complexSYcomputational(3)

NAME
complexSYcomputational - complex

SYNOPSIS

  Functions
subroutine chesv_aa_2stage (uplo, n, nrhs, a, lda, tb, ltb, ipiv, ipiv2, b, ldb, work, lwork, info)
CHESV_AA_2STAGE computes the solution to system of linear equations A * X = B for HE matrices
subroutine chetrf_aa_2stage (uplo, n, a, lda, tb, ltb, ipiv, ipiv2, work, lwork, info)
CHETRF_AA_2STAGE
subroutine chetrs_aa_2stage (uplo, n, nrhs, a, lda, tb, ltb, ipiv, ipiv2, b, ldb, info)
CHETRS_AA_2STAGE
subroutine cla_syamv (uplo, n, alpha, a, lda, x, incx, beta, y, incy)
CLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.
real function cla_syrcond_c (uplo, n, a, lda, af, ldaf, ipiv, c, capply, info, work, rwork)
CLA_SYRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices.
real function cla_syrcond_x (uplo, n, a, lda, af, ldaf, ipiv, x, info, work, rwork)
CLA_SYRCOND_X computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices.
subroutine cla_syrfsx_extended (prec_type, uplo, n, nrhs, a, lda, af, ldaf, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms, err_bnds_norm, err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise, info)
CLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
real function cla_syrpvgrw (uplo, n, info, a, lda, af, ldaf, ipiv, work)
CLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
subroutine clahef_aa (uplo, j1, m, nb, a, lda, ipiv, h, ldh, work)
CLAHEF_AA
subroutine clasyf (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
CLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
subroutine clasyf_aa (uplo, j1, m, nb, a, lda, ipiv, h, ldh, work)
CLASYF_AA
subroutine clasyf_rk (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)
CLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
subroutine clasyf_rook (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
CLASYF_ROOK computes a partial factorization of a complex symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
subroutine csycon (uplo, n, a, lda, ipiv, anorm, rcond, work, info)
CSYCON
subroutine csycon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, info)
CSYCON_3
subroutine csycon_rook (uplo, n, a, lda, ipiv, anorm, rcond, work, info)
CSYCON_ROOK
subroutine csyconv (uplo, way, n, a, lda, ipiv, e, info)
CSYCONV
subroutine csyconvf (uplo, way, n, a, lda, e, ipiv, info)
CSYCONVF
subroutine csyconvf_rook (uplo, way, n, a, lda, e, ipiv, info)
CSYCONVF_ROOK
subroutine csyequb (uplo, n, a, lda, s, scond, amax, work, info)
CSYEQUB
subroutine csyrfs (uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CSYRFS
subroutine csyrfsx (uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv, s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
CSYRFSX
subroutine csysv_aa_2stage (uplo, n, nrhs, a, lda, tb, ltb, ipiv, ipiv2, b, ldb, work, lwork, info)
CSYSV_AA_2STAGE computes the solution to system of linear equations A * X = B for SY matrices
subroutine csytf2 (uplo, n, a, lda, ipiv, info)
CSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
subroutine csytf2_rk (uplo, n, a, lda, e, ipiv, info)
CSYTF2_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
subroutine csytf2_rook (uplo, n, a, lda, ipiv, info)
CSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).
subroutine csytrf (uplo, n, a, lda, ipiv, work, lwork, info)
CSYTRF
subroutine csytrf_aa (uplo, n, a, lda, ipiv, work, lwork, info)
CSYTRF_AA
subroutine csytrf_aa_2stage (uplo, n, a, lda, tb, ltb, ipiv, ipiv2, work, lwork, info)
CSYTRF_AA_2STAGE
subroutine csytrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info)
CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
subroutine csytrf_rook (uplo, n, a, lda, ipiv, work, lwork, info)
CSYTRF_ROOK
subroutine csytri (uplo, n, a, lda, ipiv, work, info)
CSYTRI
subroutine csytri2 (uplo, n, a, lda, ipiv, work, lwork, info)
CSYTRI2
subroutine csytri2x (uplo, n, a, lda, ipiv, work, nb, info)
CSYTRI2X
subroutine csytri_3 (uplo, n, a, lda, e, ipiv, work, lwork, info)
CSYTRI_3
subroutine csytri_3x (uplo, n, a, lda, e, ipiv, work, nb, info)
CSYTRI_3X
subroutine csytri_rook (uplo, n, a, lda, ipiv, work, info)
CSYTRI_ROOK
subroutine csytrs (uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
CSYTRS
subroutine csytrs2 (uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info)
CSYTRS2
subroutine csytrs_3 (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, info)
CSYTRS_3
subroutine csytrs_aa (uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info)
CSYTRS_AA
subroutine csytrs_aa_2stage (uplo, n, nrhs, a, lda, tb, ltb, ipiv, ipiv2, b, ldb, info)
CSYTRS_AA_2STAGE
subroutine csytrs_rook (uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
CSYTRS_ROOK
subroutine ctgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
CTGSYL
subroutine ctrsyl (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, info)
CTRSYL
subroutine ctrsyl3 (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, swork, ldswork, info)
CTRSYL3

Detailed Description

This is the group of complex computational functions for SY matrices

Function Documentation

  subroutine chesv_aa_2stage (character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tb, integer ltb, integer, dimension( * ) ipiv, integer, dimension( * ) ipiv2, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) work, integer lwork, integer info)

CHESV_AA_2STAGE computes the solution to system of linear equations A * X = B for HE matrices

Purpose:

     CHESV_AA_2STAGE computes the solution to a complex system of
     linear equations
        A * X = B,
     where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
     matrices.
     Aasen's 2-stage algorithm is used to factor A as
        A = U**H * T * U,  if UPLO = 'U', or
        A = L * T * L**H,  if UPLO = 'L',
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, and T is Hermitian and band. The matrix T is
     then LU-factored with partial pivoting. The factored form of A
     is then used to solve the system of equations A * X = B.
     This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the hermitian 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, L is stored below (or above) the subdiaonal blocks,
              when UPLO  is 'L' (or 'U').

    LDA

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

    TB

              TB is COMPLEX array, dimension (LTB)
              On exit, details of the LU factorization of the band matrix.

    LTB

              LTB is INTEGER
              The size of the array TB. LTB >= 4*N, internally
              used to select NB such that LTB >= (3*NB+1)*N.
              If LTB = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of LTB,
              returns this value as the first entry of TB, and
              no error message related to LTB is issued by XERBLA.

    IPIV

              IPIV is INTEGER array, dimension (N)
              On exit, it contains the details of the interchanges, i.e.,
              the row and column k of A were interchanged with the
              row and column IPIV(k).

    IPIV2

              IPIV2 is INTEGER array, dimension (N)
              On exit, it contains the details of the interchanges, i.e.,
              the row and column k of T were interchanged with the
              row and column IPIV(k).

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    WORK

              WORK is COMPLEX workspace of size LWORK

    LWORK

              LWORK is INTEGER
              The size of WORK. LWORK >= N, internally used to select NB
              such that LWORK >= N*NB.
              If LWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the WORK array,
              returns this value as the first entry of the WORK array, and
              no error message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = i, band LU factorization failed on i-th column

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 183 of file chesv_aa_2stage.f.

  subroutine chetrf_aa_2stage (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tb, integer ltb, integer, dimension( * ) ipiv, integer, dimension( * ) ipiv2, complex, dimension( * ) work, integer lwork, integer info)

CHETRF_AA_2STAGE

Purpose:

     CHETRF_AA_2STAGE computes the factorization of a real hermitian matrix A
     using the Aasen's algorithm.  The form of the factorization is
        A = U**T*T*U  or  A = L*T*L**T
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, and T is a hermitian band matrix with the
     bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is
     LU factorized with partial pivoting).
     This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the hermitian 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, L is stored below (or above) the subdiaonal blocks,
              when UPLO  is 'L' (or 'U').

    LDA

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

    TB

              TB is COMPLEX array, dimension (LTB)
              On exit, details of the LU factorization of the band matrix.

    LTB

              LTB is INTEGER
              The size of the array TB. LTB >= 4*N, internally
              used to select NB such that LTB >= (3*NB+1)*N.
              If LTB = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of LTB,
              returns this value as the first entry of TB, and
              no error message related to LTB is issued by XERBLA.

    IPIV

              IPIV is INTEGER array, dimension (N)
              On exit, it contains the details of the interchanges, i.e.,
              the row and column k of A were interchanged with the
              row and column IPIV(k).

    IPIV2

              IPIV2 is INTEGER array, dimension (N)
              On exit, it contains the details of the interchanges, i.e.,
              the row and column k of T were interchanged with the
              row and column IPIV(k).

    WORK

              WORK is COMPLEX workspace of size LWORK

    LWORK

              LWORK is INTEGER
              The size of WORK. LWORK >= N, internally used to select NB
              such that LWORK >= N*NB.
              If LWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the WORK array,
              returns this value as the first entry of the WORK array, and
              no error message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = i, band LU factorization failed on i-th column

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 158 of file chetrf_aa_2stage.f.

  subroutine chetrs_aa_2stage (character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tb, integer ltb, integer, dimension( * ) ipiv, integer, dimension( * ) ipiv2, complex, dimension( ldb, * ) b, integer ldb, integer info)

CHETRS_AA_2STAGE

Purpose:

     CHETRS_AA_2STAGE solves a system of linear equations A*X = B with a real
     hermitian matrix A using the factorization A = U**T*T*U or
     A = L*T*L**T computed by CHETRF_AA_2STAGE.

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**T*T*U;
              = 'L':  Lower triangular, form is A = L*T*L**T.

    N

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

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              Details of factors computed by CHETRF_AA_2STAGE.

    LDA

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

    TB

              TB is COMPLEX array, dimension (LTB)
              Details of factors computed by CHETRF_AA_2STAGE.

    LTB

              LTB is INTEGER
              The size of the array TB. LTB >= 4*N.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges as computed by
              CHETRF_AA_2STAGE.

    IPIV2

              IPIV2 is INTEGER array, dimension (N)
              Details of the interchanges as computed by
              CHETRF_AA_2STAGE.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 139 of file chetrs_aa_2stage.f.

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

CLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.

Purpose:

     CLA_SYAMV  performs the matrix-vector operation
             y := alpha*abs(A)*abs(x) + beta*abs(y),
     where alpha and beta are scalars, x and y are vectors and A is an
     n by n symmetric matrix.
     This function is primarily used in calculating error bounds.
     To protect against underflow during evaluation, components in
     the resulting vector are perturbed away from zero by (N+1)
     times the underflow threshold.  To prevent unnecessarily large
     errors for block-structure embedded in general matrices,
     'symbolically' zero components are not perturbed.  A zero
     entry is considered 'symbolic' if all multiplications involved
     in computing that entry have at least one zero multiplicand.

Parameters
    UPLO

              UPLO is INTEGER
               On entry, UPLO specifies whether the upper or lower
               triangular part of the array A is to be referenced as
               follows:
                  UPLO = BLAS_UPPER   Only the upper triangular part of A
                                      is to be referenced.
                  UPLO = BLAS_LOWER   Only the lower triangular part of A
                                      is to be referenced.
               Unchanged on exit.

    N

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

    ALPHA

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

    A

              A is COMPLEX array, dimension ( LDA, n ).
               Before entry, the leading m by n part of the array A must
               contain the matrix of coefficients.
               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
               ( 1 + ( n - 1 )*abs( INCX ) )
               Before entry, the incremented array X must contain the
               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 REAL .
               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 REAL array, dimension
               ( 1 + ( n - 1 )*abs( INCY ) )
               Before entry with BETA non-zero, the incremented array Y
               must contain the 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.

