realSYcomputational(3)

NAME
realSYcomputational - real

SYNOPSIS

  Functions
subroutine sla_syamv (uplo, n, alpha, a, lda, x, incx, beta, y, incy)
SLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.
real function sla_syrcond (uplo, n, a, lda, af, ldaf, ipiv, cmode, c, info, work, iwork)
SLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix.
subroutine sla_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)
SLA_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 sla_syrpvgrw (uplo, n, info, a, lda, af, ldaf, ipiv, work)
SLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
subroutine slasyf (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
SLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
subroutine slasyf_aa (uplo, j1, m, nb, a, lda, ipiv, h, ldh, work)
SLASYF_AA
subroutine slasyf_rook (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
SLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
subroutine ssycon (uplo, n, a, lda, ipiv, anorm, rcond, work, iwork, info)
SSYCON
subroutine ssycon_rook (uplo, n, a, lda, ipiv, anorm, rcond, work, iwork, info)
SSYCON_ROOK
subroutine ssyconv (uplo, way, n, a, lda, ipiv, e, info)
SSYCONV
subroutine ssyequb (uplo, n, a, lda, s, scond, amax, work, info)
SSYEQUB
subroutine ssygs2 (itype, uplo, n, a, lda, b, ldb, info)
SSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).
subroutine ssygst (itype, uplo, n, a, lda, b, ldb, info)
SSYGST
subroutine ssyrfs (uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SSYRFS
subroutine ssyrfsx (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, iwork, info)
SSYRFSX
subroutine ssytd2 (uplo, n, a, lda, d, e, tau, info)
SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
subroutine ssytf2 (uplo, n, a, lda, ipiv, info)
SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
subroutine ssytf2_rook (uplo, n, a, lda, ipiv, info)
SSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).
subroutine ssytrd (uplo, n, a, lda, d, e, tau, work, lwork, info)
SSYTRD
subroutine ssytrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
SSYTRD_2STAGE
subroutine ssytrd_sy2sb (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)
SSYTRD_SY2SB
subroutine ssytrf (uplo, n, a, lda, ipiv, work, lwork, info)
SSYTRF
subroutine ssytrf_aa (uplo, n, a, lda, ipiv, work, lwork, info)
SSYTRF_AA
subroutine ssytrf_aa_2stage (uplo, n, a, lda, tb, ltb, ipiv, ipiv2, work, lwork, info)
SSYTRF_AA_2STAGE
subroutine ssytrf_rook (uplo, n, a, lda, ipiv, work, lwork, info)
SSYTRF_ROOK
subroutine ssytri (uplo, n, a, lda, ipiv, work, info)
SSYTRI
subroutine ssytri2 (uplo, n, a, lda, ipiv, work, lwork, info)
SSYTRI2
subroutine ssytri2x (uplo, n, a, lda, ipiv, work, nb, info)
SSYTRI2X
subroutine ssytri_rook (uplo, n, a, lda, ipiv, work, info)
SSYTRI_ROOK
subroutine ssytrs (uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
SSYTRS
subroutine ssytrs2 (uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info)
SSYTRS2
subroutine ssytrs_aa (uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info)
SSYTRS_AA
subroutine ssytrs_aa_2stage (uplo, n, nrhs, a, lda, tb, ltb, ipiv, ipiv2, b, ldb, info)
SSYTRS_AA_2STAGE
subroutine ssytrs_rook (uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
SSYTRS_ROOK
subroutine stgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
STGSYL
subroutine strsyl (trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, info)
STRSYL

Detailed Description

This is the group of real computational functions for SY matrices

Function Documentation

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

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

Purpose:

     SLA_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 REAL 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 REAL 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 175 of file sla_syamv.f.

  real function sla_syrcond (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, integer cmode, real, dimension( * ) c, integer info, real, dimension( * ) work, integer, dimension( * ) iwork)

SLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix.

