complex16SYauxiliary(3)

NAME
complex16SYauxiliary

SYNOPSIS

  Functions
subroutine zlaesy (A, B, C, RT1, RT2, EVSCAL, CS1, SN1)
ZLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.
double precision function zlansy (NORM, UPLO, N, A, LDA, WORK)
ZLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix.
subroutine zlaqsy (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
ZLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
subroutine zsymv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZSYMV computes a matrix-vector product for a complex symmetric matrix.
subroutine zsyr (UPLO, N, ALPHA, X, INCX, A, LDA)
ZSYR performs the symmetric rank-1 update of a complex symmetric matrix.
subroutine zsyswapr (UPLO, N, A, LDA, I1, I2)
ZSYSWAPR
subroutine ztgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO)
ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Detailed Description

This is the group of complex16 auxiliary functions for SY matrices

Function Documentation

  subroutine zlaesy (complex*16 A, complex*16 B, complex*16 C, complex*16 RT1, complex*16 RT2, complex*16 EVSCAL, complex*16 CS1, complex*16 SN1)

ZLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.

Purpose:

     ZLAESY computes the eigendecomposition of a 2-by-2 symmetric matrix
        ( ( A, B );( B, C ) )
     provided the norm of the matrix of eigenvectors is larger than
     some threshold value.
     RT1 is the eigenvalue of larger absolute value, and RT2 of
     smaller absolute value.  If the eigenvectors are computed, then
     on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence
     [  CS1     SN1   ] . [ A  B ] . [ CS1    -SN1   ] = [ RT1  0  ]
     [ -SN1     CS1   ]   [ B  C ]   [ SN1     CS1   ]   [  0  RT2 ]

Parameters:
    A

              A is COMPLEX*16
              The ( 1, 1 ) element of input matrix.

    B

              B is COMPLEX*16
              The ( 1, 2 ) element of input matrix.  The ( 2, 1 ) element
              is also given by B, since the 2-by-2 matrix is symmetric.

    C

              C is COMPLEX*16
              The ( 2, 2 ) element of input matrix.

    RT1

              RT1 is COMPLEX*16
              The eigenvalue of larger modulus.

    RT2

              RT2 is COMPLEX*16
              The eigenvalue of smaller modulus.

    EVSCAL

              EVSCAL is COMPLEX*16
              The complex value by which the eigenvector matrix was scaled
              to make it orthonormal.  If EVSCAL is zero, the eigenvectors
              were not computed.  This means one of two things:  the 2-by-2
              matrix could not be diagonalized, or the norm of the matrix
              of eigenvectors before scaling was larger than the threshold
              value THRESH (set below).

    CS1

              CS1 is COMPLEX*16

    SN1

              SN1 is COMPLEX*16
              If EVSCAL .NE. 0,  ( CS1, SN1 ) is the unit right eigenvector
              for RT1.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 117 of file zlaesy.f.

  double precision function zlansy (character NORM, character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) WORK)

ZLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix.

Purpose:

     ZLANSY  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 symmetric matrix A.

Returns:
    ZLANSY

        ZLANSY = ( 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 ZLANSY as described
              above.

    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              symmetric matrix A is to be referenced.
              = 'U':  Upper triangular part of A is referenced
              = 'L':  Lower triangular part of A is referenced

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.  When N = 0, ZLANSY is
              set to zero.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              The symmetric matrix A.  If UPLO = 'U', the leading n by n
              upper triangular part of A contains the upper triangular part
              of the matrix A, and the strictly lower triangular part of A
              is not referenced.  If UPLO = 'L', the leading n by n lower
              triangular part of A contains the lower triangular part of
              the matrix A, and the strictly upper triangular part of A is
              not referenced.

    LDA

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

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
              where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
              WORK is not referenced.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 125 of file zlansy.f.

  subroutine zlaqsy (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)

ZLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.

Purpose:

     ZLAQSY equilibrates a symmetric matrix A using the scaling factors
     in the vector S.

Parameters:
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              symmetric matrix A is stored.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = 'U', the leading
              n by n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n by n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, if EQUED = 'Y', the equilibrated matrix:
              diag(S) * A * diag(S).

