LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

Functions/Subroutines  
subroutine  sla_syamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) 
SLA_SYAMV computes a matrixvector 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 extraprecise 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 diagonal pivoting method.  
subroutine  ssycon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO) 
SSYCON  
subroutine  ssyconv (UPLO, WAY, N, A, LDA, IPIV, WORK, 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  ssytrd (UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO) 
SSYTRD  
subroutine  ssytrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) 
SSYTRF  
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  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  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 
This is the group of real computational functions for SY matrices
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 matrixvector product using a symmetric indefinite matrix to calculate error bounds.
Download SLA_SYAMV + dependencies [TGZ] [ZIP] [TXT]SLA_SYAMV performs the matrixvector 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 blockstructure 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.
[in]  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. 
[in]  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. 
[in]  ALPHA  ALPHA is REAL . On entry, ALPHA specifies the scalar alpha. Unchanged on exit. 
[in]  A  A is REAL array of DIMENSION ( LDA, n ). Before entry, the leading m by n part of the array A must contain the matrix of coefficients. Unchanged on exit. 
[in]  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. 
[in]  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. 
[in]  INCX  INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit. 
[in]  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. 
[in,out]  Y  Y is REAL array, dimension ( 1 + ( n  1 )*abs( INCY ) ) Before entry with BETA nonzero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y. 
[in]  INCY  INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit. 
Level 2 Blas routine.  Written on 22October1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs.  Modified for the absolutevalue product, April 2006 Jason Riedy, UC Berkeley
Definition at line 177 of file sla_syamv.f.
REAL function sla_syrcond  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldaf, * )  AF,  
integer  LDAF,  
integer, dimension( * )  IPIV,  
integer  CMODE,  
real, dimension( * )  C,  
integer  INFO,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK  
) 
SLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix.
Download SLA_SYRCOND + dependencies [TGZ] [ZIP] [TXT]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 infinitynorm condition number.
[in]  UPLO  UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 
[in]  N  N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. 
[in]  A  A is REAL array, dimension (LDA,N) On entry, the NbyN matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  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. 
[in]  LDAF  LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF. 
[in]  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) 
[in]  C  C is REAL array, dimension (N) The vector C in the formula op(A) * op2(C). 
[out]  INFO  INFO is INTEGER = 0: Successful exit. i > 0: The ith argument is invalid. 
[in]  WORK  WORK is REAL array, dimension (3*N). Workspace. 
[in]  IWORK  IWORK is INTEGER array, dimension (N). Workspace. 
Definition at line 146 of file sla_syrcond.f.
subroutine sla_syrfsx_extended  (  integer  PREC_TYPE, 
character  UPLO,  
integer  N,  
integer  NRHS,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldaf, * )  AF,  
integer  LDAF,  
integer, dimension( * )  IPIV,  
logical  COLEQU,  
real, dimension( * )  C,  
real, dimension( ldb, * )  B,  
integer  LDB,  
real, dimension( ldy, * )  Y,  
integer  LDY,  
real, dimension( * )  BERR_OUT,  
integer  N_NORMS,  
real, dimension( nrhs, * )  ERR_BNDS_NORM,  
real, dimension( nrhs, * )  ERR_BNDS_COMP,  
real, dimension( * )  RES,  
real, dimension( * )  AYB,  
real, dimension( * )  DY,  
real, dimension( * )  Y_TAIL,  
real  RCOND,  
integer  ITHRESH,  
real  RTHRESH,  
real  DZ_UB,  
logical  IGNORE_CWISE,  
integer  INFO  
) 
SLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extraprecise iterative refinement and provides error bounds and backward error estimates for the solution.
Download SLA_SYRFSX_EXTENDED + dependencies [TGZ] [ZIP] [TXT]SLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations by performing extraprecise 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.
[in]  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 
[in]  UPLO  UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 
[in]  N  N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of righthandsides, i.e., the number of columns of the matrix B. 
[in]  A  A is REAL array, dimension (LDA,N) On entry, the NbyN matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  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. 
[in]  LDAF  LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF. 
[in]  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. 
[in]  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. 