Purpose:

        SLA_SYRCOND estimates the Skeel condition number of  op(A) * op2(C)
        where op2 is determined by CMODE as follows
        CMODE =  1    op2(C) = C
        CMODE =  0    op2(C) = I
        CMODE = -1    op2(C) = inv(C)
        The Skeel condition number cond(A) = norminf( |inv(A)||A| )
        is computed by computing scaling factors R such that
        diag(R)*A*op2(C) is row equilibrated and computing the standard
        infinity-norm condition number.

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

    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 SSYTRF.

    CMODE

              CMODE is INTEGER
         Determines op2(C) in the formula op(A) * op2(C) as follows:
         CMODE =  1    op2(C) = C
         CMODE =  0    op2(C) = I
         CMODE = -1    op2(C) = inv(C)

    C

              C is REAL array, dimension (N)
         The vector C in the formula op(A) * op2(C).

    INFO

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

    WORK

              WORK is REAL array, dimension (3*N).
         Workspace.

    IWORK

              IWORK is INTEGER array, dimension (N).
         Workspace.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 144 of file sla_syrcond.f.

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

SLA_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:

     SLA_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 SSYRFSX 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 REAL 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 REAL array, dimension (LDAF,N)
         The block diagonal matrix D and the multipliers used to
         obtain the factor U or L as computed by SSYTRF.

    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 SSYTRF.

    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 REAL 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 REAL array, dimension (LDY,NRHS)
         On entry, the solution matrix X, as computed by SSYTRS.
         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 SLA_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 REAL array, dimension (N)
         Workspace to hold the intermediate residual.

    AYB

              AYB is REAL array, dimension (N)
         Workspace. This can be the same workspace passed for Y_TAIL.

    DY

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

    Y_TAIL

              Y_TAIL is REAL 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 SLA_SYRFSX_EXTENDED had an illegal
                 value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 389 of file sla_syrfsx_extended.f.

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

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

Purpose:

     SLA_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 SSYTRF, .i.e., the pivot in
         column INFO is exactly 0.

    A

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

    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 SSYTRF.

    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 120 of file sla_syrpvgrw.f.

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

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

Purpose:

     SLASYF computes a partial factorization of a real 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.
     SLASYF is an auxiliary routine called by SSYTRF. 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 REAL 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 REAL 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 175 of file slasyf.f.

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

SLASYF_AA

Purpose:

     DLATRF_AA factorizes a panel of a real 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 SSYTRF_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 REAL 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 REAL workspace, dimension (LDH,NB).

    LDH

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

    WORK

              WORK is REAL workspace, dimension (M).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 142 of file slasyf_aa.f.

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

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

Purpose:

     SLASYF_ROOK computes a partial factorization of a real 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.
     SLASYF_ROOK is an auxiliary routine called by SSYTRF_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 REAL 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 REAL 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 slasyf_rook.f.

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

SSYCON

Purpose:

     SSYCON estimates the reciprocal of the condition number (in the
     1-norm) of a real symmetric matrix A using the factorization
     A = U*D*U**T or A = L*D*L**T computed by SSYTRF.
     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 REAL array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by SSYTRF.

    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 SSYTRF.

    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 REAL array, dimension (2*N)

    IWORK

              IWORK is INTEGER 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 128 of file ssycon.f.

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

SSYCON_ROOK

Purpose:

     SSYCON_ROOK estimates the reciprocal of the condition number (in the
     1-norm) of a real symmetric matrix A using the factorization
     A = U*D*U**T or A = L*D*L**T computed by SSYTRF_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 REAL array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by SSYTRF_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 SSYTRF_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 REAL array, dimension (2*N)

    IWORK

              IWORK is INTEGER 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.

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 142 of file ssycon_rook.f.

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

SSYCONV

Purpose:

     SSYCONV 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 REAL array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by SSYTRF.

    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 SSYTRF.

    E

              E is REAL 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 ssyconv.f.

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

SSYEQUB

Purpose:

     SSYEQUB 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 REAL 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 REAL 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 130 of file ssyequb.f.

  subroutine ssygs2 (integer itype, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, integer info)

SSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).

