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

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', '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 .LT. 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 .LT. 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 .LE. 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 .LT. 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, that
                     store the Householder representation of the stage2
                     band to tridiagonal.

           LHOUS2

                     LHOUS2 is INTEGER
                     The dimension of the array HOUS2. LHOUS2 = MAX(1, dimension)
                     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.
                     LHOUS2 = MAX(1, dimension) where
                     dimension = 4*N if VECT='N'
                     not available now if VECT='H'

           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*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 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*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).

   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*U**T  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*U**T  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

                     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

                     IPIV 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

                     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 NB 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 NB 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 LDWORK = -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 LDWORK 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*U**T 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*U**T;
                     = '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 DOUBLE array, dimension (MAX(1,LWORK))

           LWORK

                     LWORK is INTEGER, LWORK >= MAX(1,3*N-2).

            aram[out] INFO
            batim
                     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*U**T 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*U**T;
                     = '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

                     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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