[in]  B  B is REAL array, dimension (LDB,NRHS) The righthandside matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in,out]  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. 
[in]  LDY  LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N). 
[out]  BERR_OUT  BERR_OUT is REAL array, dimension (NRHS) On exit, BERR_OUT(j) contains the componentwise relative backward error for righthandside 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. 
[in]  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. 
[in,out]  ERR_BNDS_NORM  ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) For each righthand 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 righthand 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. 
[in,out]  ERR_BNDS_COMP  ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) For each righthand 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 righthand 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 righthand 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 righthand 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 righthand 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. 
[in]  RES  RES is REAL array, dimension (N) Workspace to hold the intermediate residual. 
[in]  AYB  AYB is REAL array, dimension (N) Workspace. This can be the same workspace passed for Y_TAIL. 
[in]  DY  DY is REAL array, dimension (N) Workspace to hold the intermediate solution. 
[in]  Y_TAIL  Y_TAIL is REAL array, dimension (N) Workspace to hold the trailing bits of the intermediate solution. 
[in]  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. 
[in]  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. 
[in]  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 illconditioned matrices. See LAWN 165 for more details. 
[in]  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. 
[in]  IGNORE_CWISE  IGNORE_CWISE is LOGICAL If .TRUE. then ignore componentwise convergence. Default value is .FALSE.. 
[out]  INFO  INFO is INTEGER = 0: Successful exit. < 0: if INFO = i, the ith argument to SLA_SYRFSX_EXTENDED had an illegal value 
Definition at line 391 of file sla_syrfsx_extended.f.
REAL function sla_syrpvgrw  (  character*1  UPLO, 
integer  N,  
integer  INFO,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldaf, * )  AF,  
integer  LDAF,  
integer, dimension( * )  IPIV,  
real, dimension( * )  WORK  
) 
SLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
Download SLA_SYRPVGRW + dependencies [TGZ] [ZIP] [TXT]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.
[in]  UPLO  UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 
[in]  N  N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. 
[in]  INFO  INFO is INTEGER The value of INFO returned from SSYTRF, .i.e., the pivot in column INFO is exactly 0. 
[in]  A  A is REAL array, dimension (LDA,N) On entry, the NbyN matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  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. 
[in]  LDAF  LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF. 
[in]  WORK  WORK is REAL array, dimension (2*N) 
Definition at line 122 of file sla_syrpvgrw.f.
subroutine slasyf  (  character  UPLO, 
integer  N,  
integer  NB,  
integer  KB,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
real, dimension( ldw, * )  W,  
integer  LDW,  
integer  INFO  
) 
SLASYF computes a partial factorization of a real symmetric matrix, using the diagonal pivoting method.
Download SLASYF + dependencies [TGZ] [ZIP] [TXT]SLASYF computes a partial factorization of a real symmetric matrix A using the BunchKaufman 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 NB1, 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').
[in]  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 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  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 2by2 pivot blocks. 
[out]  KB  KB is INTEGER The number of columns of A that were actually factored. KB is either NB1 or NB, or N if N <= NB. 
[in,out]  A  A is REAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading nbyn 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 nbyn 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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  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 UPLO = 'L', only the first KB elements are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1by1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k1) < 0, then rows and columns k1 and IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2 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 2by2 diagonal block. 
[out]  W  W is REAL array, dimension (LDW,NB) 
[in]  LDW  LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N). 
[out]  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. 
Definition at line 157 of file slasyf.f.
subroutine ssycon  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
real  ANORM,  
real  RCOND,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
SSYCON
Download SSYCON + dependencies [TGZ] [ZIP] [TXT]SSYCON estimates the reciprocal of the condition number (in the 1norm) 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))).
[in]  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. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF. 
[in]  ANORM  ANORM is REAL The 1norm of the original matrix A. 
[out]  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 1norm of inv(A) computed in this routine. 