Purpose:

     SSYGS2 reduces a real symmetric-definite generalized eigenproblem
     to standard form.
     If ITYPE = 1, the problem is A*x = lambda*B*x,
     and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
     If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
     B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T *A*L.
     B must have been previously factorized as U**T *U or L*L**T by SPOTRF.

Parameters
    ITYPE

              ITYPE is INTEGER
              = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
              = 2 or 3: compute U*A*U**T or L**T *A*L.

    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              symmetric matrix A is stored, and how B has been factorized.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = 'U', the leading
              n by n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n by n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, if INFO = 0, the transformed matrix, stored in the
              same format as A.

    LDA

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

    B

              B is REAL array, dimension (LDB,N)
              The triangular factor from the Cholesky factorization of B,
              as returned by SPOTRF.

    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 126 of file ssygs2.f.

  subroutine ssygst (integer itype, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, integer info)

SSYGST

Purpose:

     SSYGST reduces a real symmetric-definite generalized eigenproblem
     to standard form.
     If ITYPE = 1, the problem is A*x = lambda*B*x,
     and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
     If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
     B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
     B must have been previously factorized as U**T*U or L*L**T by SPOTRF.

Parameters
    ITYPE

              ITYPE is INTEGER
              = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
              = 2 or 3: compute U*A*U**T or L**T*A*L.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored and B is factored as
                      U**T*U;
              = 'L':  Lower triangle of A is stored and B is factored as
                      L*L**T.

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, if INFO = 0, the transformed matrix, stored in the
              same format as A.

    LDA

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

    B

              B is REAL array, dimension (LDB,N)
              The triangular factor from the Cholesky factorization of B,
              as returned by SPOTRF.

    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 126 of file ssygst.f.

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

SSYRFS

Purpose:

     SSYRFS 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 REAL 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 REAL 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 SSYTRF.

    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 SSYTRF.

    B

              B is REAL 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 REAL array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by SSYTRS.
              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 REAL array, dimension (3*N)

    IWORK

              IWORK is INTEGER 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 189 of file ssyrfs.f.

  subroutine ssyrfsx (character uplo, character equed, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, real, dimension( * ) s, real, dimension( ldb, * ) b, integer ldb, real, 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, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SSYRFSX

Purpose:

        SSYRFSX 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 REAL 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 REAL 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 SSYTRF.

    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 SSYTRF.

    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 REAL 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 REAL array, dimension (LDX,NRHS)
         On entry, the solution matrix X, as computed by SGETRS.
         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 REAL array, dimension (4*N)

    IWORK

              IWORK is INTEGER array, dimension (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 ssyrfsx.f.

  subroutine ssytd2 (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tau, integer info)

SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).

Purpose:

     SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
     form T by an orthogonal similarity transformation: Q**T * A * Q = T.

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 REAL array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = 'U', the leading
              n-by-n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n-by-n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, if UPLO = 'U', the diagonal and first superdiagonal
              of A are overwritten by the corresponding elements of the
              tridiagonal matrix T, and the elements above the first
              superdiagonal, with the array TAU, represent the orthogonal
              matrix Q as a product of elementary reflectors; if UPLO
              = 'L', the diagonal and first subdiagonal of A are over-
              written by the corresponding elements of the tridiagonal
              matrix T, and the elements below the first subdiagonal, with
              the array TAU, represent the orthogonal matrix Q as a product
              of elementary reflectors. See Further Details.

    LDA

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

    D

              D is REAL array, dimension (N)
              The diagonal elements of the tridiagonal matrix T:
              D(i) = A(i,i).

    E

              E is REAL array, dimension (N-1)
              The off-diagonal elements of the tridiagonal matrix T:
              E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

    TAU

              TAU is REAL array, dimension (N-1)
              The scalar factors of the elementary reflectors (see Further
              Details).

    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.

