complexSYauxiliary(3)

NAME
complexSYauxiliary - complex

SYNOPSIS

  Functions
subroutine claesy (a, b, c, rt1, rt2, evscal, cs1, sn1)
CLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.
real function clansy (norm, uplo, n, a, lda, work)
CLANSY 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 claqsy (uplo, n, a, lda, s, scond, amax, equed)
CLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
subroutine csymv (uplo, n, alpha, a, lda, x, incx, beta, y, incy)
CSYMV computes a matrix-vector product for a complex symmetric matrix.
subroutine csyr (uplo, n, alpha, x, incx, a, lda)
CSYR performs the symmetric rank-1 update of a complex symmetric matrix.
subroutine csyswapr (uplo, n, a, lda, i1, i2)
CSYSWAPR
subroutine ctgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, info)
CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Detailed Description

This is the group of complex auxiliary functions for SY matrices

Function Documentation

  subroutine claesy (complex a, complex b, complex c, complex rt1, complex rt2, complex evscal, complex cs1, complex sn1)

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

Purpose:

     CLAESY 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
              The ( 1, 1 ) element of input matrix.

    B

              B is COMPLEX
              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
              The ( 2, 2 ) element of input matrix.

    RT1

              RT1 is COMPLEX
              The eigenvalue of larger modulus.

    RT2

              RT2 is COMPLEX
              The eigenvalue of smaller modulus.

    EVSCAL

              EVSCAL is COMPLEX
              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

    SN1

              SN1 is COMPLEX
              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.

Definition at line 114 of file claesy.f.

  real function clansy (character norm, character uplo, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) work)

CLANSY 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:

     CLANSY  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
    CLANSY

        CLANSY = ( 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 CLANSY 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, CLANSY is
              set to zero.

    A

              A is COMPLEX 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 REAL 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.

Definition at line 122 of file clansy.f.

  subroutine claqsy (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) s, real scond, real amax, character equed)

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

Purpose:

     CLAQSY 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 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 REAL array, dimension (N)
              The scale factors for A.

    SCOND

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

    AMAX

              AMAX is REAL
              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.

Definition at line 133 of file claqsy.f.

  subroutine csymv (character uplo, integer n, complex alpha, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) x, integer incx, complex beta, complex, dimension( * ) y, integer incy)

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

Purpose:

     CSYMV  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
               On entry, ALPHA specifies the scalar alpha.
               Unchanged on exit.

    A

              A is COMPLEX 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 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
               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 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.

Definition at line 156 of file csymv.f.

  subroutine csyr (character uplo, integer n, complex alpha, complex, dimension( * ) x, integer incx, complex, dimension( lda, * ) a, integer lda)

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

Purpose:

     CSYR   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
               On entry, ALPHA specifies the scalar alpha.
               Unchanged on exit.

    X

              X is COMPLEX 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 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.

Definition at line 134 of file csyr.f.

  subroutine csyswapr (character uplo, integer n, complex, dimension( lda, n ) a, integer lda, integer i1, integer i2)

CSYSWAPR

Purpose:

     CSYSWAPR 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 array, dimension (LDA,N)
              On entry, the N-by-N matrix A. On exit, the permuted matrix
              where the rows I1 and I2 and columns I1 and I2 are interchanged.
              If UPLO = 'U', the interchanges are applied to the upper
              triangular part and the strictly lower triangular part of A is
              not referenced; if UPLO = 'L', the interchanges are applied to
              the lower triangular part 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.

Definition at line 99 of file csyswapr.f.

  subroutine ctgsy2 (character trans, integer ijob, integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldc, * ) c, integer ldc, complex, dimension( ldd, * ) d, integer ldd, complex, dimension( lde, * ) e, integer lde, complex, dimension( ldf, * ) f, integer ldf, real scale, real rdsum, real rdscal, integer info)

CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:

     CTGSY2 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 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 communication with CLACON.
     CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL
     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
     CTGSYL.

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. (SGECON 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 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 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 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 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 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 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 REAL
              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 REAL
              On entry, the sum of squares of computed contributions to
              the Dif-estimate under computation by CTGSYL, 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 CTGSY2 is called by
              CTGSYL.

    RDSCAL

              RDSCAL is REAL
              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 CTGSY2 is called by
              CTGSYL.

    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.

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

Definition at line 256 of file ctgsy2.f.

Author

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