Further Details:

      Level 2 Blas routine.
      -- Written on 22-October-1986.
         Jack Dongarra, Argonne National Lab.
         Jeremy Du Croz, Nag Central Office.
         Sven Hammarling, Nag Central Office.
         Richard Hanson, Sandia National Labs.
      -- Modified for the absolute-value product, April 2006
         Jason Riedy, UC Berkeley

Definition at line 177 of file cla_syamv.f.

  real function cla_syrcond_c (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, real, dimension( * ) c, logical capply, integer info, complex, dimension( * ) work, real, dimension( * ) rwork)

CLA_SYRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices.

Purpose:

        CLA_SYRCOND_C Computes the infinity norm condition number of
        op(A) * inv(diag(C)) where C is a REAL vector.

Parameters
    UPLO

              UPLO is CHARACTER*1
           = 'U':  Upper triangle of A is stored;
           = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
         The number of linear equations, i.e., the order of the
         matrix A.  N >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
         On entry, the N-by-N matrix A

    LDA

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

    AF

              AF is COMPLEX array, dimension (LDAF,N)
         The block diagonal matrix D and the multipliers used to
         obtain the factor U or L as computed by CSYTRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
         Details of the interchanges and the block structure of D
         as determined by CSYTRF.

    C

              C is REAL array, dimension (N)
         The vector C in the formula op(A) * inv(diag(C)).

    CAPPLY

              CAPPLY is LOGICAL
         If .TRUE. then access the vector C in the formula above.

    INFO

              INFO is INTEGER
           = 0:  Successful exit.
         i > 0:  The ith argument is invalid.

    WORK

              WORK is COMPLEX array, dimension (2*N).
         Workspace.

    RWORK

              RWORK is REAL array, dimension (N).
         Workspace.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 136 of file cla_syrcond_c.f.

  real function cla_syrcond_x (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex, dimension( * ) x, integer info, complex, dimension( * ) work, real, dimension( * ) rwork)

CLA_SYRCOND_X computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices.

Purpose:

        CLA_SYRCOND_X Computes the infinity norm condition number of
        op(A) * diag(X) where X is a COMPLEX vector.

Parameters
    UPLO

              UPLO is CHARACTER*1
           = 'U':  Upper triangle of A is stored;
           = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
         The number of linear equations, i.e., the order of the
         matrix A.  N >= 0.

    A

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

    LDA

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

    AF

              AF is COMPLEX array, dimension (LDAF,N)
         The block diagonal matrix D and the multipliers used to
         obtain the factor U or L as computed by CSYTRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
         Details of the interchanges and the block structure of D
         as determined by CSYTRF.

    X

              X is COMPLEX array, dimension (N)
         The vector X in the formula op(A) * diag(X).

    INFO

              INFO is INTEGER
           = 0:  Successful exit.
         i > 0:  The ith argument is invalid.

    WORK

              WORK is COMPLEX array, dimension (2*N).
         Workspace.

    RWORK

              RWORK is REAL array, dimension (N).
         Workspace.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 129 of file cla_syrcond_x.f.

  subroutine cla_syrfsx_extended (integer prec_type, character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, logical colequ, real, dimension( * ) c, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldy, * ) y, integer ldy, real, dimension( * ) berr_out, integer n_norms, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, complex, dimension( * ) res, real, dimension( * ) ayb, complex, dimension( * ) dy, complex, dimension( * ) y_tail, real rcond, integer ithresh, real rthresh, real dz_ub, logical ignore_cwise, integer info)

CLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

     CLA_SYRFSX_EXTENDED improves the computed solution to a system of
     linear equations by performing extra-precise iterative refinement
     and provides error bounds and backward error estimates for the solution.
     This subroutine is called by CSYRFSX to perform iterative refinement.
     In addition to normwise error bound, the code provides maximum
     componentwise error bound if possible. See comments for ERR_BNDS_NORM
     and ERR_BNDS_COMP for details of the error bounds. Note that this
     subroutine is only responsible for setting the second fields of
     ERR_BNDS_NORM and ERR_BNDS_COMP.

Parameters
    PREC_TYPE

              PREC_TYPE is INTEGER
         Specifies the intermediate precision to be used in refinement.
         The value is defined by ILAPREC(P) where P is a CHARACTER and P
              = 'S':  Single
              = 'D':  Double
              = 'I':  Indigenous
              = 'X' or 'E':  Extra

    UPLO

              UPLO is CHARACTER*1
           = 'U':  Upper triangle of A is stored;
           = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
         The number of linear equations, i.e., the order of the
         matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
         The number of right-hand-sides, i.e., the number of columns of the
         matrix B.

    A

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

    LDA

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

    AF

              AF is COMPLEX array, dimension (LDAF,N)
         The block diagonal matrix D and the multipliers used to
         obtain the factor U or L as computed by CSYTRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
         Details of the interchanges and the block structure of D
         as determined by CSYTRF.

    COLEQU

              COLEQU is LOGICAL
         If .TRUE. then column equilibration was done to A before calling
         this routine. This is needed to compute the solution and error
         bounds correctly.

    C

              C is REAL array, dimension (N)
         The column scale factors for A. If COLEQU = .FALSE., C
         is not accessed. If C is input, each element of C should be a power
         of the radix to ensure a reliable solution and error estimates.
         Scaling by powers of the radix does not cause rounding errors unless
         the result underflows or overflows. Rounding errors during scaling
         lead to refining with a matrix that is not equivalent to the
         input matrix, producing error estimates that may not be
         reliable.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
         The right-hand-side matrix B.

    LDB

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

    Y

              Y is COMPLEX array, dimension (LDY,NRHS)
         On entry, the solution matrix X, as computed by CSYTRS.
         On exit, the improved solution matrix Y.

    LDY

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

    BERR_OUT

              BERR_OUT is REAL array, dimension (NRHS)
         On exit, BERR_OUT(j) contains the componentwise relative backward
         error for right-hand-side j from the formula
             max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
         where abs(Z) is the componentwise absolute value of the matrix
         or vector Z. This is computed by CLA_LIN_BERR.

    N_NORMS

              N_NORMS is INTEGER
         Determines which error bounds to return (see ERR_BNDS_NORM
         and ERR_BNDS_COMP).
         If N_NORMS >= 1 return normwise error bounds.
         If N_NORMS >= 2 return componentwise error bounds.

    ERR_BNDS_NORM

              ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         normwise relative error, which is defined as follows:
         Normwise relative error in the ith solution vector:
                 max_j (abs(XTRUE(j,i) - X(j,i)))
                ------------------------------
                      max_j abs(X(j,i))
         The array is indexed by the type of error information as described
         below. There currently are up to three pieces of information
         returned.
         The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
         right-hand side.
         The second index in ERR_BNDS_NORM(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * slamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated normwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * slamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*A, where S scales each row by a power of the
                  radix so all absolute row sums of Z are approximately 1.
         This subroutine is only responsible for setting the second field
         above.
         See Lapack Working Note 165 for further details and extra
         cautions.

    ERR_BNDS_COMP

              ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         componentwise relative error, which is defined as follows:
         Componentwise relative error in the ith solution vector:
                        abs(XTRUE(j,i) - X(j,i))
                 max_j ----------------------
                             abs(X(j,i))
         The array is indexed by the right-hand side i (on which the
         componentwise relative error depends), and the type of error
         information as described below. There currently are up to three
         pieces of information returned for each right-hand side. If
         componentwise accuracy is not requested (PARAMS(3) = 0.0), then
         ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
         the first (:,N_ERR_BNDS) entries are returned.
         The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
         right-hand side.
         The second index in ERR_BNDS_COMP(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * slamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated componentwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * slamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*(A*diag(x)), where x is the solution for the
                  current right-hand side and S scales each row of
                  A*diag(x) by a power of the radix so all absolute row
                  sums of Z are approximately 1.
         This subroutine is only responsible for setting the second field
         above.
         See Lapack Working Note 165 for further details and extra
         cautions.

    RES

              RES is COMPLEX array, dimension (N)
         Workspace to hold the intermediate residual.

    AYB

              AYB is REAL array, dimension (N)
         Workspace.

    DY

              DY is COMPLEX array, dimension (N)
         Workspace to hold the intermediate solution.

    Y_TAIL

              Y_TAIL is COMPLEX array, dimension (N)
         Workspace to hold the trailing bits of the intermediate solution.

    RCOND

              RCOND is REAL
         Reciprocal scaled condition number.  This is an estimate of the
         reciprocal Skeel condition number of the matrix A after
         equilibration (if done).  If this is less than the machine
         precision (in particular, if it is zero), the matrix is singular
         to working precision.  Note that the error may still be small even
         if this number is very small and the matrix appears ill-
         conditioned.

    ITHRESH

              ITHRESH is INTEGER
         The maximum number of residual computations allowed for
         refinement. The default is 10. For 'aggressive' set to 100 to
         permit convergence using approximate factorizations or
         factorizations other than LU. If the factorization uses a
         technique other than Gaussian elimination, the guarantees in
         ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.

    RTHRESH

              RTHRESH is REAL
         Determines when to stop refinement if the error estimate stops
         decreasing. Refinement will stop when the next solution no longer
         satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
         the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
         default value is 0.5. For 'aggressive' set to 0.9 to permit
         convergence on extremely ill-conditioned matrices. See LAWN 165
         for more details.

    DZ_UB

              DZ_UB is REAL
         Determines when to start considering componentwise convergence.
         Componentwise convergence is only considered after each component
         of the solution Y is stable, which we define as the relative
         change in each component being less than DZ_UB. The default value
         is 0.25, requiring the first bit to be stable. See LAWN 165 for
         more details.

    IGNORE_CWISE

              IGNORE_CWISE is LOGICAL
         If .TRUE. then ignore componentwise convergence. Default value
         is .FALSE..

    INFO

              INFO is INTEGER
           = 0:  Successful exit.
           < 0:  if INFO = -i, the ith argument to CLA_SYRFSX_EXTENDED had an illegal
                 value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 388 of file cla_syrfsx_extended.f.

  real function cla_syrpvgrw (character*1 uplo, integer n, integer info, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, real, dimension( * ) work)

CLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.

Purpose:

     CLA_SYRPVGRW computes the reciprocal pivot growth factor
     norm(A)/norm(U). The 'max absolute element' norm is used. If this is
     much less than 1, the stability of the LU factorization of the
     (equilibrated) matrix A could be poor. This also means that the
     solution X, estimated condition numbers, and error bounds could be
     unreliable.

Parameters
    UPLO

              UPLO is CHARACTER*1
           = 'U':  Upper triangle of A is stored;
           = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
         The number of linear equations, i.e., the order of the
         matrix A.  N >= 0.

    INFO

              INFO is INTEGER
         The value of INFO returned from CSYTRF, .i.e., the pivot in
         column INFO is exactly 0.

    A

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

    LDA

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

    AF

              AF is COMPLEX array, dimension (LDAF,N)
         The block diagonal matrix D and the multipliers used to
         obtain the factor U or L as computed by CSYTRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
         Details of the interchanges and the block structure of D
         as determined by CSYTRF.

    WORK

              WORK is REAL array, dimension (2*N)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 121 of file cla_syrpvgrw.f.

  subroutine clahef_aa (character uplo, integer j1, integer m, integer nb, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( * ) work)

CLAHEF_AA

Purpose:

     CLAHEF_AA factorizes a panel of a complex hermitian matrix A using
     the Aasen's algorithm. The panel consists of a set of NB rows of A
     when UPLO is U, or a set of NB columns when UPLO is L.
     In order to factorize the panel, the Aasen's algorithm requires the
     last row, or column, of the previous panel. The first row, or column,
     of A is set to be the first row, or column, of an identity matrix,
     which is used to factorize the first panel.
     The resulting J-th row of U, or J-th column of L, is stored in the
     (J-1)-th row, or column, of A (without the unit diagonals), while
     the diagonal and subdiagonal of A are overwritten by those of T.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    J1

              J1 is INTEGER
              The location of the first row, or column, of the panel
              within the submatrix of A, passed to this routine, e.g.,
              when called by CHETRF_AA, for the first panel, J1 is 1,
              while for the remaining panels, J1 is 2.

