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realSYcomputational(3) LAPACK realSYcomputational(3)

NAME

realSYcomputational

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.

Date

June 2017

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

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.

Date

December 2016

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 resonsible 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 definte 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.

Date

December 2016

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.

Date

December 2016

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.

Date

November 2013

Contributors:

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

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.

Date

November 2017

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.

Date

November 2013

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

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.

Date

December 2016

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.

Date

December 2016

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

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.

Date

December 2016

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.

Date

November 2017

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

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.

Date

December 2016

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.

Date

December 2016

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.

Date

December 2016

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.

Date

April 2012

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.

Date

December 2016

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).

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.

Date

December 2016

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

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.

Date

November 2013

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

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.

Date

December 2016

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).

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.

Date

November 2017

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 

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.

Date

November 2017

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
 

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.

Date

December 2016

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).

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.

Date

November 2017

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.

Date

November 2017

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.

Date

June 2016

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

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.

Date

December 2016

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.

Date

November 2017

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.

Date

June 2017

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.

Date

April 2012

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

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.

Date

December 2016

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.

Date

December 2016

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.

Date

November 2017

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.

Date

November 2017

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.

Date

April 2012

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

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.

Date

December 2016

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.

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.

Date

December 2016

Author

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