Provided by: liblapack-doc_3.8.0-2_all bug

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 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.

Author

       Generated automatically by Doxygen for LAPACK from the source code.