[out]  WORK  WORK is REAL array, dimension (2*N) 
[out]  IWORK  IWORK is INTEGER array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
Definition at line 130 of file ssycon.f.
subroutine ssyconv  (  character  UPLO, 
character  WAY,  
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
real, dimension( * )  WORK,  
integer  INFO  
) 
SSYCONV
Download SSYCONV + dependencies [TGZ] [ZIP] [TXT]SSYCONV convert A given by TRF into L and D and viceversa. Get Nondiag elements of D (returned in workspace) and apply or reverse permutation done in TRF.
[in]  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. 
[in]  WAY  WAY is CHARACTER*1 = 'C': Convert = 'R': Revert 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF. 
[out]  WORK  WORK is REAL array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
Definition at line 113 of file ssyconv.f.
subroutine ssyequb  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( * )  S,  
real  SCOND,  
real  AMAX,  
real, dimension( * )  WORK,  
integer  INFO  
) 
SSYEQUB
Download SSYEQUB + dependencies [TGZ] [ZIP] [TXT]SSYEQUB computes row and column scalings intended to equilibrate a symmetric matrix A and reduce its condition number (with respect to the twonorm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.
[in]  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. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  A  A is REAL array, dimension (LDA,N) The NbyN symmetric matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  S  S is REAL array, dimension (N) If INFO = 0, S contains the scale factors for A. 
[out]  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. 
[out]  AMAX  AMAX is REAL Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. 
[out]  WORK  WORK is REAL array, dimension (3*N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, the ith diagonal element is nonpositive. 
Definition at line 136 of file ssyequb.f.
subroutine ssygs2  (  integer  ITYPE, 
character  UPLO,  
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldb, * )  B,  
integer  LDB,  
integer  INFO  
) 
SSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).
Download SSYGS2 + dependencies [TGZ] [ZIP] [TXT]SSYGS2 reduces a real symmetricdefinite 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.
[in]  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. 
[in]  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 
[in]  N  N is INTEGER The order of the matrices A and B. N >= 0. 
[in,out]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  B  B is REAL array, dimension (LDB,N) The triangular factor from the Cholesky factorization of B, as returned by SPOTRF. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. 
Definition at line 128 of file ssygs2.f.
subroutine ssygst  (  integer  ITYPE, 
character  UPLO,  
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldb, * )  B,  
integer  LDB,  
integer  INFO  
) 
SSYGST
Download SSYGST + dependencies [TGZ] [ZIP] [TXT]SSYGST reduces a real symmetricdefinite 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.
[in]  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. 
[in]  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. 
[in]  N  N is INTEGER The order of the matrices A and B. N >= 0. 
[in,out]  A  A is REAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading NbyN 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 NbyN 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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  B  B is REAL array, dimension (LDB,N) The triangular factor from the Cholesky factorization of B, as returned by SPOTRF. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
Definition at line 128 of file ssygst.f.
subroutine ssyrfs  (  character  UPLO, 
integer  N,  
integer  NRHS,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldaf, * )  AF,  
integer  LDAF,  
integer, dimension( * )  IPIV,  
real, dimension( ldb, * )  B,  
integer  LDB,  
real, dimension( ldx, * )  X,  
integer  LDX,  
real, dimension( * )  FERR,  
real, dimension( * )  BERR,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
SSYRFS
Download SSYRFS + dependencies [TGZ] [ZIP] [TXT]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.
[in]  UPLO  UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. 
[in]  A  A is REAL array, dimension (LDA,N) The symmetric matrix A. If UPLO = 'U', the leading NbyN 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 NbyN 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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  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. 
[in]  LDAF  LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF. 
[in]  B  B is REAL array, dimension (LDB,NRHS) The right hand side matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in,out]  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. 
[in]  LDX  LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). 
[out]  FERR  FERR is REAL array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the jth 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. 
[out]  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). 
[out]  WORK  WORK is REAL array, dimension (3*N) 
[out]  IWORK  IWORK is INTEGER array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
ITMAX is the maximum number of steps of iterative refinement.