    LDA

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

    S

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

    SCOND

              SCOND is DOUBLE PRECISION
              Ratio of the smallest S(i) to the largest S(i).

    AMAX

              AMAX is DOUBLE PRECISION
              Absolute value of largest matrix entry.

    EQUED

              EQUED is CHARACTER*1
              Specifies whether or not equilibration was done.
              = 'N':  No equilibration.
              = 'Y':  Equilibration was done, i.e., A has been replaced by
                      diag(S) * A * diag(S).

Internal Parameters:

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

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 136 of file zlaqsy.f.

  subroutine zsymv (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) X, integer INCX, complex*16 BETA, complex*16, dimension( * ) Y, integer INCY)

ZSYMV computes a matrix-vector product for a complex symmetric matrix.

Purpose:

     ZSYMV  performs the matrix-vector  operation
        y := alpha*A*x + beta*y,
     where alpha and beta are scalars, x and y are n element vectors and
     A is an n by n symmetric matrix.

Parameters:
    UPLO

              UPLO is CHARACTER*1
               On entry, UPLO specifies whether the upper or lower
               triangular part of the array A is to be referenced as
               follows:
                  UPLO = 'U' or 'u'   Only the upper triangular part of A
                                      is to be referenced.
                  UPLO = 'L' or 'l'   Only the lower triangular part of A
                                      is to be referenced.
               Unchanged on exit.

    N

              N is INTEGER
               On entry, N specifies the order of the matrix A.
               N must be at least zero.
               Unchanged on exit.

    ALPHA

              ALPHA is COMPLEX*16
               On entry, ALPHA specifies the scalar alpha.
               Unchanged on exit.

    A

              A is COMPLEX*16 array, dimension ( LDA, N )
               Before entry, with  UPLO = 'U' or 'u', the leading n by n
               upper triangular part of the array A must contain the upper
               triangular part of the symmetric matrix and the strictly
               lower triangular part of A is not referenced.
               Before entry, with UPLO = 'L' or 'l', the leading n by n
               lower triangular part of the array A must contain the lower
               triangular part of the symmetric matrix and the strictly
               upper triangular part of A is not referenced.
               Unchanged on exit.

    LDA

              LDA is INTEGER
               On entry, LDA specifies the first dimension of A as declared
               in the calling (sub) program. LDA must be at least
               max( 1, N ).
               Unchanged on exit.

    X

              X is COMPLEX*16 array, dimension at least
               ( 1 + ( N - 1 )*abs( INCX ) ).
               Before entry, the incremented array X must contain the N-
               element vector x.
               Unchanged on exit.

    INCX

              INCX is INTEGER
               On entry, INCX specifies the increment for the elements of
               X. INCX must not be zero.
               Unchanged on exit.

    BETA

              BETA is COMPLEX*16
               On entry, BETA specifies the scalar beta. When BETA is
               supplied as zero then Y need not be set on input.
               Unchanged on exit.

    Y

              Y is COMPLEX*16 array, dimension at least
               ( 1 + ( N - 1 )*abs( INCY ) ).
               Before entry, the incremented array Y must contain the n
               element vector y. On exit, Y is overwritten by the updated
               vector y.

    INCY

              INCY is INTEGER
               On entry, INCY specifies the increment for the elements of
               Y. INCY must not be zero.
               Unchanged on exit.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 159 of file zsymv.f.

  subroutine zsyr (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension( * ) X, integer INCX, complex*16, dimension( lda, * ) A, integer LDA)

ZSYR performs the symmetric rank-1 update of a complex symmetric matrix.

Purpose:

     ZSYR   performs the symmetric rank 1 operation
        A := alpha*x*x**H + A,
     where alpha is a complex scalar, x is an n element vector and A is an
     n by n symmetric matrix.

Parameters:
    UPLO

              UPLO is CHARACTER*1
               On entry, UPLO specifies whether the upper or lower
               triangular part of the array A is to be referenced as
               follows:
                  UPLO = 'U' or 'u'   Only the upper triangular part of A
                                      is to be referenced.
                  UPLO = 'L' or 'l'   Only the lower triangular part of A
                                      is to be referenced.
               Unchanged on exit.