    M

              M is INTEGER
              The dimension of the submatrix. M >= 0.

    NB

              NB is INTEGER
              The dimension of the panel to be facotorized.

    A

              A is COMPLEX array, dimension (LDA,M) for
              the first panel, while dimension (LDA,M+1) for the
              remaining panels.
              On entry, A contains the last row, or column, of
              the previous panel, and the trailing submatrix of A
              to be factorized, except for the first panel, only
              the panel is passed.
              On exit, the leading panel is factorized.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the row and column interchanges,
              the row and column k were interchanged with the row and
              column IPIV(k).

    H

              H is COMPLEX workspace, dimension (LDH,NB).

    LDH

              LDH is INTEGER
              The leading dimension of the workspace H. LDH >= max(1,M).

    WORK

              WORK is COMPLEX workspace, dimension (M).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 142 of file clahef_aa.f.

  subroutine clasyf (character uplo, integer n, integer nb, integer kb, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( ldw, * ) w, integer ldw, integer info)

CLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.

Purpose:

     CLASYF computes a partial factorization of a complex symmetric matrix
     A using the Bunch-Kaufman diagonal pivoting method. The partial
     factorization has the form:
     A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
           ( 0  U22 ) (  0   D  ) ( U12**T U22**T )
     A  =  ( L11  0 ) ( D    0  ) ( L11**T L21**T )  if UPLO = 'L'
           ( L21  I ) ( 0   A22 ) (  0       I    )
     where the order of D is at most NB. The actual order is returned in
     the argument KB, and is either NB or NB-1, or N if N <= NB.
     Note that U**T denotes the transpose of U.
     CLASYF is an auxiliary routine called by CSYTRF. It uses blocked code
     (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
     A22 (if UPLO = 'L').

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.

    NB

              NB is INTEGER
              The maximum number of columns of the matrix A that should be
              factored.  NB should be at least 2 to allow for 2-by-2 pivot
              blocks.

    KB

              KB is INTEGER
              The number of columns of A that were actually factored.
              KB is either NB-1 or NB, or N if N <= NB.

    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, A contains details of the partial factorization.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If UPLO = 'U':
                 Only the last KB elements of IPIV are set.
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) = IPIV(k-1) < 0, then rows and columns
                 k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
                 is a 2-by-2 diagonal block.
              If UPLO = 'L':
                 Only the first KB elements of IPIV are set.
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) = IPIV(k+1) < 0, then rows and columns
                 k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
                 is a 2-by-2 diagonal block.

    W

              W is COMPLEX array, dimension (LDW,NB)

    LDW

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

    INFO

              INFO is INTEGER
              = 0: successful exit
              > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
                   has been completed, but the block diagonal matrix D is
                   exactly singular.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      November 2013,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley

Definition at line 176 of file clasyf.f.

  subroutine clasyf_aa (character uplo, integer j1, integer m, integer nb, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( * ) work)

CLASYF_AA

Purpose:

     DLATRF_AA factorizes a panel of a complex symmetric matrix A using
     the Aasen's algorithm. The panel consists of a set of NB rows of A
     when UPLO is U, or a set of NB columns when UPLO is L.
     In order to factorize the panel, the Aasen's algorithm requires the
     last row, or column, of the previous panel. The first row, or column,
     of A is set to be the first row, or column, of an identity matrix,
     which is used to factorize the first panel.
     The resulting J-th row of U, or J-th column of L, is stored in the
     (J-1)-th row, or column, of A (without the unit diagonals), while
     the diagonal and subdiagonal of A are overwritten by those of T.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    J1

              J1 is INTEGER
              The location of the first row, or column, of the panel
              within the submatrix of A, passed to this routine, e.g.,
              when called by CSYTRF_AA, for the first panel, J1 is 1,
              while for the remaining panels, J1 is 2.

    M

              M is INTEGER
              The dimension of the submatrix. M >= 0.

    NB

              NB is INTEGER
              The dimension of the panel to be facotorized.

    A

              A is COMPLEX array, dimension (LDA,M) for
              the first panel, while dimension (LDA,M+1) for the
              remaining panels.
              On entry, A contains the last row, or column, of
              the previous panel, and the trailing submatrix of A
              to be factorized, except for the first panel, only
              the panel is passed.
              On exit, the leading panel is factorized.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (M)
              Details of the row and column interchanges,
              the row and column k were interchanged with the row and
              column IPIV(k).

    H

              H is COMPLEX workspace, dimension (LDH,NB).

    LDH

              LDH is INTEGER
              The leading dimension of the workspace H. LDH >= max(1,M).

    WORK

              WORK is COMPLEX workspace, dimension (M).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 142 of file clasyf_aa.f.

  subroutine clasyf_rk (character uplo, integer n, integer nb, integer kb, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( ldw, * ) w, integer ldw, integer info)

CLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.

Purpose:

     CLASYF_RK computes a partial factorization of a complex symmetric
     matrix A using the bounded Bunch-Kaufman (rook) diagonal
     pivoting method. The partial factorization has the form:
     A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
           ( 0  U22 ) (  0   D  ) ( U12**T U22**T )
     A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L',
           ( L21  I ) (  0  A22 ) (  0       I    )
     where the order of D is at most NB. The actual order is returned in
     the argument KB, and is either NB or NB-1, or N if N <= NB.
     CLASYF_RK is an auxiliary routine called by CSYTRF_RK. It uses
     blocked code (calling Level 3 BLAS) to update the submatrix
     A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

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.

    NB

              NB is INTEGER
              The maximum number of columns of the matrix A that should be
              factored.  NB should be at least 2 to allow for 2-by-2 pivot
              blocks.

    KB

              KB is INTEGER
              The number of columns of A that were actually factored.
              KB is either NB-1 or NB, or N if N <= NB.

    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, contains:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    are stored on exit in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is COMPLEX array, dimension (N)
              On exit, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
              If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is set to 0 in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              IPIV describes the permutation matrix P in the factorization
              of matrix A as follows. The absolute value of IPIV(k)
              represents the index of row and column that were
              interchanged with the k-th row and column. The value of UPLO
              describes the order in which the interchanges were applied.
              Also, the sign of IPIV represents the block structure of
              the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
              diagonal blocks which correspond to 1 or 2 interchanges
              at each factorization step.
              If UPLO = 'U',
              ( in factorization order, k decreases from N to 1 ):
                a) A single positive entry IPIV(k) > 0 means:
                   D(k,k) is a 1-by-1 diagonal block.
                   If IPIV(k) != k, rows and columns k and IPIV(k) were
                   interchanged in the submatrix A(1:N,N-KB+1:N);
                   If IPIV(k) = k, no interchange occurred.
                b) A pair of consecutive negative entries
                   IPIV(k) < 0 and IPIV(k-1) < 0 means:
                   D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
                   (NOTE: negative entries in IPIV appear ONLY in pairs).
                   1) If -IPIV(k) != k, rows and columns
                      k and -IPIV(k) were interchanged
                      in the matrix A(1:N,N-KB+1:N).
                      If -IPIV(k) = k, no interchange occurred.
                   2) If -IPIV(k-1) != k-1, rows and columns
                      k-1 and -IPIV(k-1) were interchanged
                      in the submatrix A(1:N,N-KB+1:N).
                      If -IPIV(k-1) = k-1, no interchange occurred.
                c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
                d) NOTE: Any entry IPIV(k) is always NONZERO on output.
              If UPLO = 'L',
              ( in factorization order, k increases from 1 to N ):
                a) A single positive entry IPIV(k) > 0 means:
                   D(k,k) is a 1-by-1 diagonal block.
                   If IPIV(k) != k, rows and columns k and IPIV(k) were
                   interchanged in the submatrix A(1:N,1:KB).
                   If IPIV(k) = k, no interchange occurred.
                b) A pair of consecutive negative entries
                   IPIV(k) < 0 and IPIV(k+1) < 0 means:
                   D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
                   (NOTE: negative entries in IPIV appear ONLY in pairs).
                   1) If -IPIV(k) != k, rows and columns
                      k and -IPIV(k) were interchanged
                      in the submatrix A(1:N,1:KB).
                      If -IPIV(k) = k, no interchange occurred.
                   2) If -IPIV(k+1) != k+1, rows and columns
                      k-1 and -IPIV(k-1) were interchanged
                      in the submatrix A(1:N,1:KB).
                      If -IPIV(k+1) = k+1, no interchange occurred.
                c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
                d) NOTE: Any entry IPIV(k) is always NONZERO on output.

    W

              W is COMPLEX array, dimension (LDW,NB)

    LDW

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

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: If INFO = -k, the k-th argument had an illegal value
              > 0: If INFO = k, the matrix A is singular, because:
                     If UPLO = 'U': column k in the upper
                     triangular part of A contains all zeros.
                     If UPLO = 'L': column k in the lower
                     triangular part of A contains all zeros.
                   Therefore D(k,k) is exactly zero, and superdiagonal
                   elements of column k of U (or subdiagonal elements of
                   column k of L ) are all zeros. The factorization has
                   been completed, but the block diagonal matrix D is
                   exactly singular, and division by zero will occur if
                   it is used to solve a system of equations.
                   NOTE: INFO only stores the first occurrence of
                   a singularity, any subsequent occurrence of singularity
                   is not stored in INFO even though the factorization
                   always completes.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      December 2016,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 260 of file clasyf_rk.f.

  subroutine clasyf_rook (character uplo, integer n, integer nb, integer kb, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( ldw, * ) w, integer ldw, integer info)

CLASYF_ROOK computes a partial factorization of a complex symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.

Purpose:

     CLASYF_ROOK computes a partial factorization of a complex symmetric
     matrix A using the bounded Bunch-Kaufman ('rook') diagonal
     pivoting method. The partial factorization has the form:
     A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
           ( 0  U22 ) (  0   D  ) ( U12**T U22**T )
     A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L'
           ( L21  I ) (  0  A22 ) (  0       I    )
     where the order of D is at most NB. The actual order is returned in
     the argument KB, and is either NB or NB-1, or N if N <= NB.
     CLASYF_ROOK is an auxiliary routine called by CSYTRF_ROOK. It uses
     blocked code (calling Level 3 BLAS) to update the submatrix
     A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

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.

    NB

              NB is INTEGER
              The maximum number of columns of the matrix A that should be
              factored.  NB should be at least 2 to allow for 2-by-2 pivot
              blocks.

    KB

              KB is INTEGER
              The number of columns of A that were actually factored.
              KB is either NB-1 or NB, or N if N <= NB.

    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, A contains details of the partial factorization.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If UPLO = 'U':
                 Only the last KB elements of IPIV are set.
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
                 columns k and -IPIV(k) were interchanged and rows and
                 columns k-1 and -IPIV(k-1) were inerchaged,
                 D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
              If UPLO = 'L':
                 Only the first KB elements of IPIV are set.
                 If IPIV(k) > 0, then rows and columns k and IPIV(k)
                 were interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
                 columns k and -IPIV(k) were interchanged and rows and
                 columns k+1 and -IPIV(k+1) were inerchaged,
                 D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

    W

              W is COMPLEX array, dimension (LDW,NB)

    LDW

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

    INFO

              INFO is INTEGER
              = 0: successful exit
              > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
                   has been completed, but the block diagonal matrix D is
                   exactly singular.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      November 2013,     Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 182 of file clasyf_rook.f.

  subroutine csycon (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, real anorm, real rcond, complex, dimension( * ) work, integer info)

CSYCON

Purpose:

     CSYCON estimates the reciprocal of the condition number (in the
     1-norm) of a complex symmetric matrix A using the factorization
     A = U*D*U**T or A = L*D*L**T computed by CSYTRF.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

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)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by CSYTRF.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF.