Definition at line 191 of file ssyrfs.f.
subroutine ssyrfsx  (  character  UPLO, 
character  EQUED,  
integer  N,  
integer  NRHS,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldaf, * )  AF,  
integer  LDAF,  
integer, dimension( * )  IPIV,  
real, dimension( * )  S,  
real, dimension( ldb, * )  B,  
integer  LDB,  
real, dimension( ldx, * )  X,  
integer  LDX,  
real  RCOND,  
real, dimension( * )  BERR,  
integer  N_ERR_BNDS,  
real, dimension( nrhs, * )  ERR_BNDS_NORM,  
real, dimension( nrhs, * )  ERR_BNDS_COMP,  
integer  NPARAMS,  
real, dimension( * )  PARAMS,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
SSYRFSX
Download SSYRFSX + dependencies [TGZ] [ZIP] [TXT]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.
[in]  UPLO  UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 
[in]  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. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. 
[in]  A  A is REAL array, dimension (LDA,N) The symmetric matrix A. If UPLO = 'U', the leading NbyN 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 NbyN 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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  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. 
[in]  LDAF  LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF. 
[in,out]  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. 
[in]  B  B is REAL array, dimension (LDB,NRHS) The right hand side matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in,out]  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. 
[in]  LDX  LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). 
[out]  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. 
[out]  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). 
[in]  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. 
[out]  ERR_BNDS_NORM  ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) For each righthand 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 righthand 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. 
[out]  ERR_BNDS_COMP  ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) For each righthand 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 righthand 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 righthand 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 righthand 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 righthand 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. 
[in]  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. 
[in,out]  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 highernumbered 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 doubleprecision refinement algorithm, possibly with doubledsingle 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 doubleprecision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence) 
[out]  WORK  WORK is REAL array, dimension (4*N) 
[out]  IWORK  IWORK is INTEGER array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: Successful exit. The solution to every righthand side is guaranteed. < 0: If INFO = i, the ith 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 righthand 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 righthand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth righthand 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 righthand 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 righthand sides check ERR_BNDS_NORM or ERR_BNDS_COMP. 
Definition at line 400 of file ssyrfsx.f.
subroutine ssytd2  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( * )  D,  
real, dimension( * )  E,  
real, dimension( * )  TAU,  
integer  INFO  
) 
SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
Download SSYTD2 + dependencies [TGZ] [ZIP] [TXT]SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.
[in]  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 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  A  A is REAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading nbyn 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 nbyn 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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  D  D is REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). 
[out]  E  E is REAL array, dimension (N1) The offdiagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. 
[out]  TAU  TAU is REAL array, dimension (N1) The scalar factors of the elementary reflectors (see Further Details). 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. 
If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n1) . . . 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:i1) is stored on exit in A(1:i1,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(n1). 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 offdiagonal elements of T, and vi denotes an element of the vector defining H(i).
Definition at line 174 of file ssytd2.f.
subroutine ssytf2  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
integer  INFO  
) 
SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
Download SSYTF2 + dependencies [TGZ] [ZIP] [TXT]SSYTF2 computes the factorization of a real symmetric matrix A using the BunchKaufman 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 1by1 and 2by2 diagonal blocks. This is the unblocked version of the algorithm, calling Level 2 BLAS.
[in]  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 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  A  A is REAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading nbyn 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 nbyn 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). 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  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 1by1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k1) < 0, then rows and columns k1 and IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2 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 2by2 diagonal block. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = k, the kth 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. 
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 1by1 and 2by2 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 ) ks U(k) = ( 0 I 0 ) s ( 0 0 I ) nk ks s nk If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k1,k). If s = 2, the upper triangle of D(k) overwrites A(k1,k1), A(k1,k), and A(k,k), and v overwrites A(1:k2,k1: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 1by1 and 2by2 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 ) k1 L(k) = ( 0 I 0 ) s ( 0 v I ) nks+1 k1 s nks+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).
092906  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 010196  Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA 196  Based on modifications by J. Lewis, Boeing Computer Services Company
Definition at line 187 of file ssytf2.f.
subroutine ssytrd  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( * )  D,  
real, dimension( * )  E,  
real, dimension( * )  TAU,  
real, dimension( * )  WORK,  
integer  LWORK,  
integer  INFO  
) 
SSYTRD
Download SSYTRD + dependencies [TGZ] [ZIP] [TXT]SSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T.
[in]  UPLO  UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  A  A is REAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading NbyN 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 NbyN 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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  D  D is REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). 
