Scroll to navigation

complex16SYauxiliary(3) LAPACK 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

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

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

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

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

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

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 communication 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.

Author

Generated automatically by Doxygen for LAPACK from the source code.
Sat Aug 1 2020 Version 3.9.0