    N

              N is INTEGER
               On entry, N specifies the order of the matrix A.
               N must be at least zero.
               Unchanged on exit.

    ALPHA

              ALPHA is COMPLEX*16
               On entry, ALPHA specifies the scalar alpha.
               Unchanged on exit.

    X

              X is COMPLEX*16 array, dimension at least
               ( 1 + ( N - 1 )*abs( INCX ) ).
               Before entry, the incremented array X must contain the N-
               element vector x.
               Unchanged on exit.

    INCX

              INCX is INTEGER
               On entry, INCX specifies the increment for the elements of
               X. INCX must not be zero.
               Unchanged on exit.

    A

              A is COMPLEX*16 array, dimension ( LDA, N )
               Before entry, with  UPLO = 'U' or 'u', the leading n by n
               upper triangular part of the array A must contain the upper
               triangular part of the symmetric matrix and the strictly
               lower triangular part of A is not referenced. On exit, the
               upper triangular part of the array A is overwritten by the
               upper triangular part of the updated matrix.
               Before entry, with UPLO = 'L' or 'l', the leading n by n
               lower triangular part of the array A must contain the lower
               triangular part of the symmetric matrix and the strictly
               upper triangular part of A is not referenced. On exit, the
               lower triangular part of the array A is overwritten by the
               lower triangular part of the updated matrix.

    LDA

              LDA is INTEGER
               On entry, LDA specifies the first dimension of A as declared
               in the calling (sub) program. LDA must be at least
               max( 1, N ).
               Unchanged on exit.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 137 of file zsyr.f.

  subroutine zsyswapr (character UPLO, integer N, complex*16, dimension( lda, n ) A, integer LDA, integer I1, integer I2)

ZSYSWAPR

Purpose:

     ZSYSWAPR applies an elementary permutation on the rows and the columns of
     a symmetric matrix.

Parameters:
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the details of the factorization are stored
              as an upper or lower triangular matrix.
              = 'U':  Upper triangular, form is A = U*D*U**T;
              = 'L':  Lower triangular, form is A = L*D*L**T.

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the NB diagonal matrix D and the multipliers
              used to obtain the factor U or L as computed by ZSYTRF.
              On exit, if INFO = 0, the (symmetric) inverse of the original
              matrix.  If UPLO = 'U', the upper triangular part of the
              inverse is formed and the part of A below the diagonal is not
              referenced; if UPLO = 'L' the lower triangular part of the
              inverse is formed and the part of A above the diagonal is
              not referenced.

    LDA

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

    I1

              I1 is INTEGER
              Index of the first row to swap

    I2

              I2 is INTEGER
              Index of the second row to swap

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 104 of file zsyswapr.f.

  subroutine ztgsy2 (character TRANS, integer IJOB, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( ldd, * ) D, integer LDD, complex*16, dimension( lde, * ) E, integer LDE, complex*16, dimension( ldf, * ) F, integer LDF, double precision SCALE, double precision RDSUM, double precision RDSCAL, integer INFO)

ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:

     ZTGSY2 solves the generalized Sylvester equation
                 A * R - L * B = scale * C               (1)
                 D * R - L * E = scale * F
     using Level 1 and 2 BLAS, where R and L are unknown M-by-N matrices,
     (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
     N-by-N and M-by-N, respectively. A, B, D and E are upper triangular
     (i.e., (A,D) and (B,E) in generalized Schur form).
     The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
     scaling factor chosen to avoid overflow.
     In matrix notation solving equation (1) corresponds to solve
     Zx = scale * b, where Z is defined as
            Z = [ kron(In, A)  -kron(B**H, Im) ]             (2)
                [ kron(In, D)  -kron(E**H, Im) ],
     Ik is the identity matrix of size k and X**H is the conjuguate transpose of X.
     kron(X, Y) is the Kronecker product between the matrices X and Y.
     If TRANS = 'C', y in the conjugate transposed system Z**H*y = scale*b
     is solved for, which is equivalent to solve for R and L in
                 A**H * R  + D**H * L   = scale * C           (3)
                 R  * B**H + L  * E**H  = scale * -F
     This case is used to compute an estimate of Dif[(A, D), (B, E)] =
     = sigma_min(Z) using reverse communicaton with ZLACON.
     ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL
     of an upper bound on the separation between to matrix pairs. Then
     the input (A, D), (B, E) are sub-pencils of two matrix pairs in
     ZTGSYL.