Further Details:

      If UPLO = 'U', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(n-1) . . . H(2) H(1).
      Each H(i) has the form
         H(i) = I - tau * v * v**T
      where tau is a real scalar, and v is a real vector with
      v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
      A(1:i-1,i+1), and tau in TAU(i).
      If UPLO = 'L', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(1) H(2) . . . H(n-1).
      Each H(i) has the form
         H(i) = I - tau * v * v**T
      where tau is a real scalar, and v is a real vector with
      v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
      and tau in TAU(i).
      The contents of A on exit are illustrated by the following examples
      with n = 5:
      if UPLO = 'U':                       if UPLO = 'L':
        (  d   e   v2  v3  v4 )              (  d                  )
        (      d   e   v3  v4 )              (  e   d              )
        (          d   e   v4 )              (  v1  e   d          )
        (              d   e  )              (  v1  v2  e   d      )
        (                  d  )              (  v1  v2  v3  e   d  )
      where d and e denote diagonal and off-diagonal elements of T, and vi
      denotes an element of the vector defining H(i).

Definition at line 172 of file ssytd2.f.

  subroutine ssytf2 (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

Purpose:

     SSYTF2 computes the factorization of a real 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 REAL 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.204 and l.372
             IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
        by
             IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
      01-01-96 - Based on modifications by
        J. Lewis, Boeing Computer Services Company
        A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
      1-96 - Based on modifications by J. Lewis, Boeing Computer Services
             Company

Definition at line 194 of file ssytf2.f.

  subroutine ssytf2_rook (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

SSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).

Purpose:

     SSYTF2_ROOK computes the factorization of a real 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 REAL 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 ssytf2_rook.f.

  subroutine ssytrd (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SSYTRD

Purpose:

     SSYTRD reduces a real symmetric matrix A to real symmetric
     tridiagonal form T by an orthogonal similarity transformation:
     Q**T * A * Q = T.

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 REAL array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, if UPLO = 'U', the diagonal and first superdiagonal
              of A are overwritten by the corresponding elements of the
              tridiagonal matrix T, and the elements above the first
              superdiagonal, with the array TAU, represent the orthogonal
              matrix Q as a product of elementary reflectors; if UPLO
              = 'L', the diagonal and first subdiagonal of A are over-
              written by the corresponding elements of the tridiagonal
              matrix T, and the elements below the first subdiagonal, with
              the array TAU, represent the orthogonal matrix Q as a product
              of elementary reflectors. See Further Details.

    LDA

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

    D

              D is REAL array, dimension (N)
              The diagonal elements of the tridiagonal matrix T:
              D(i) = A(i,i).

    E

              E is REAL array, dimension (N-1)
              The off-diagonal elements of the tridiagonal matrix T:
              E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

    TAU

              TAU is REAL array, dimension (N-1)
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is REAL 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.
              For optimum performance LWORK >= N*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.

Further Details:

      If UPLO = 'U', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(n-1) . . . H(2) H(1).
      Each H(i) has the form
         H(i) = I - tau * v * v**T
      where tau is a real scalar, and v is a real vector with
      v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
      A(1:i-1,i+1), and tau in TAU(i).
      If UPLO = 'L', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(1) H(2) . . . H(n-1).
      Each H(i) has the form
         H(i) = I - tau * v * v**T
      where tau is a real scalar, and v is a real vector with
      v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
      and tau in TAU(i).
      The contents of A on exit are illustrated by the following examples
      with n = 5:
      if UPLO = 'U':                       if UPLO = 'L':
        (  d   e   v2  v3  v4 )              (  d                  )
        (      d   e   v3  v4 )              (  e   d              )
        (          d   e   v4 )              (  v1  e   d          )
        (              d   e  )              (  v1  v2  e   d      )
        (                  d  )              (  v1  v2  v3  e   d  )
      where d and e denote diagonal and off-diagonal elements of T, and vi
      denotes an element of the vector defining H(i).