    ANORM

              ANORM is REAL
              The 1-norm of the original matrix A.

    RCOND

              RCOND is REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is COMPLEX array, dimension (2*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 123 of file csycon.f.

  subroutine csycon_3 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, real anorm, real rcond, complex, dimension( * ) work, integer info)

CSYCON_3

Purpose:

     CSYCON_3 estimates the reciprocal of the condition number (in the
     1-norm) of a complex symmetric matrix A using the factorization
     computed by CSYTRF_RK or CSYTRF_BK:
        A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**T (or L**T) is the transpose of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is symmetric and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
     This routine uses BLAS3 solver CSYTRS_3.

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 = P*U*D*(U**T)*(P**T);
              = 'L':  Lower triangular, form is A = P*L*D*(L**T)*(P**T).

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              Diagonal of the block diagonal matrix D and factors U or L
              as computed by CSYTRF_RK and CSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    should be provided on entry in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is COMPLEX array, dimension (N)
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is not referenced in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF_RK or CSYTRF_BK.

    ANORM

              ANORM is REAL
              The 1-norm of the original matrix A.

    RCOND

              RCOND is REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is COMPLEX array, dimension (2*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      June 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 164 of file csycon_3.f.

  subroutine csycon_rook (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, real anorm, real rcond, complex, dimension( * ) work, integer info)

CSYCON_ROOK

Purpose:

     CSYCON_ROOK estimates the reciprocal of the condition number (in the
     1-norm) of a complex symmetric matrix A using the factorization
     A = U*D*U**T or A = L*D*L**T computed by CSYTRF_ROOK.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

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)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by CSYTRF_ROOK.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF_ROOK.

    ANORM

              ANORM is REAL
              The 1-norm of the original matrix A.

    RCOND

              RCOND is REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is COMPLEX array, dimension (2*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

       April 2012, Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 137 of file csycon_rook.f.

  subroutine csyconv (character uplo, character way, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( * ) e, integer info)

CSYCONV

Purpose:

     CSYCONV convert A given by TRF into L and D and vice-versa.
     Get Non-diag elements of D (returned in workspace) and
     apply or reverse permutation done in TRF.

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.

    WAY

              WAY is CHARACTER*1
              = 'C': Convert
              = 'R': Revert

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by CSYTRF.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF.

    E

              E is COMPLEX array, dimension (N)
              E stores the supdiagonal/subdiagonal of the symmetric 1-by-1
              or 2-by-2 block diagonal matrix D in LDLT.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 113 of file csyconv.f.

  subroutine csyconvf (character uplo, character way, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, integer info)

CSYCONVF

Purpose:

     If parameter WAY = 'C':
     CSYCONVF converts the factorization output format used in
     CSYTRF provided on entry in parameter A into the factorization
     output format used in CSYTRF_RK (or CSYTRF_BK) that is stored
     on exit in parameters A and E. It also converts in place details of
     the intechanges stored in IPIV from the format used in CSYTRF into
     the format used in CSYTRF_RK (or CSYTRF_BK).
     If parameter WAY = 'R':
     CSYCONVF performs the conversion in reverse direction, i.e.
     converts the factorization output format used in CSYTRF_RK
     (or CSYTRF_BK) provided on entry in parameters A and E into
     the factorization output format used in CSYTRF that is stored
     on exit in parameter A. It also converts in place details of
     the intechanges stored in IPIV from the format used in CSYTRF_RK
     (or CSYTRF_BK) into the format used in CSYTRF.
     CSYCONVF can also convert in Hermitian matrix case, i.e. between
     formats used in CHETRF and CHETRF_RK (or CHETRF_BK).

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the details of the factorization are
              stored as an upper or lower triangular matrix A.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    WAY

              WAY is CHARACTER*1
              = 'C': Convert
              = 'R': Revert

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              1) If WAY ='C':
              On entry, contains factorization details in format used in
              CSYTRF:
                a) all elements of the symmetric block diagonal
                   matrix D on the diagonal of A and on superdiagonal
                   (or subdiagonal) of A, and
                b) If UPLO = 'U': multipliers used to obtain factor U
                   in the superdiagonal part of A.
                   If UPLO = 'L': multipliers used to obtain factor L
                   in the superdiagonal part of A.
              On exit, contains factorization details in format used in
              CSYTRF_RK or CSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    are stored on exit in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.
              2) If WAY = 'R':
              On entry, contains factorization details in format used in
              CSYTRF_RK or CSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    are stored on exit in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.
              On exit, contains factorization details in format used in
              CSYTRF:
                a) all elements of the symmetric block diagonal
                   matrix D on the diagonal of A and on superdiagonal
                   (or subdiagonal) of A, and
                b) If UPLO = 'U': multipliers used to obtain factor U
                   in the superdiagonal part of A.
                   If UPLO = 'L': multipliers used to obtain factor L
                   in the superdiagonal part of A.

    LDA

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

    E

              E is COMPLEX array, dimension (N)
              1) If WAY ='C':
              On entry, just a workspace.
              On exit, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
              If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
              2) If WAY = 'R':
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              On exit, is not changed

    IPIV

              IPIV is INTEGER array, dimension (N)
              1) If WAY ='C':
              On entry, details of the interchanges and the block
              structure of D in the format used in CSYTRF.
              On exit, details of the interchanges and the block
              structure of D in the format used in CSYTRF_RK
              ( or CSYTRF_BK).
              1) If WAY ='R':
              On entry, details of the interchanges and the block
              structure of D in the format used in CSYTRF_RK
              ( or CSYTRF_BK).
              On exit, details of the interchanges and the block
              structure of D in the format used in CSYTRF.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      November 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley

Definition at line 208 of file csyconvf.f.

  subroutine csyconvf_rook (character uplo, character way, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, integer info)

CSYCONVF_ROOK

Purpose:

     If parameter WAY = 'C':
     CSYCONVF_ROOK converts the factorization output format used in
     CSYTRF_ROOK provided on entry in parameter A into the factorization
     output format used in CSYTRF_RK (or CSYTRF_BK) that is stored
     on exit in parameters A and E. IPIV format for CSYTRF_ROOK and
     CSYTRF_RK (or CSYTRF_BK) is the same and is not converted.
     If parameter WAY = 'R':
     CSYCONVF_ROOK performs the conversion in reverse direction, i.e.
     converts the factorization output format used in CSYTRF_RK
     (or CSYTRF_BK) provided on entry in parameters A and E into
     the factorization output format used in CSYTRF_ROOK that is stored
     on exit in parameter A. IPIV format for CSYTRF_ROOK and
     CSYTRF_RK (or CSYTRF_BK) is the same and is not converted.
     CSYCONVF_ROOK can also convert in Hermitian matrix case, i.e. between
     formats used in CHETRF_ROOK and CHETRF_RK (or CHETRF_BK).

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the details of the factorization are
              stored as an upper or lower triangular matrix A.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    WAY

              WAY is CHARACTER*1
              = 'C': Convert
              = 'R': Revert

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              1) If WAY ='C':
              On entry, contains factorization details in format used in
              CSYTRF_ROOK:
                a) all elements of the symmetric block diagonal
                   matrix D on the diagonal of A and on superdiagonal
                   (or subdiagonal) of A, and
                b) If UPLO = 'U': multipliers used to obtain factor U
                   in the superdiagonal part of A.
                   If UPLO = 'L': multipliers used to obtain factor L
                   in the superdiagonal part of A.
              On exit, contains factorization details in format used in
              CSYTRF_RK or CSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    are stored on exit in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.
              2) If WAY = 'R':
              On entry, contains factorization details in format used in
              CSYTRF_RK or CSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    are stored on exit in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.
              On exit, contains factorization details in format used in
              CSYTRF_ROOK:
                a) all elements of the symmetric block diagonal
                   matrix D on the diagonal of A and on superdiagonal
                   (or subdiagonal) of A, and
                b) If UPLO = 'U': multipliers used to obtain factor U
                   in the superdiagonal part of A.
                   If UPLO = 'L': multipliers used to obtain factor L
                   in the superdiagonal part of A.

    LDA

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

    E

              E is COMPLEX array, dimension (N)
              1) If WAY ='C':
              On entry, just a workspace.
              On exit, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
              If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
              2) If WAY = 'R':
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              On exit, is not changed

    IPIV

              IPIV is INTEGER array, dimension (N)
              On entry, details of the interchanges and the block
              structure of D as determined:
              1) by CSYTRF_ROOK, if WAY ='C';
              2) by CSYTRF_RK (or CSYTRF_BK), if WAY ='R'.
              The IPIV format is the same for all these routines.
              On exit, is not changed.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      November 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley

Definition at line 199 of file csyconvf_rook.f.

  subroutine csyequb (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) s, real scond, real amax, complex, dimension( * ) work, integer info)

CSYEQUB

Purpose:

     CSYEQUB computes row and column scalings intended to equilibrate a
     symmetric matrix A (with respect to the Euclidean norm) and reduce
     its condition number. The scale factors S are computed by the BIN
     algorithm (see references) so that the scaled matrix B with elements
     B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
     the smallest possible condition number over all possible diagonal
     scalings.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              The N-by-N symmetric matrix whose scaling factors are to be
              computed.

    LDA

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

    S

              S is REAL array, dimension (N)
              If INFO = 0, S contains the scale factors for A.

    SCOND

              SCOND is REAL
              If INFO = 0, S contains the ratio of the smallest S(i) to
              the largest S(i). If SCOND >= 0.1 and AMAX is neither too
              large nor too small, it is not worth scaling by S.

    AMAX

              AMAX is REAL
              Largest absolute value of any matrix element. If AMAX is
              very close to overflow or very close to underflow, the
              matrix should be scaled.

    WORK

              WORK is COMPLEX array, dimension (2*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the i-th diagonal element is nonpositive.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

References:
    Livne, O.E. and Golub, G.H., 'Scaling by Binormalization',
     Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
     DOI 10.1023/B:NUMA.0000016606.32820.69
     Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679

Definition at line 131 of file csyequb.f.

  subroutine csyrfs (character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CSYRFS

Purpose:

     CSYRFS improves the computed solution to a system of linear
     equations when the coefficient matrix is symmetric indefinite, and
     provides error bounds and backward error estimates for the solution.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X.  NRHS >= 0.

    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(1,N).

    AF

              AF is COMPLEX array, dimension (LDAF,N)
              The factored form of the matrix A.  AF contains the block
              diagonal matrix D and the multipliers used to obtain the
              factor U or L from the factorization A = U*D*U**T or
              A = L*D*L**T as computed by CSYTRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              The right hand side matrix B.

    LDB

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

    X

              X is COMPLEX array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by CSYTRS.
              On exit, the improved solution matrix X.

    LDX

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

    FERR

              FERR is REAL array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j), FERR(j)
              is an estimated upper bound for the magnitude of the largest
              element in (X(j) - XTRUE) divided by the magnitude of the
              largest element in X(j).  The estimate is as reliable as
              the estimate for RCOND, and is almost always a slight
              overestimate of the true error.

    BERR

              BERR is REAL array, dimension (NRHS)
              The componentwise relative backward error of each solution
              vector X(j) (i.e., the smallest relative change in
              any element of A or B that makes X(j) an exact solution).

    WORK

              WORK is COMPLEX array, dimension (2*N)

    RWORK

              RWORK is REAL array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Internal Parameters:

      ITMAX is the maximum number of steps of iterative refinement.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 190 of file csyrfs.f.

  subroutine csyrfsx (character uplo, character equed, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, real, dimension( * ) s, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer nparams, real, dimension( * ) params, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CSYRFSX

Purpose:

        CSYRFSX improves the computed solution to a system of linear
        equations when the coefficient matrix is symmetric indefinite, and
        provides error bounds and backward error estimates for the
        solution.  In addition to normwise error bound, the code provides
        maximum componentwise error bound if possible.  See comments for
        ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds.
        The original system of linear equations may have been equilibrated
        before calling this routine, as described by arguments EQUED and S
        below. In this case, the solution and error bounds returned are
        for the original unequilibrated system.

