# REALSYCOMPUTATIONAL(3) - Linux man page online | Library functions

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Chapter

Tue Nov 28 2017

**realSYcomputational(3) LAPACK realSYcomputational(3)**

## NAME

realSYcomputational## SYNOPSIS

Functions subroutine sla_syamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) SLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds. real function sla_syrcond (UPLO, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK) SLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix. subroutine sla_syrfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO) SLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. real function sla_syrpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK) SLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix. subroutine slasyf (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) SLASYF computes a partial factorization of a real symmetric matrix using the Bunch- Kaufman diagonal pivoting method. subroutine slasyf_aa (UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK) SLASYF_AA subroutine slasyf_rook (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) SLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. subroutine ssycon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO) SSYCON subroutine ssycon_rook (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO) SSYCON_ROOK subroutine ssyconv (UPLO, WAY, N, A, LDA, IPIV, E, INFO) SSYCONV subroutine ssyequb (UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO) SSYEQUB subroutine ssygs2 (ITYPE, UPLO, N, A, LDA, B, LDB, INFO) SSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm). subroutine ssygst (ITYPE, UPLO, N, A, LDA, B, LDB, INFO) SSYGST subroutine ssyrfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO) SSYRFS subroutine ssyrfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) SSYRFSX subroutine ssytd2 (UPLO, N, A, LDA, D, E, TAU, INFO) SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). subroutine ssytf2 (UPLO, N, A, LDA, IPIV, INFO) SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). subroutine ssytf2_rook (UPLO, N, A, LDA, IPIV, INFO) SSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm). subroutine ssytrd (UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO) SSYTRD subroutine ssytrd_2stage (VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO) SSYTRD_2STAGE subroutine ssytrd_sy2sb (UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO) SSYTRD_SY2SB subroutine ssytrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) SSYTRF subroutine ssytrf_aa (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) SSYTRF_AA subroutine ssytrf_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_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: June 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: December 2016 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: June 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: June 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: December 2016 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: December 2016 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: June 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 Generated automatically by Doxygen for LAPACK from the source code.**Version 3.7.1 Tue Nov 28 2017 realSYcomputational(3)**