Parameters:
    TRANS

              TRANS is CHARACTER*1
              = 'N', solve the generalized Sylvester equation (1).
              = 'T': solve the 'transposed' system (3).

    IJOB

              IJOB is INTEGER
              Specifies what kind of functionality to be performed.
              =0: solve (1) only.
              =1: A contribution from this subsystem to a Frobenius
                  norm-based estimate of the separation between two matrix
                  pairs is computed. (look ahead strategy is used).
              =2: A contribution from this subsystem to a Frobenius
                  norm-based estimate of the separation between two matrix
                  pairs is computed. (DGECON on sub-systems is used.)
              Not referenced if TRANS = 'T'.

    M

              M is INTEGER
              On entry, M specifies the order of A and D, and the row
              dimension of C, F, R and L.

    N

              N is INTEGER
              On entry, N specifies the order of B and E, and the column
              dimension of C, F, R and L.

    A

              A is COMPLEX*16 array, dimension (LDA, M)
              On entry, A contains an upper triangular matrix.

    LDA

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

    B

              B is COMPLEX*16 array, dimension (LDB, N)
              On entry, B contains an upper triangular matrix.

    LDB

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

    C

              C is COMPLEX*16 array, dimension (LDC, N)
              On entry, C contains the right-hand-side of the first matrix
              equation in (1).
              On exit, if IJOB = 0, C has been overwritten by the solution
              R.

    LDC

              LDC is INTEGER
              The leading dimension of the matrix C. LDC >= max(1, M).

    D

              D is COMPLEX*16 array, dimension (LDD, M)
              On entry, D contains an upper triangular matrix.

    LDD

              LDD is INTEGER
              The leading dimension of the matrix D. LDD >= max(1, M).

    E

              E is COMPLEX*16 array, dimension (LDE, N)
              On entry, E contains an upper triangular matrix.

    LDE

              LDE is INTEGER
              The leading dimension of the matrix E. LDE >= max(1, N).

    F

              F is COMPLEX*16 array, dimension (LDF, N)
              On entry, F contains the right-hand-side of the second matrix
              equation in (1).
              On exit, if IJOB = 0, F has been overwritten by the solution
              L.

    LDF

              LDF is INTEGER
              The leading dimension of the matrix F. LDF >= max(1, M).

    SCALE

              SCALE is DOUBLE PRECISION
              On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
              R and L (C and F on entry) will hold the solutions to a
              slightly perturbed system but the input matrices A, B, D and
              E have not been changed. If SCALE = 0, R and L will hold the
              solutions to the homogeneous system with C = F = 0.
              Normally, SCALE = 1.

    RDSUM

              RDSUM is DOUBLE PRECISION
              On entry, the sum of squares of computed contributions to
              the Dif-estimate under computation by ZTGSYL, where the
              scaling factor RDSCAL (see below) has been factored out.
              On exit, the corresponding sum of squares updated with the
              contributions from the current sub-system.
              If TRANS = 'T' RDSUM is not touched.
              NOTE: RDSUM only makes sense when ZTGSY2 is called by
              ZTGSYL.

    RDSCAL

              RDSCAL is DOUBLE PRECISION
              On entry, scaling factor used to prevent overflow in RDSUM.
              On exit, RDSCAL is updated w.r.t. the current contributions
              in RDSUM.
              If TRANS = 'T', RDSCAL is not touched.
              NOTE: RDSCAL only makes sense when ZTGSY2 is called by
              ZTGSYL.

    INFO

              INFO is INTEGER
              On exit, if INFO is set to
                =0: Successful exit
                <0: If INFO = -i, input argument number i is illegal.
                >0: The matrix pairs (A, D) and (B, E) have common or very
                    close eigenvalues.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

Definition at line 261 of file ztgsy2.f.

Author

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