Definition at line 191 of file ssytrd.f.

  subroutine ssytrd_2stage (character vect, character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tau, real, dimension( * ) hous2, integer lhous2, real, dimension( * ) work, integer lwork, integer info)

SSYTRD_2STAGE

Purpose:

     SSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric
     tridiagonal form T by a orthogonal similarity transformation:
     Q1**T Q2**T* A * Q2 * Q1 = T.

Parameters
    VECT

              VECT is CHARACTER*1
              = 'N':  No need for the Housholder representation,
                      in particular for the second stage (Band to
                      tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
              = 'V':  the Householder representation is needed to
                      either generate Q1 Q2 or to apply Q1 Q2,
                      then LHOUS2 is to be queried and computed.
                      (NOT AVAILABLE IN THIS RELEASE).

    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 REAL array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, if UPLO = 'U', the band superdiagonal
              of A are overwritten by the corresponding elements of the
              internal band-diagonal matrix AB, and the elements above
              the KD superdiagonal, with the array TAU, represent the orthogonal
              matrix Q1 as a product of elementary reflectors; if UPLO
              = 'L', the diagonal and band subdiagonal of A are over-
              written by the corresponding elements of the internal band-diagonal
              matrix AB, and the elements below the KD subdiagonal, with
              the array TAU, represent the orthogonal matrix Q1 as a product
              of elementary reflectors. See Further Details.

    LDA

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

    D

              D is REAL array, dimension (N)
              The diagonal elements of the tridiagonal matrix T.

    E

              E is REAL array, dimension (N-1)
              The off-diagonal elements of the tridiagonal matrix T.

    TAU

              TAU is REAL array, dimension (N-KD)
              The scalar factors of the elementary reflectors of
              the first stage (see Further Details).

    HOUS2

              HOUS2 is REAL array, dimension (LHOUS2)
              Stores the Householder representation of the stage2
              band to tridiagonal.

    LHOUS2

              LHOUS2 is INTEGER
              The dimension of the array HOUS2.
              If LWORK = -1, or LHOUS2 = -1,
              then a query is assumed; the routine
              only calculates the optimal size of the HOUS2 array, returns
              this value as the first entry of the HOUS2 array, and no error
              message related to LHOUS2 is issued by XERBLA.
              If VECT='N', LHOUS2 = max(1, 4*n);
              if VECT='V', option not yet available.

    WORK

              WORK is REAL array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK = MAX(1, dimension)
              If LWORK = -1, or LHOUS2=-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.
              LWORK = MAX(1, dimension) where
              dimension   = max(stage1,stage2) + (KD+1)*N
                          = N*KD + N*max(KD+1,FACTOPTNB)
                            + max(2*KD*KD, KD*NTHREADS)
                            + (KD+1)*N
              where KD is the blocking size of the reduction,
              FACTOPTNB is the blocking used by the QR or LQ
              algorithm, usually FACTOPTNB=128 is a good choice
              NTHREADS is the number of threads used when
              openMP compilation is enabled, otherwise =1.

    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.

Further Details:

      Implemented by Azzam Haidar.
      All details are available on technical report, SC11, SC13 papers.
      Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
      Parallel reduction to condensed forms for symmetric eigenvalue problems
      using aggregated fine-grained and memory-aware kernels. In Proceedings
      of 2011 International Conference for High Performance Computing,
      Networking, Storage and Analysis (SC '11), New York, NY, USA,
      Article 8 , 11 pages.
      http://doi.acm.org/10.1145/2063384.2063394
      A. Haidar, J. Kurzak, P. Luszczek, 2013.
      An improved parallel singular value algorithm and its implementation
      for multicore hardware, In Proceedings of 2013 International Conference
      for High Performance Computing, Networking, Storage and Analysis (SC '13).
      Denver, Colorado, USA, 2013.
      Article 90, 12 pages.
      http://doi.acm.org/10.1145/2503210.2503292
      A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
      A novel hybrid CPU-GPU generalized eigensolver for electronic structure
      calculations based on fine-grained memory aware tasks.
      International Journal of High Performance Computing Applications.
      Volume 28 Issue 2, Pages 196-209, May 2014.
      http://hpc.sagepub.com/content/28/2/196

Definition at line 222 of file ssytrd_2stage.f.

  subroutine ssytrd_sy2sb (character uplo, integer n, integer kd, real, dimension( lda, * ) a, integer lda, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SSYTRD_SY2SB

Purpose:

     SSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric
     band-diagonal form AB by a orthogonal similarity transformation:
     Q**T * A * Q = AB.

