex2 (logical WANTQ, logical WANTZ, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer J1, integer INFO)
ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.
Purpose:

ZTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22) in an upper triangular matrix pair (A, B) by an unitary equivalence transformation. (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular. Optionally, the matrices Q and Z of generalized Schur vectors are updated. Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H
Parameters:

WANTQ
WANTQ is LOGICAL .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q.
WANTZWANTZ is LOGICAL .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z.
NN is INTEGER The order of the matrices A and B. N >= 0.
AA is COMPLEX*16 arrays, dimensions (LDA,N) On entry, the matrix A in the pair (A, B). On exit, the updated matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
BB is COMPLEX*16 arrays, dimensions (LDB,N) On entry, the matrix B in the pair (A, B). On exit, the updated matrix B.
LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
QQ is COMPLEX*16 array, dimension (LDZ,N) If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE..
LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= 1; If WANTQ = .TRUE., LDQ >= N.
ZZ is COMPLEX*16 array, dimension (LDZ,N) If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE..
LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1; If WANTZ = .TRUE., LDZ >= N.
J1J1 is INTEGER The index to the first block (A11, B11).
INFOINFO is INTEGER =0: Successful exit. =1: The transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is ill conditioned.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 September 2012
Further Details:
 In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details.
Contributors:
 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S901 87 Umea, Sweden.
References:

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and RealTime Applications, Kluwer Academic Publ. 1993, pp 195218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF94.04, Department of Computing Science, Umea University, S901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
Author
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