         Some optional parameters are bundled in the PARAMS array.  These
         settings determine how refinement is performed, but often the
         defaults are acceptable.  If the defaults are acceptable, users
         can pass NPARAMS = 0 which prevents the source code from accessing
         the PARAMS argument.

Parameters
    UPLO

              UPLO is CHARACTER*1
           = 'U':  Upper triangle of A is stored;
           = 'L':  Lower triangle of A is stored.

    EQUED

              EQUED is CHARACTER*1
         Specifies the form of equilibration that was done to A
         before calling this routine. This is needed to compute
         the solution and error bounds correctly.
           = 'N':  No equilibration
           = 'Y':  Both row and column equilibration, i.e., A has been
                   replaced by diag(S) * A * diag(S).
                   The right hand side B has been changed accordingly.

    N

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

    NRHS

              NRHS is INTEGER
         The number of right hand sides, i.e., the number of columns
         of the matrices B and X.  NRHS >= 0.

    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(1,N).

    AF

              AF is COMPLEX array, dimension (LDAF,N)
         The factored form of the matrix A.  AF contains the block
         diagonal matrix D and the multipliers used to obtain the
         factor U or L from the factorization A = U*D*U**T or A =
         L*D*L**T as computed by CSYTRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
         Details of the interchanges and the block structure of D
         as determined by CSYTRF.

    S

              S is REAL array, dimension (N)
         The scale factors for A.  If EQUED = 'Y', A is multiplied on
         the left and right by diag(S).  S is an input argument if FACT =
         'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
         = 'Y', each element of S must be positive.  If S is output, each
         element of S is a power of the radix. If S is input, each element
         of S should be a power of the radix to ensure a reliable solution
         and error estimates. Scaling by powers of the radix does not cause
         rounding errors unless the result underflows or overflows.
         Rounding errors during scaling lead to refining with a matrix that
         is not equivalent to the input matrix, producing error estimates
         that may not be reliable.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
         The right hand side matrix B.

    LDB

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

    X

              X is COMPLEX array, dimension (LDX,NRHS)
         On entry, the solution matrix X, as computed by CGETRS.
         On exit, the improved solution matrix X.

    LDX

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

    RCOND

              RCOND is REAL
         Reciprocal scaled condition number.  This is an estimate of the
         reciprocal Skeel condition number of the matrix A after
         equilibration (if done).  If this is less than the machine
         precision (in particular, if it is zero), the matrix is singular
         to working precision.  Note that the error may still be small even
         if this number is very small and the matrix appears ill-
         conditioned.

    BERR

              BERR is REAL array, dimension (NRHS)
         Componentwise relative backward error.  This is the
         componentwise relative backward error of each solution vector X(j)
         (i.e., the smallest relative change in any element of A or B that
         makes X(j) an exact solution).

    N_ERR_BNDS

              N_ERR_BNDS is INTEGER
         Number of error bounds to return for each right hand side
         and each type (normwise or componentwise).  See ERR_BNDS_NORM and
         ERR_BNDS_COMP below.

    ERR_BNDS_NORM

              ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         normwise relative error, which is defined as follows:
         Normwise relative error in the ith solution vector:
                 max_j (abs(XTRUE(j,i) - X(j,i)))
                ------------------------------
                      max_j abs(X(j,i))
         The array is indexed by the type of error information as described
         below. There currently are up to three pieces of information
         returned.
         The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
         right-hand side.
         The second index in ERR_BNDS_NORM(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * slamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated normwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * slamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*A, where S scales each row by a power of the
                  radix so all absolute row sums of Z are approximately 1.
         See Lapack Working Note 165 for further details and extra
         cautions.

    ERR_BNDS_COMP

              ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         componentwise relative error, which is defined as follows:
         Componentwise relative error in the ith solution vector:
                        abs(XTRUE(j,i) - X(j,i))
                 max_j ----------------------
                             abs(X(j,i))
         The array is indexed by the right-hand side i (on which the
         componentwise relative error depends), and the type of error
         information as described below. There currently are up to three
         pieces of information returned for each right-hand side. If
         componentwise accuracy is not requested (PARAMS(3) = 0.0), then
         ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
         the first (:,N_ERR_BNDS) entries are returned.
         The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
         right-hand side.
         The second index in ERR_BNDS_COMP(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * slamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated componentwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * slamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*(A*diag(x)), where x is the solution for the
                  current right-hand side and S scales each row of
                  A*diag(x) by a power of the radix so all absolute row
                  sums of Z are approximately 1.
         See Lapack Working Note 165 for further details and extra
         cautions.

    NPARAMS

              NPARAMS is INTEGER
         Specifies the number of parameters set in PARAMS.  If <= 0, the
         PARAMS array is never referenced and default values are used.

    PARAMS

              PARAMS is REAL array, dimension NPARAMS
         Specifies algorithm parameters.  If an entry is < 0.0, then
         that entry will be filled with default value used for that
         parameter.  Only positions up to NPARAMS are accessed; defaults
         are used for higher-numbered parameters.
           PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
                refinement or not.
             Default: 1.0
                = 0.0:  No refinement is performed, and no error bounds are
                        computed.
                = 1.0:  Use the double-precision refinement algorithm,
                        possibly with doubled-single computations if the
                        compilation environment does not support DOUBLE
                        PRECISION.
                  (other values are reserved for future use)
           PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
                computations allowed for refinement.
             Default: 10
             Aggressive: Set to 100 to permit convergence using approximate
                         factorizations or factorizations other than LU. If
                         the factorization uses a technique other than
                         Gaussian elimination, the guarantees in
                         err_bnds_norm and err_bnds_comp may no longer be
                         trustworthy.
           PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
                will attempt to find a solution with small componentwise
                relative error in the double-precision algorithm.  Positive
                is true, 0.0 is false.
             Default: 1.0 (attempt componentwise convergence)

    WORK

              WORK is COMPLEX array, dimension (2*N)

    RWORK

              RWORK is REAL array, dimension (2*N)

    INFO

              INFO is INTEGER
           = 0:  Successful exit. The solution to every right-hand side is
             guaranteed.
           < 0:  If INFO = -i, the i-th argument had an illegal value
           > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
             has been completed, but the factor U is exactly singular, so
             the solution and error bounds could not be computed. RCOND = 0
             is returned.
           = N+J: The solution corresponding to the Jth right-hand side is
             not guaranteed. The solutions corresponding to other right-
             hand sides K with K > J may not be guaranteed as well, but
             only the first such right-hand side is reported. If a small
             componentwise error is not requested (PARAMS(3) = 0.0) then
             the Jth right-hand side is the first with a normwise error
             bound that is not guaranteed (the smallest J such
             that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
             the Jth right-hand side is the first with either a normwise or
             componentwise error bound that is not guaranteed (the smallest
             J such that either ERR_BNDS_NORM(J,1) = 0.0 or
             ERR_BNDS_COMP(J,1) = 0.0). See the definition of
             ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
             about all of the right-hand sides check ERR_BNDS_NORM or
             ERR_BNDS_COMP.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 398 of file csyrfsx.f.

  subroutine csysv_aa_2stage (character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tb, integer ltb, integer, dimension( * ) ipiv, integer, dimension( * ) ipiv2, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) work, integer lwork, integer info)

CSYSV_AA_2STAGE computes the solution to system of linear equations A * X = B for SY matrices

Purpose:

     CSYSV_AA_2STAGE computes the solution to a complex system of
     linear equations
        A * X = B,
     where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
     matrices.
     Aasen's 2-stage algorithm is used to factor A as
        A = U**T * T * U,  if UPLO = 'U', or
        A = L * T * L**T,  if UPLO = 'L',
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, and T is symmetric and band. The matrix T is
     then LU-factored with partial pivoting. The factored form of A
     is then used to solve the system of equations A * X = B.
     This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 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, L is stored below (or above) the subdiaonal blocks,
              when UPLO  is 'L' (or 'U').

    LDA

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

    TB

              TB is COMPLEX array, dimension (LTB)
              On exit, details of the LU factorization of the band matrix.

    LTB

              LTB is INTEGER
              The size of the array TB. LTB >= 4*N, internally
              used to select NB such that LTB >= (3*NB+1)*N.
              If LTB = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of LTB,
              returns this value as the first entry of TB, and
              no error message related to LTB is issued by XERBLA.

    IPIV

              IPIV is INTEGER array, dimension (N)
              On exit, it contains the details of the interchanges, i.e.,
              the row and column k of A were interchanged with the
              row and column IPIV(k).

    IPIV2

              IPIV2 is INTEGER array, dimension (N)
              On exit, it contains the details of the interchanges, i.e.,
              the row and column k of T were interchanged with the
              row and column IPIV(k).

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    WORK

              WORK is COMPLEX workspace of size LWORK

    LWORK

              LWORK is INTEGER
              The size of WORK. LWORK >= N, internally used to select NB
              such that LWORK >= N*NB.
              If LWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the WORK array,
              returns this value as the first entry of the WORK array, and
              no error message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = i, band LU factorization failed on i-th column

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 183 of file csysv_aa_2stage.f.

  subroutine csytf2 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

CSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

Purpose:

     CSYTF2 computes the factorization of a complex symmetric matrix A
     using the Bunch-Kaufman diagonal pivoting method:
        A = U*D*U**T  or  A = L*D*L**T
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, U**T is the transpose of U, and D is symmetric and
     block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     This is the unblocked version of the algorithm, calling Level 2 BLAS.

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, the block diagonal matrix D and the multipliers used
              to obtain the factor U or L (see below for further details).

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If UPLO = 'U':
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) = IPIV(k-1) < 0, then rows and columns
                 k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
                 is a 2-by-2 diagonal block.
              If UPLO = 'L':
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) = IPIV(k+1) < 0, then rows and columns
                 k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
                 is a 2-by-2 diagonal block.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -k, the k-th argument had an illegal value
              > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
                   has been completed, but the block diagonal matrix D is
                   exactly singular, and division by zero will occur if it
                   is used to solve a system of equations.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      If UPLO = 'U', then A = U*D*U**T, where
         U = P(n)*U(n)* ... *P(k)U(k)* ...,
      i.e., U is a product of terms P(k)*U(k), where k decreases from n to
      1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
      and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
      defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
      that if the diagonal block D(k) is of order s (s = 1 or 2), then
                 (   I    v    0   )   k-s
         U(k) =  (   0    I    0   )   s
                 (   0    0    I   )   n-k
                    k-s   s   n-k
      If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
      If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
      and A(k,k), and v overwrites A(1:k-2,k-1:k).
      If UPLO = 'L', then A = L*D*L**T, where
         L = P(1)*L(1)* ... *P(k)*L(k)* ...,
      i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
      n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
      and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
      defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
      that if the diagonal block D(k) is of order s (s = 1 or 2), then
                 (   I    0     0   )  k-1
         L(k) =  (   0    I     0   )  s
                 (   0    v     I   )  n-k-s+1
                    k-1   s  n-k-s+1
      If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
      If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
      and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:

      09-29-06 - patch from
        Bobby Cheng, MathWorks
        Replace l.209 and l.377
             IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
        by
             IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
      1-96 - Based on modifications by J. Lewis, Boeing Computer Services
             Company

Definition at line 190 of file csytf2.f.

  subroutine csytf2_rk (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, integer info)

CSYTF2_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

Purpose:

     CSYTF2_RK computes the factorization of a complex symmetric matrix A
     using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
        A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**T (or L**T) is the transpose of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is symmetric and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     This is the unblocked version of the algorithm, calling Level 2 BLAS.
     For more information see Further Details section.