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.

    KD

              KD is INTEGER
              The number of superdiagonals of the reduced matrix if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
              The reduced matrix is stored in the array AB.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, if UPLO = 'U', the diagonal and first superdiagonal
              of A are overwritten by the corresponding elements of the
              tridiagonal matrix T, and the elements above the first
              superdiagonal, with the array TAU, represent the orthogonal
              matrix Q as a product of elementary reflectors; if UPLO
              = 'L', the diagonal and first subdiagonal of A are over-
              written by the corresponding elements of the tridiagonal
              matrix T, and the elements below the first subdiagonal, with
              the array TAU, represent the orthogonal matrix Q as a product
              of elementary reflectors. See Further Details.

    LDA

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

    AB

              AB is REAL array, dimension (LDAB,N)
              On exit, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    TAU

              TAU is REAL array, dimension (N-KD)
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is REAL array, dimension (LWORK)
              On exit, if INFO = 0, or if LWORK=-1,
              WORK(1) returns the size of LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK which should be calculated
              by a workspace query. LWORK = MAX(1, LWORK_QUERY)
              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.
              LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
              where FACTOPTNB is the blocking used by the QR or LQ
              algorithm, usually FACTOPTNB=128 is a good choice otherwise
              putting LWORK=-1 will provide the size of WORK.

    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.

Further Details:

      Implemented by Azzam Haidar.
      All details are available on technical report, SC11, SC13 papers.
      Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
      Parallel reduction to condensed forms for symmetric eigenvalue problems
      using aggregated fine-grained and memory-aware kernels. In Proceedings
      of 2011 International Conference for High Performance Computing,
      Networking, Storage and Analysis (SC '11), New York, NY, USA,
      Article 8 , 11 pages.
      http://doi.acm.org/10.1145/2063384.2063394
      A. Haidar, J. Kurzak, P. Luszczek, 2013.
      An improved parallel singular value algorithm and its implementation
      for multicore hardware, In Proceedings of 2013 International Conference
      for High Performance Computing, Networking, Storage and Analysis (SC '13).
      Denver, Colorado, USA, 2013.
      Article 90, 12 pages.
      http://doi.acm.org/10.1145/2503210.2503292
      A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
      A novel hybrid CPU-GPU generalized eigensolver for electronic structure
      calculations based on fine-grained memory aware tasks.
      International Journal of High Performance Computing Applications.
      Volume 28 Issue 2, Pages 196-209, May 2014.
      http://hpc.sagepub.com/content/28/2/196

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd.
  Each H(i) has the form
     H(i) = I - tau * v * v**T
  where tau is a real scalar, and v is a real vector with
  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
  A(i,i+kd+1:n), and tau in TAU(i).
  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(1) H(2) . . . H(k), where k = n-kd.
  Each H(i) has the form
     H(i) = I - tau * v * v**T
  where tau is a real scalar, and v is a real vector with
  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
  A(i+kd+2:n,i), and tau in TAU(i).
  The contents of A on exit are illustrated by the following examples
  with n = 5:
  if UPLO = 'U':                       if UPLO = 'L':
    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
    (                            ab    )              (  v1     v2     v3     ab/v4 ab )
  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i)..fi

Definition at line 241 of file ssytrd_sy2sb.f.

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

SSYTRF

Purpose:

     SSYTRF computes the factorization of a real symmetric matrix A using
     the Bunch-Kaufman diagonal pivoting method.  The form of the
     factorization is
        A = U**T*D*U  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 REAL 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 REAL 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**T*D*U, 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 ssytrf.f.

