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NAME

       complex16GEauxiliary

SYNOPSIS

   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 n-by-n
           matrix.
       double precision function zlange (NORM, M, N, A, LDA, WORK)
           ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, 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 N-by-N 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.

           A

                     A is COMPLEX*16 array, dimension (LDA, N)
                     On entry, the  LU part of the factorization of the n-by-n
                     matrix A computed by ZGETC2:  A = P * L * U * Q

           LDA

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

           RHS

                     RHS is COMPLEX*16 array, dimension N.
                     On entry, the right hand side vector b.
                     On exit, the solution vector X.

           IPIV

                     IPIV is INTEGER array, dimension (N).
                     The pivot indices; for 1 <= i <= N, row i of the
                     matrix has been interchanged with row IPIV(i).

           JPIV

                     JPIV is INTEGER array, dimension (N).
                     The pivot indices; for 1 <= j <= N, column j of the
                     matrix has been interchanged with column JPIV(j).

           SCALE

                     SCALE 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:
           November 2017

       Contributors:
           Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901
           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 n-by-n matrix.

       Purpose:

            ZGETC2 computes an LU factorization, using complete pivoting, of the
            n-by-n 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.

           A

                     A is COMPLEX*16 array, dimension (LDA, N)
                     On entry, the n-by-n 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.

           LDA

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

           IPIV

                     IPIV is INTEGER array, dimension (N).
                     The pivot indices; for 1 <= i <= N, row i of the
                     matrix has been interchanged with row IPIV(i).

           JPIV

                     JPIV is INTEGER array, dimension (N).
                     The pivot indices; for 1 <= j <= N, column j of the
                     matrix has been interchanged with column JPIV(j).

           INFO

                     INFO 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, S-901
           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 1-norm, Frobenius norm, infinity-norm, 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.

           M

                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.  When M = 0,
                     ZLANGE is set to zero.

           N

                     N is INTEGER
                     The number of columns of the matrix A.  N >= 0.  When N = 0,
                     ZLANGE is set to zero.

           A

                     A is COMPLEX*16 array, dimension (LDA,N)
                     The m by n matrix A.

           LDA

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

           WORK

                     WORK 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:
           December 2016

   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.

           N

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

           A

                     A 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.

           LDA

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

           R

                     R is DOUBLE PRECISION array, dimension (M)
                     The row scale factors for A.

           C

                     C is DOUBLE PRECISION array, dimension (N)
                     The column scale factors for A.

           ROWCND

                     ROWCND is DOUBLE PRECISION
                     Ratio of the smallest R(i) to the largest R(i).

           COLCND

                     COLCND is DOUBLE PRECISION
                     Ratio of the smallest C(i) to the largest C(i).

           AMAX

                     AMAX is DOUBLE PRECISION
                     Absolute value of largest matrix entry.

           EQUED

                     EQUED 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:
           December 2016

   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.

           WANTZ

                     WANTZ is LOGICAL
                     .TRUE. : update the right transformation matrix Z;
                     .FALSE.: do not update Z.

           N

                     N is INTEGER
                     The order of the matrices A and B. N >= 0.

           A

                     A is COMPLEX*16 array, dimensions (LDA,N)
                     On entry, the matrix A in the pair (A, B).
                     On exit, the updated matrix A.

           LDA

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

           B

                     B is COMPLEX*16 array, dimensions (LDB,N)
                     On entry, the matrix B in the pair (A, B).
                     On exit, the updated matrix B.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,N).

           Q

                     Q is COMPLEX*16 array, dimension (LDQ,N)
                     If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,
                     the updated matrix Q.
                     Not referenced if WANTQ = .FALSE..

           LDQ

                     LDQ is INTEGER
                     The leading dimension of the array Q. LDQ >= 1;
                     If WANTQ = .TRUE., LDQ >= N.

           Z

                     Z 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..

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z. LDZ >= 1;
                     If WANTZ = .TRUE., LDZ >= N.

           J1

                     J1 is INTEGER
                     The index to the first block (A11, B11).

           INFO

                     INFO 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:
           June 2017

       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, S-901
           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 Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
            [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 UMINF-94.04, Department of Computing Science, Umea University, S-901 87 Umea,
           Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.

Author

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