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, contains:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    are stored on exit in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is COMPLEX array, dimension (N)
              On exit, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
              If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is set to 0 in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              IPIV describes the permutation matrix P in the factorization
              of matrix A as follows. The absolute value of IPIV(k)
              represents the index of row and column that were
              interchanged with the k-th row and column. The value of UPLO
              describes the order in which the interchanges were applied.
              Also, the sign of IPIV represents the block structure of
              the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
              diagonal blocks which correspond to 1 or 2 interchanges
              at each factorization step. For more info see Further
              Details section.
              If UPLO = 'U',
              ( in factorization order, k decreases from N to 1 ):
                a) A single positive entry IPIV(k) > 0 means:
                   D(k,k) is a 1-by-1 diagonal block.
                   If IPIV(k) != k, rows and columns k and IPIV(k) were
                   interchanged in the matrix A(1:N,1:N);
                   If IPIV(k) = k, no interchange occurred.
                b) A pair of consecutive negative entries
                   IPIV(k) < 0 and IPIV(k-1) < 0 means:
                   D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
                   (NOTE: negative entries in IPIV appear ONLY in pairs).
                   1) If -IPIV(k) != k, rows and columns
                      k and -IPIV(k) were interchanged
                      in the matrix A(1:N,1:N).
                      If -IPIV(k) = k, no interchange occurred.
                   2) If -IPIV(k-1) != k-1, rows and columns
                      k-1 and -IPIV(k-1) were interchanged
                      in the matrix A(1:N,1:N).
                      If -IPIV(k-1) = k-1, no interchange occurred.
                c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
                d) NOTE: Any entry IPIV(k) is always NONZERO on output.
              If UPLO = 'L',
              ( in factorization order, k increases from 1 to N ):
                a) A single positive entry IPIV(k) > 0 means:
                   D(k,k) is a 1-by-1 diagonal block.
                   If IPIV(k) != k, rows and columns k and IPIV(k) were
                   interchanged in the matrix A(1:N,1:N).
                   If IPIV(k) = k, no interchange occurred.
                b) A pair of consecutive negative entries
                   IPIV(k) < 0 and IPIV(k+1) < 0 means:
                   D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
                   (NOTE: negative entries in IPIV appear ONLY in pairs).
                   1) If -IPIV(k) != k, rows and columns
                      k and -IPIV(k) were interchanged
                      in the matrix A(1:N,1:N).
                      If -IPIV(k) = k, no interchange occurred.
                   2) If -IPIV(k+1) != k+1, rows and columns
                      k-1 and -IPIV(k-1) were interchanged
                      in the matrix A(1:N,1:N).
                      If -IPIV(k+1) = k+1, no interchange occurred.
                c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
                d) NOTE: Any entry IPIV(k) is always NONZERO on output.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: If INFO = -k, the k-th argument had an illegal value
              > 0: If INFO = k, the matrix A is singular, because:
                     If UPLO = 'U': column k in the upper
                     triangular part of A contains all zeros.
                     If UPLO = 'L': column k in the lower
                     triangular part of A contains all zeros.
                   Therefore D(k,k) is exactly zero, and superdiagonal
                   elements of column k of U (or subdiagonal elements of
                   column k of L ) are all zeros. The factorization has
                   been completed, but the block diagonal matrix D is
                   exactly singular, and division by zero will occur if
                   it is used to solve a system of equations.
                   NOTE: INFO only stores the first occurrence of
                   a singularity, any subsequent occurrence of singularity
                   is not stored in INFO even though the factorization
                   always completes.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

     TODO: put further details

Contributors:

      December 2016,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester
      01-01-96 - Based on modifications by
        J. Lewis, Boeing Computer Services Company
        A. Petitet, Computer Science Dept.,
                    Univ. of Tenn., Knoxville abd , USA

Definition at line 240 of file csytf2_rk.f.

  subroutine csytf2_rook (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

CSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).

Purpose:

     CSYTF2_ROOK computes the factorization of a complex symmetric matrix A
     using the bounded Bunch-Kaufman ('rook') diagonal pivoting method:
        A = U*D*U**T  or  A = L*D*L**T
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, U**T is the transpose of U, and D is symmetric and
     block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     This is the unblocked version of the algorithm, calling Level 2 BLAS.

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, the block diagonal matrix D and the multipliers used
              to obtain the factor U or L (see below for further details).

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If UPLO = 'U':
                 If IPIV(k) > 0, then rows and columns k and IPIV(k)
                 were interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
                 columns k and -IPIV(k) were interchanged and rows and
                 columns k-1 and -IPIV(k-1) were inerchaged,
                 D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
              If UPLO = 'L':
                 If IPIV(k) > 0, then rows and columns k and IPIV(k)
                 were interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
                 columns k and -IPIV(k) were interchanged and rows and
                 columns k+1 and -IPIV(k+1) were inerchaged,
                 D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -k, the k-th argument had an illegal value
              > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
                   has been completed, but the block diagonal matrix D is
                   exactly singular, and division by zero will occur if it
                   is used to solve a system of equations.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      If UPLO = 'U', then A = U*D*U**T, where
         U = P(n)*U(n)* ... *P(k)U(k)* ...,
      i.e., U is a product of terms P(k)*U(k), where k decreases from n to
      1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
      and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
      defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
      that if the diagonal block D(k) is of order s (s = 1 or 2), then
                 (   I    v    0   )   k-s
         U(k) =  (   0    I    0   )   s
                 (   0    0    I   )   n-k
                    k-s   s   n-k
      If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
      If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
      and A(k,k), and v overwrites A(1:k-2,k-1:k).
      If UPLO = 'L', then A = L*D*L**T, where
         L = P(1)*L(1)* ... *P(k)*L(k)* ...,
      i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
      n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
      and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
      defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
      that if the diagonal block D(k) is of order s (s = 1 or 2), then
                 (   I    0     0   )  k-1
         L(k) =  (   0    I     0   )  s
                 (   0    v     I   )  n-k-s+1
                    k-1   s  n-k-s+1
      If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
      If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
      and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:

      November 2013,     Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester
      01-01-96 - Based on modifications by
        J. Lewis, Boeing Computer Services Company
        A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

Definition at line 193 of file csytf2_rook.f.

  subroutine csytrf (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer lwork, integer info)

CSYTRF

Purpose:

     CSYTRF computes the factorization of a complex symmetric matrix A
     using the Bunch-Kaufman diagonal pivoting method.  The form of the
     factorization is
        A = U*D*U**T  or  A = L*D*L**T
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, and D is symmetric and block diagonal with
     1-by-1 and 2-by-2 diagonal blocks.
     This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    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, the block diagonal matrix D and the multipliers used
              to obtain the factor U or L (see below for further details).

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If IPIV(k) > 0, then rows and columns k and IPIV(k) were
              interchanged and D(k,k) is a 1-by-1 diagonal block.
              If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
              columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
              is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
              IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
              interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

    WORK

              WORK is COMPLEX array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of WORK.  LWORK >=1.  For best performance
              LWORK >= N*NB, where NB is the block size returned by ILAENV.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
                    has been completed, but the block diagonal matrix D is
                    exactly singular, and division by zero will occur if it
                    is used to solve a system of equations.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      If UPLO = 'U', then A = U*D*U**T, where
         U = P(n)*U(n)* ... *P(k)U(k)* ...,
      i.e., U is a product of terms P(k)*U(k), where k decreases from n to
      1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
      and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
      defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
      that if the diagonal block D(k) is of order s (s = 1 or 2), then
                 (   I    v    0   )   k-s
         U(k) =  (   0    I    0   )   s
                 (   0    0    I   )   n-k
                    k-s   s   n-k
      If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
      If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
      and A(k,k), and v overwrites A(1:k-2,k-1:k).
      If UPLO = 'L', then A = L*D*L**T, where
         L = P(1)*L(1)* ... *P(k)*L(k)* ...,
      i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
      n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
      and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
      defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
      that if the diagonal block D(k) is of order s (s = 1 or 2), then
                 (   I    0     0   )  k-1
         L(k) =  (   0    I     0   )  s
                 (   0    v     I   )  n-k-s+1
                    k-1   s  n-k-s+1
      If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
      If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
      and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Definition at line 181 of file csytrf.f.

  subroutine csytrf_aa (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer lwork, integer info)

CSYTRF_AA

Purpose:

     CSYTRF_AA computes the factorization of a complex symmetric matrix A
     using the Aasen's algorithm.  The form of the factorization is
        A = U**T*T*U  or  A = L*T*L**T
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, and T is a complex symmetric tridiagonal matrix.
     This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    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, the tridiagonal matrix is stored in the diagonals
              and the subdiagonals of A just below (or above) the diagonals,
              and L is stored below (or above) the subdiaonals, when UPLO
              is 'L' (or 'U').

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              On exit, it contains the details of the interchanges, i.e.,
              the row and column k of A were interchanged with the
              row and column IPIV(k).

    WORK

              WORK is COMPLEX array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of WORK. LWORK >= MAX(1,2*N). For optimum performance
              LWORK >= N*(1+NB), where NB is the optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 131 of file csytrf_aa.f.

  subroutine csytrf_aa_2stage (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tb, integer ltb, integer, dimension( * ) ipiv, integer, dimension( * ) ipiv2, complex, dimension( * ) work, integer lwork, integer info)

CSYTRF_AA_2STAGE

Purpose:

     CSYTRF_AA_2STAGE computes the factorization of a complex symmetric matrix A
     using the Aasen's algorithm.  The form of the factorization is
        A = U**T*T*U  or  A = L*T*L**T
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, and T is a complex symmetric band matrix with the
     bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is
     LU factorized with partial pivoting).
     This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the hermitian 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, L is stored below (or above) the subdiaonal blocks,
              when UPLO  is 'L' (or 'U').

    LDA

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

    TB

              TB is COMPLEX array, dimension (LTB)
              On exit, details of the LU factorization of the band matrix.

    LTB

              LTB is INTEGER
              The size of the array TB. LTB >= 4*N, internally
              used to select NB such that LTB >= (3*NB+1)*N.
              If LTB = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of LTB,
              returns this value as the first entry of TB, and
              no error message related to LTB is issued by XERBLA.

    IPIV

              IPIV is INTEGER array, dimension (N)
              On exit, it contains the details of the interchanges, i.e.,
              the row and column k of A were interchanged with the
              row and column IPIV(k).

    IPIV2

              IPIV2 is INTEGER array, dimension (N)
              On exit, it contains the details of the interchanges, i.e.,
              the row and column k of T were interchanged with the
              row and column IPIV(k).

    WORK

              WORK is COMPLEX workspace of size LWORK

    LWORK

              LWORK is INTEGER
              The size of WORK. LWORK >= N, internally used to select NB
              such that LWORK >= N*NB.
              If LWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the WORK array,
              returns this value as the first entry of the WORK array, and
              no error message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = i, band LU factorization failed on i-th column

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 158 of file csytrf_aa_2stage.f.

  subroutine csytrf_rk (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer lwork, integer info)

CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Purpose:

     CSYTRF_RK computes the factorization of a complex symmetric matrix A
     using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
        A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**T (or L**T) is the transpose of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is symmetric and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     This is the blocked version of the algorithm, calling Level 3 BLAS.
     For more information see Further Details section.