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

SSYTRF_AA

Purpose:

     SSYTRF_AA computes the factorization of a real 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 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 REAL 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 REAL 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 ssytrf_aa.f.

  subroutine ssytrf_aa_2stage (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tb, integer ltb, integer, dimension( * ) ipiv, integer, dimension( * ) ipiv2, real, dimension( * ) work, integer lwork, integer info)

SSYTRF_AA_2STAGE

Purpose:

     SSYTRF_AA_2STAGE computes the factorization of a real 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 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 REAL 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 REAL 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 REAL 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 ssytrf_aa_2stage.f.

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

SSYTRF_ROOK

Purpose:

     SSYTRF_ROOK computes the factorization of a real 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 REAL 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 REAL 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 ssytrf_rook.f.

  subroutine ssytri (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, real, dimension( * ) work, integer info)

SSYTRI

Purpose:

     SSYTRI 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
     SSYTRF.

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 REAL 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 SSYTRF.
              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 SSYTRF.

    WORK

              WORK is REAL 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.

Definition at line 113 of file ssytri.f.

  subroutine ssytri2 (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, real, dimension( * ) work, integer lwork, integer info)

SSYTRI2

Purpose:

     SSYTRI2 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
     SSYTRF. SSYTRI2 sets the LEADING DIMENSION of the workspace
     before calling SSYTRI2X 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 REAL 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 SSYTRF.
              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 SSYTRF.

    WORK

              WORK is REAL 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 ssytri2.f.

  subroutine ssytri2x (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, real, dimension( n+nb+1,* ) work, integer nb, integer info)

SSYTRI2X

Purpose:

     SSYTRI2X 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
     SSYTRF.

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 REAL 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 SSYTRF.
              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 SSYTRF.

    WORK

              WORK is REAL 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 ssytri2x.f.

  subroutine ssytri_rook (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, real, dimension( * ) work, integer info)

SSYTRI_ROOK

Purpose:

     SSYTRI_ROOK computes the inverse of a real symmetric
     matrix A using the factorization A = U*D*U**T or A = L*D*L**T
     computed by SSYTRF_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 REAL 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 SSYTRF_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 SSYTRF_ROOK.

    WORK

              WORK is REAL 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:

       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 128 of file ssytri_rook.f.

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

SSYTRS

Purpose:

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

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

    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 SSYTRF.

    B

              B is REAL 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 ssytrs.f.

  subroutine ssytrs2 (character uplo, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) work, integer info)

SSYTRS2

Purpose:

     SSYTRS2 solves a system of linear equations A*X = B with a real
     symmetric matrix A using the factorization A = U*D*U**T or
     A = L*D*L**T computed by SSYTRF and converted by SSYCONV.

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

    B

              B is REAL 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 REAL 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 ssytrs2.f.

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

SSYTRS_AA

Purpose:

     SSYTRS_AA solves a system of linear equations A*X = B with a real
     symmetric matrix A using the factorization A = U**T*T*U or
     A = L*T*L**T computed by SSYTRF_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 REAL array, dimension (LDA,N)
              Details of factors computed by SSYTRF_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 SSYTRF_AA.

    B

              B is REAL 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 REAL 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 ssytrs_aa.f.

  subroutine ssytrs_aa_2stage (character uplo, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tb, integer ltb, integer, dimension( * ) ipiv, integer, dimension( * ) ipiv2, real, dimension( ldb, * ) b, integer ldb, integer info)

SSYTRS_AA_2STAGE

Purpose:

     SSYTRS_AA_2STAGE solves a system of linear equations A*X = B with a real
     symmetric matrix A using the factorization A = U**T*T*U or
     A = L*T*L**T computed by SSYTRF_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 REAL array, dimension (LDA,N)
              Details of factors computed by SSYTRF_AA_2STAGE.

    LDA

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

    TB

              TB is REAL array, dimension (LTB)
              Details of factors computed by SSYTRF_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
              SSYTRF_AA_2STAGE.

    IPIV2

              IPIV2 is INTEGER array, dimension (N)
              Details of the interchanges as computed by
              SSYTRF_AA_2STAGE.

    B

              B is REAL 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 ssytrs_aa_2stage.f.

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

SSYTRS_ROOK

Purpose:

     SSYTRS_ROOK solves a system of linear equations A*X = B with
     a real symmetric matrix A using the factorization A = U*D*U**T or
     A = L*D*L**T computed by SSYTRF_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 REAL array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by SSYTRF_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 SSYTRF_ROOK.

    B

              B is REAL 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:

       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 134 of file ssytrs_rook.f.

  subroutine stgsyl (character trans, integer ijob, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldc, * ) c, integer ldc, real, dimension( ldd, * ) d, integer ldd, real, dimension( lde, * ) e, integer lde, real, dimension( ldf, * ) f, integer ldf, real scale, real dif, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

STGSYL

Purpose:

     STGSYL 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 real entries. (A, D) and (B, E) must be in
     generalized (real) Schur canonical form, i.e. A, B are upper quasi
     triangular and D, E are upper triangular.
     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**T, Im) ]         (2)
                    [ kron(In, D)  -kron(E**T, Im) ].
     Here Ik is the identity matrix of size k and X**T is the transpose of
     X. kron(X, Y) is the Kronecker product between the matrices X and Y.
     If TRANS = 'T', STGSYL solves the transposed system Z**T*y = scale*b,
     which is equivalent to solve for R and L in
                 A**T * R + D**T * L = scale * C           (3)
                 R * B**T + L * E**T = scale * -F
     This case (TRANS = 'T') 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 SLACON.
     If IJOB >= 1, STGSYL 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. See [1-2] for more
     information.
     This is a level 3 BLAS algorithm.

Parameters
    TRANS

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

    IJOB

              IJOB is INTEGER
              Specifies what kind of functionality to be performed.
              = 0: solve (1) only.
              = 1: 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 IJOB  = 1 is used).
              = 4: Only an estimate of Dif[(A,D), (B,E)] is computed.
                   ( SGECON on sub-systems is used ).
              Not referenced if TRANS = 'T'.

    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 REAL array, dimension (LDA, M)
              The upper quasi triangular matrix A.

    LDA

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

    B

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

    LDB

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

    C

              C is REAL 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 REAL 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 REAL 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 REAL 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 = 'T', DIF is not touched.

    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, C and F hold the
              solutions R and L, respectively, to the homogeneous system
              with C = F = 0. Normally, SCALE = 1.

    WORK

              WORK is REAL 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+6)

    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 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 296 of file stgsyl.f.

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

STRSYL

Purpose:

     STRSYL solves the real 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**T, and  A and B are both upper quasi-
     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.
     A and B must be in Schur canonical form (as returned by SHSEQR), that
     is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
     each 2-by-2 diagonal block has its diagonal elements equal and its
     off-diagonal elements of opposite sign.

Parameters
    TRANA

              TRANA is CHARACTER*1
              Specifies the option op(A):
              = 'N': op(A) = A    (No transpose)
              = 'T': op(A) = A**T (Transpose)
              = 'C': op(A) = A**H (Conjugate transpose = Transpose)

    TRANB

              TRANB is CHARACTER*1
              Specifies the option op(B):
              = 'N': op(B) = B    (No transpose)
              = 'T': op(B) = B**T (Transpose)
              = 'C': op(B) = B**H (Conjugate transpose = 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 REAL array, dimension (LDA,M)
              The upper quasi-triangular matrix A, in Schur canonical form.

    LDA

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

    B

              B is REAL array, dimension (LDB,N)
              The upper quasi-triangular matrix B, in Schur canonical form.

    LDB

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

    C

              C is REAL 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 162 of file strsyl.f.

Author

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