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, contains:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    are stored on exit in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is COMPLEX array, dimension (N)
              On exit, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
              If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is set to 0 in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              IPIV describes the permutation matrix P in the factorization
              of matrix A as follows. The absolute value of IPIV(k)
              represents the index of row and column that were
              interchanged with the k-th row and column. The value of UPLO
              describes the order in which the interchanges were applied.
              Also, the sign of IPIV represents the block structure of
              the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
              diagonal blocks which correspond to 1 or 2 interchanges
              at each factorization step. For more info see Further
              Details section.
              If UPLO = 'U',
              ( in factorization order, k decreases from N to 1 ):
                a) A single positive entry IPIV(k) > 0 means:
                   D(k,k) is a 1-by-1 diagonal block.
                   If IPIV(k) != k, rows and columns k and IPIV(k) were
                   interchanged in the matrix A(1:N,1:N);
                   If IPIV(k) = k, no interchange occurred.
                b) A pair of consecutive negative entries
                   IPIV(k) < 0 and IPIV(k-1) < 0 means:
                   D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
                   (NOTE: negative entries in IPIV appear ONLY in pairs).
                   1) If -IPIV(k) != k, rows and columns
                      k and -IPIV(k) were interchanged
                      in the matrix A(1:N,1:N).
                      If -IPIV(k) = k, no interchange occurred.
                   2) If -IPIV(k-1) != k-1, rows and columns
                      k-1 and -IPIV(k-1) were interchanged
                      in the matrix A(1:N,1:N).
                      If -IPIV(k-1) = k-1, no interchange occurred.
                c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
                d) NOTE: Any entry IPIV(k) is always NONZERO on output.
              If UPLO = 'L',
              ( in factorization order, k increases from 1 to N ):
                a) A single positive entry IPIV(k) > 0 means:
                   D(k,k) is a 1-by-1 diagonal block.
                   If IPIV(k) != k, rows and columns k and IPIV(k) were
                   interchanged in the matrix A(1:N,1:N).
                   If IPIV(k) = k, no interchange occurred.
                b) A pair of consecutive negative entries
                   IPIV(k) < 0 and IPIV(k+1) < 0 means:
                   D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
                   (NOTE: negative entries in IPIV appear ONLY in pairs).
                   1) If -IPIV(k) != k, rows and columns
                      k and -IPIV(k) were interchanged
                      in the matrix A(1:N,1:N).
                      If -IPIV(k) = k, no interchange occurred.
                   2) If -IPIV(k+1) != k+1, rows and columns
                      k-1 and -IPIV(k-1) were interchanged
                      in the matrix A(1:N,1:N).
                      If -IPIV(k+1) = k+1, no interchange occurred.
                c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
                d) NOTE: Any entry IPIV(k) is always NONZERO on output.

    WORK

              WORK is COMPLEX array, dimension ( MAX(1,LWORK) ).
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of WORK.  LWORK >=1.  For best performance
              LWORK >= N*NB, where NB is the block size returned
              by ILAENV.
              If LWORK = -1, then a workspace query is assumed;
              the routine only calculates the optimal size of the WORK
              array, returns this value as the first entry of the WORK
              array, and no error message related to LWORK is issued
              by XERBLA.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: If INFO = -k, the k-th argument had an illegal value
              > 0: If INFO = k, the matrix A is singular, because:
                     If UPLO = 'U': column k in the upper
                     triangular part of A contains all zeros.
                     If UPLO = 'L': column k in the lower
                     triangular part of A contains all zeros.
                   Therefore D(k,k) is exactly zero, and superdiagonal
                   elements of column k of U (or subdiagonal elements of
                   column k of L ) are all zeros. The factorization has
                   been completed, but the block diagonal matrix D is
                   exactly singular, and division by zero will occur if
                   it is used to solve a system of equations.
                   NOTE: INFO only stores the first occurrence of
                   a singularity, any subsequent occurrence of singularity
                   is not stored in INFO even though the factorization
                   always completes.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

     TODO: put correct description

Contributors:

      December 2016,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 257 of file csytrf_rk.f.

  subroutine csytrf_rook (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer lwork, integer info)

CSYTRF_ROOK

Purpose:

     CSYTRF_ROOK computes the factorization of a complex symmetric matrix A
     using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
     The form of the factorization is
        A = U*D*U**T  or  A = L*D*L**T
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, and D is symmetric and block diagonal with
     1-by-1 and 2-by-2 diagonal blocks.
     This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    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, the block diagonal matrix D and the multipliers used
              to obtain the factor U or L (see below for further details).

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If UPLO = 'U':
                   If IPIV(k) > 0, then rows and columns k and IPIV(k)
                   were interchanged and D(k,k) is a 1-by-1 diagonal block.
                   If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
                   columns k and -IPIV(k) were interchanged and rows and
                   columns k-1 and -IPIV(k-1) were inerchaged,
                   D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
              If UPLO = 'L':
                   If IPIV(k) > 0, then rows and columns k and IPIV(k)
                   were interchanged and D(k,k) is a 1-by-1 diagonal block.
                   If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
                   columns k and -IPIV(k) were interchanged and rows and
                   columns k+1 and -IPIV(k+1) were inerchaged,
                   D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

    WORK

              WORK is COMPLEX array, dimension (MAX(1,LWORK)).
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of WORK.  LWORK >=1.  For best performance
              LWORK >= N*NB, where NB is the block size returned by ILAENV.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
                    has been completed, but the block diagonal matrix D is
                    exactly singular, and division by zero will occur if it
                    is used to solve a system of equations.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      If UPLO = 'U', then A = U*D*U**T, where
         U = P(n)*U(n)* ... *P(k)U(k)* ...,
      i.e., U is a product of terms P(k)*U(k), where k decreases from n to
      1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
      and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
      defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
      that if the diagonal block D(k) is of order s (s = 1 or 2), then
                 (   I    v    0   )   k-s
         U(k) =  (   0    I    0   )   s
                 (   0    0    I   )   n-k
                    k-s   s   n-k
      If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
      If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
      and A(k,k), and v overwrites A(1:k-2,k-1:k).
      If UPLO = 'L', then A = L*D*L**T, where
         L = P(1)*L(1)* ... *P(k)*L(k)* ...,
      i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
      n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
      and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
      defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
      that if the diagonal block D(k) is of order s (s = 1 or 2), then
                 (   I    0     0   )  k-1
         L(k) =  (   0    I     0   )  s
                 (   0    v     I   )  n-k-s+1
                    k-1   s  n-k-s+1
      If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
      If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
      and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:

       June 2016, Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 207 of file csytrf_rook.f.

  subroutine csytri (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer info)

CSYTRI

Purpose:

     CSYTRI computes the inverse of a complex symmetric indefinite matrix
     A using the factorization A = U*D*U**T or A = L*D*L**T computed by
     CSYTRF.

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 block diagonal matrix D and the multipliers
              used to obtain the factor U or L as computed by CSYTRF.
              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).

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF.

    WORK

              WORK is COMPLEX array, dimension (2*N)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                   inverse could not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 113 of file csytri.f.

  subroutine csytri2 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer lwork, integer info)

CSYTRI2

Purpose:

     CSYTRI2 computes the inverse of a COMPLEX symmetric indefinite matrix
     A using the factorization A = U*D*U**T or A = L*D*L**T computed by
     CSYTRF. CSYTRI2 sets the LEADING DIMENSION of the workspace
     before calling CSYTRI2X that actually computes the inverse.

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 block diagonal matrix D and the multipliers
              used to obtain the factor U or L as computed by CSYTRF.
              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).

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF.

    WORK

              WORK is COMPLEX array, dimension (N+NB+1)*(NB+3)

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              WORK is size >= (N+NB+1)*(NB+3)
              If LWORK = -1, then a workspace query is assumed; the routine
               calculates:
                  - the optimal size of the WORK array, returns
              this value as the first entry of the WORK array,
                  - and no error message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                   inverse could not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 126 of file csytri2.f.

  subroutine csytri2x (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( n+nb+1,* ) work, integer nb, integer info)

CSYTRI2X

Purpose:

     CSYTRI2X computes the inverse of a real symmetric indefinite matrix
     A using the factorization A = U*D*U**T or A = L*D*L**T computed by
     CSYTRF.

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 NNB diagonal matrix D and the multipliers
              used to obtain the factor U or L as computed by CSYTRF.
              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).

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the NNB structure of D
              as determined by CSYTRF.

    WORK

              WORK is COMPLEX array, dimension (N+NB+1,NB+3)

    NB

              NB is INTEGER
              Block size

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                   inverse could not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 119 of file csytri2x.f.

  subroutine csytri_3 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer lwork, integer info)

CSYTRI_3

Purpose:

     CSYTRI_3 computes the inverse of a complex symmetric indefinite
     matrix A using the factorization computed by CSYTRF_RK or CSYTRF_BK:
         A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**T (or L**T) is the transpose of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is symmetric and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     CSYTRI_3 sets the leading dimension of the workspace  before calling
     CSYTRI_3X that actually computes the inverse.  This is the blocked
     version of the algorithm, calling Level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the details of the factorization are
              stored as an upper or lower triangular matrix.
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, diagonal of the block diagonal matrix D and
              factors U or L as computed by CSYTRF_RK and CSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    should be provided on entry in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.
              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).

    E

              E is COMPLEX array, dimension (N)
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is not referenced in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF_RK or CSYTRF_BK.

    WORK

              WORK is COMPLEX array, dimension (N+NB+1)*(NB+3).
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of WORK. LWORK >= (N+NB+1)*(NB+3).
              If LDWORK = -1, then a workspace query is assumed;
              the routine only calculates the optimal size of the optimal
              size of the WORK array, returns this value as the first
              entry of the WORK array, and no error message related to
              LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                   inverse could not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      November 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley

Definition at line 168 of file csytri_3.f.

  subroutine csytri_3x (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( n+nb+1, * ) work, integer nb, integer info)

CSYTRI_3X

Purpose:

     CSYTRI_3X computes the inverse of a complex symmetric indefinite
     matrix A using the factorization computed by CSYTRF_RK or CSYTRF_BK:
         A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**T (or L**T) is the transpose of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is symmetric and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the details of the factorization are
              stored as an upper or lower triangular matrix.
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, diagonal of the block diagonal matrix D and
              factors U or L as computed by CSYTRF_RK and CSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    should be provided on entry in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.
              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).

    E

              E is COMPLEX array, dimension (N)
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is not referenced in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF_RK or CSYTRF_BK.

    WORK

              WORK is COMPLEX array, dimension (N+NB+1,NB+3).

    NB

              NB is INTEGER
              Block size.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                   inverse could not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      June 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley

Definition at line 158 of file csytri_3x.f.

  subroutine csytri_rook (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer info)

CSYTRI_ROOK

Purpose:

     CSYTRI_ROOK computes the inverse of a complex symmetric
     matrix A using the factorization A = U*D*U**T or A = L*D*L**T
     computed by CSYTRF_ROOK.

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 block diagonal matrix D and the multipliers
              used to obtain the factor U or L as computed by CSYTRF_ROOK.
              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).

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF_ROOK.

    WORK

              WORK is COMPLEX array, dimension (N)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                   inverse could not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

       December 2016, Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 128 of file csytri_rook.f.

  subroutine csytrs (character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, integer info)

CSYTRS

Purpose:

     CSYTRS solves a system of linear equations A*X = B with a complex
     symmetric matrix A using the factorization A = U*D*U**T or
     A = L*D*L**T computed by CSYTRF.

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.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by CSYTRF.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 119 of file csytrs.f.

  subroutine csytrs2 (character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) work, integer info)

CSYTRS2

Purpose:

     CSYTRS2 solves a system of linear equations A*X = B with a complex
     symmetric matrix A using the factorization A = U*D*U**T or
     A = L*D*L**T computed by CSYTRF and converted by CSYCONV.

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.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by CSYTRF.
              Note that A is input / output. This might be counter-intuitive,
              and one may think that A is input only. A is input / output. This
              is because, at the start of the subroutine, we permute A in a
              'better' form and then we permute A back to its original form at
              the end.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    WORK

              WORK is COMPLEX array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 130 of file csytrs2.f.

  subroutine csytrs_3 (character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, integer info)

CSYTRS_3

Purpose:

     CSYTRS_3 solves a system of linear equations A * X = B with a complex
     symmetric matrix A using the factorization computed
     by CSYTRF_RK or CSYTRF_BK:
        A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
     where U (or L) is unit upper (or lower) triangular matrix,
     U**T (or L**T) is the transpose of U (or L), P is a permutation
     matrix, P**T is the transpose of P, and D is symmetric and block
     diagonal with 1-by-1 and 2-by-2 diagonal blocks.
     This algorithm is using Level 3 BLAS.

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 = P*U*D*(U**T)*(P**T);
              = 'L':  Lower triangular, form is A = P*L*D*(L**T)*(P**T).

    N

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

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              Diagonal of the block diagonal matrix D and factors U or L
              as computed by CSYTRF_RK and CSYTRF_BK:
                a) ONLY diagonal elements of the symmetric block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    should be provided on entry in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

              E is COMPLEX array, dimension (N)
              On entry, contains the superdiagonal (or subdiagonal)
              elements of the symmetric block diagonal matrix D
              with 1-by-1 or 2-by-2 diagonal blocks, where
              If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
              If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
              NOTE: For 1-by-1 diagonal block D(k), where
              1 <= k <= N, the element E(k) is not referenced in both
              UPLO = 'U' or UPLO = 'L' cases.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF_RK or CSYTRF_BK.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

      June 2017,  Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 163 of file csytrs_3.f.

  subroutine csytrs_aa (character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) work, integer lwork, integer info)

CSYTRS_AA

Purpose:

     CSYTRS_AA solves a system of linear equations A*X = B with a complex
     symmetric matrix A using the factorization A = U**T*T*U or
     A = L*T*L**T computed by CSYTRF_AA.

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**T*T*U;
              = 'L':  Lower triangular, form is A = L*T*L**T.

    N

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

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              Details of factors computed by CSYTRF_AA.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges as computed by CSYTRF_AA.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    WORK

              WORK is COMPLEX array, dimension (MAX(1,LWORK))

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,3*N-2).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 129 of file csytrs_aa.f.

  subroutine csytrs_aa_2stage (character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tb, integer ltb, integer, dimension( * ) ipiv, integer, dimension( * ) ipiv2, complex, dimension( ldb, * ) b, integer ldb, integer info)

CSYTRS_AA_2STAGE

Purpose:

     CSYTRS_AA_2STAGE solves a system of linear equations A*X = B with a complex
     symmetric matrix A using the factorization A = U**T*T*U or
     A = L*T*L**T computed by CSYTRF_AA_2STAGE.

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**T*T*U;
              = 'L':  Lower triangular, form is A = L*T*L**T.

    N

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

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              Details of factors computed by CSYTRF_AA_2STAGE.

    LDA

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

    TB

              TB is COMPLEX array, dimension (LTB)
              Details of factors computed by CSYTRF_AA_2STAGE.

    LTB

              LTB is INTEGER
              The size of the array TB. LTB >= 4*N.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges as computed by
              CSYTRF_AA_2STAGE.

    IPIV2

              IPIV2 is INTEGER array, dimension (N)
              Details of the interchanges as computed by
              CSYTRF_AA_2STAGE.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 137 of file csytrs_aa_2stage.f.

  subroutine csytrs_rook (character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, integer info)

CSYTRS_ROOK

Purpose:

     CSYTRS_ROOK solves a system of linear equations A*X = B with
     a complex symmetric matrix A using the factorization A = U*D*U**T or
     A = L*D*L**T computed by CSYTRF_ROOK.

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.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by CSYTRF_ROOK.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by CSYTRF_ROOK.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

       December 2016, Igor Kozachenko,
                      Computer Science Division,
                      University of California, Berkeley
      September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                      School of Mathematics,
                      University of Manchester

Definition at line 134 of file csytrs_rook.f.

  subroutine ctgsyl (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 dif, complex, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

CTGSYL

Purpose:

     CTGSYL solves the generalized Sylvester equation:
                 A * R - L * B = scale * C            (1)
                 D * R - L * E = scale * F
     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, with complex entries. 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 (1) is equivalent 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) ],
     Here Ix is the identity matrix of size x and X**H is the conjugate
     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 (TRANS = 'C') is used to compute an one-norm-based estimate
     of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
     and (B,E), using CLACON.
     If IJOB >= 1, CTGSYL computes a Frobenius norm-based estimate of
     Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
     reciprocal of the smallest singular value of Z.
     This is a level-3 BLAS algorithm.

Parameters
    TRANS

              TRANS is CHARACTER*1
              = 'N': solve the generalized sylvester equation (1).
              = 'C': solve the 'conjugate transposed' system (3).

    IJOB

              IJOB is INTEGER
              Specifies what kind of functionality to be performed.
              =0: solve (1) only.
              =1: The functionality of 0 and 3.
              =2: The functionality of 0 and 4.
              =3: Only an estimate of Dif[(A,D), (B,E)] is computed.
                  (look ahead strategy is used).
              =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
                  (CGECON on sub-systems is used).
              Not referenced if TRANS = 'C'.

    M

              M is INTEGER
              The order of the matrices A and D, and the row dimension of
              the matrices C, F, R and L.

    N

              N is INTEGER
              The order of the matrices B and E, and the column dimension
              of the matrices C, F, R and L.

    A

              A is COMPLEX array, dimension (LDA, M)
              The upper triangular matrix A.

    LDA

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

    B

              B is COMPLEX array, dimension (LDB, N)
              The upper triangular matrix B.

    LDB

              LDB is INTEGER
              The leading dimension of the array 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) or (3).
              On exit, if IJOB = 0, 1 or 2, C has been overwritten by
              the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
              the solution achieved during the computation of the
              Dif-estimate.

    LDC

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

    D

              D is COMPLEX array, dimension (LDD, M)
              The upper triangular matrix D.

    LDD

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

    E

              E is COMPLEX array, dimension (LDE, N)
              The upper triangular matrix E.

    LDE

              LDE is INTEGER
              The leading dimension of the array 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) or (3).
              On exit, if IJOB = 0, 1 or 2, F has been overwritten by
              the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
              the solution achieved during the computation of the
              Dif-estimate.

    LDF

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

    DIF

              DIF is REAL
              On exit DIF is the reciprocal of a lower bound of the
              reciprocal of the Dif-function, i.e. DIF is an upper bound of
              Dif[(A,D), (B,E)] = sigma-min(Z), where Z as in (2).
              IF IJOB = 0 or TRANS = 'C', DIF is not referenced.

    SCALE

              SCALE is REAL
              On exit SCALE is the scaling factor in (1) or (3).
              If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
              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.

    WORK

              WORK is COMPLEX array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK > = 1.
              If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (M+N+2)

    INFO

              INFO is INTEGER
                =0: successful exit
                <0: If INFO = -i, the i-th argument had an illegal value.
                >0: (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.

References:
    [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
     [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. Appl., 15(4):1045-1060, 1994.
     [3] B. Kagstrom and L. Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

Definition at line 292 of file ctgsyl.f.

  subroutine ctrsyl (character trana, character tranb, integer isgn, integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldc, * ) c, integer ldc, real scale, integer info)

CTRSYL

Purpose:

     CTRSYL solves the complex Sylvester matrix equation:
        op(A)*X + X*op(B) = scale*C or
        op(A)*X - X*op(B) = scale*C,
     where op(A) = A or A**H, and A and B are both upper triangular. A is
     M-by-M and B is N-by-N; the right hand side C and the solution X are
     M-by-N; and scale is an output scale factor, set <= 1 to avoid
     overflow in X.

Parameters
    TRANA

              TRANA is CHARACTER*1
              Specifies the option op(A):
              = 'N': op(A) = A    (No transpose)
              = 'C': op(A) = A**H (Conjugate transpose)

    TRANB

              TRANB is CHARACTER*1
              Specifies the option op(B):
              = 'N': op(B) = B    (No transpose)
              = 'C': op(B) = B**H (Conjugate transpose)

    ISGN

              ISGN is INTEGER
              Specifies the sign in the equation:
              = +1: solve op(A)*X + X*op(B) = scale*C
              = -1: solve op(A)*X - X*op(B) = scale*C

    M

              M is INTEGER
              The order of the matrix A, and the number of rows in the
              matrices X and C. M >= 0.

    N

              N is INTEGER
              The order of the matrix B, and the number of columns in the
              matrices X and C. N >= 0.

    A

              A is COMPLEX array, dimension (LDA,M)
              The upper triangular matrix A.

    LDA

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

    B

              B is COMPLEX array, dimension (LDB,N)
              The upper triangular matrix B.

    LDB

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

    C

              C is COMPLEX array, dimension (LDC,N)
              On entry, the M-by-N right hand side matrix C.
              On exit, C is overwritten by the solution matrix X.

    LDC

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

    SCALE

              SCALE is REAL
              The scale factor, scale, set <= 1 to avoid overflow in X.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              = 1: A and B have common or very close eigenvalues; perturbed
                   values were used to solve the equation (but the matrices
                   A and B are unchanged).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 155 of file ctrsyl.f.

  subroutine ctrsyl3 (character trana, character tranb, integer isgn, integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldc, * ) c, integer ldc, real scale, real, dimension( ldswork, * ) swork, integer ldswork, integer info)

CTRSYL3

Purpose

      CTRSYL3 solves the complex Sylvester matrix equation:
         op(A)*X + X*op(B) = scale*C or
         op(A)*X - X*op(B) = scale*C,
      where op(A) = A or A**H, and  A and B are both upper triangular. A is
      M-by-M and B is N-by-N; the right hand side C and the solution X are
      M-by-N; and scale is an output scale factor, set <= 1 to avoid
      overflow in X.
      This is the block version of the algorithm.

Parameters
    TRANA

              TRANA is CHARACTER*1
              Specifies the option op(A):
              = 'N': op(A) = A    (No transpose)
              = 'C': op(A) = A**H (Conjugate transpose)

    TRANB

              TRANB is CHARACTER*1
              Specifies the option op(B):
              = 'N': op(B) = B    (No transpose)
              = 'C': op(B) = B**H (Conjugate transpose)

    ISGN

              ISGN is INTEGER
              Specifies the sign in the equation:
              = +1: solve op(A)*X + X*op(B) = scale*C
              = -1: solve op(A)*X - X*op(B) = scale*C

    M

              M is INTEGER
              The order of the matrix A, and the number of rows in the
              matrices X and C. M >= 0.

    N

              N is INTEGER
              The order of the matrix B, and the number of columns in the
              matrices X and C. N >= 0.

    A

              A is COMPLEX array, dimension (LDA,M)
              The upper triangular matrix A.

    LDA

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

    B

              B is COMPLEX array, dimension (LDB,N)
              The upper triangular matrix B.

    LDB

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

    C

              C is COMPLEX array, dimension (LDC,N)
              On entry, the M-by-N right hand side matrix C.
              On exit, C is overwritten by the solution matrix X.

    LDC

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

    SCALE

              SCALE is REAL
              The scale factor, scale, set <= 1 to avoid overflow in X.

    SWORK

              SWORK is REAL array, dimension (MAX(2, ROWS), MAX(1,COLS)).
              On exit, if INFO = 0, SWORK(1) returns the optimal value ROWS
              and SWORK(2) returns the optimal COLS.

    LDSWORK

              LDSWORK is INTEGER
              LDSWORK >= MAX(2,ROWS), where ROWS = ((M + NB - 1) / NB + 1)
              and NB is the optimal block size.
              If LDSWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal dimensions of the SWORK matrix,
              returns these values as the first and second entry of the SWORK
              matrix, and no error message related LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              = 1: A and B have common or very close eigenvalues; perturbed
                   values were used to solve the equation (but the matrices
                   A and B are unchanged).

Definition at line 154 of file ctrsyl3.f.

Author

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