[out]  E  E is REAL array, dimension (N1) The offdiagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. 
[out]  TAU  TAU is REAL array, dimension (N1) The scalar factors of the elementary reflectors (see Further Details). 
[out]  WORK  WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 
[in]  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. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n1) . . . 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:i1) is stored on exit in A(1:i1,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(n1). 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 offdiagonal elements of T, and vi denotes an element of the vector defining H(i).
Definition at line 193 of file ssytrd.f.
subroutine ssytrf  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
real, dimension( * )  WORK,  
integer  LWORK,  
integer  INFO  
) 
SSYTRF
Download SSYTRF + dependencies [TGZ] [ZIP] [TXT]SSYTRF computes the factorization of a real symmetric matrix A using the BunchKaufman 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 1by1 and 2by2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS.
[in]  UPLO  UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  A  A is REAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading NbyN 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 NbyN 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). 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  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 1by1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k1) < 0, then rows and columns k1 and IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2 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 2by2 diagonal block. 
[out]  WORK  WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 
[in]  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. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith 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. 
If UPLO = 'U', then A = U*D*U**T, where U = P(n)*U(n)* ... <em>P(k)U(k)</em> ..., 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 1by1 and 2by2 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 ) ks U(k) = ( 0 I 0 ) s ( 0 0 I ) nk ks s nk If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k1,k). If s = 2, the upper triangle of D(k) overwrites A(k1,k1), A(k1,k), and A(k,k), and v overwrites A(1:k2,k1:k). If UPLO = 'L', then A = L*D*L**T, where L = P(1)*L(1)* ... <em>P(k)*L(k)</em> ..., 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 1by1 and 2by2 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 ) k1 L(k) = ( 0 I 0 ) s ( 0 v I ) nks+1 k1 s nks+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
Definition at line 183 of file ssytrf.f.
subroutine ssytri  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
real, dimension( * )  WORK,  
integer  INFO  
) 
SSYTRI
Download SSYTRI + dependencies [TGZ] [ZIP] [TXT]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.
[in]  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. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF. 
[out]  WORK  WORK is REAL array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. 
Definition at line 115 of file ssytri.f.
subroutine ssytri2  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
real, dimension( * )  WORK,  
integer  LWORK,  
integer  INFO  
) 
SSYTRI2
Download SSYTRI2 + dependencies [TGZ] [ZIP] [TXT]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.
[in]  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. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the NB structure of D as determined by SSYTRF. 
[out]  WORK  WORK is REAL array, dimension (N+NB+1)*(NB+3) 
[in]  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. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. 
Definition at line 128 of file ssytri2.f.
subroutine ssytri2x  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
real, dimension( n+nb+1,* )  WORK,  
integer  NB,  
integer  INFO  
) 
SSYTRI2X
Download SSYTRI2X + dependencies [TGZ] [ZIP] [TXT]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.
[in]  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. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the NNB structure of D as determined by SSYTRF. 
[out]  WORK  WORK is REAL array, dimension (N+NNB+1,NNB+3) 
[in]  NB  NB is INTEGER Block size 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. 
Definition at line 121 of file ssytri2x.f.
subroutine ssytrs  (  character  UPLO, 
integer  N,  
integer  NRHS,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
real, dimension( ldb, * )  B,  
integer  LDB,  
integer  INFO  
) 
SSYTRS
Download SSYTRS + dependencies [TGZ] [ZIP] [TXT]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.
[in]  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. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 
[in]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF. 
[in,out]  B  B is REAL array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
Definition at line 121 of file ssytrs.f.
subroutine ssytrs2  (  character  UPLO, 
integer  N,  
integer  NRHS,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer, dimension( * )  IPIV,  
real, dimension( ldb, * )  B,  
integer  LDB,  
real, dimension( * )  WORK,  
integer  INFO  
) 
SSYTRS2
Download SSYTRS2 + dependencies [TGZ] [ZIP] [TXT]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.
[in]  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. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 
[in]  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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  IPIV  IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF. 
[in,out]  B  B is REAL array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[out]  WORK  WORK is REAL array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
Definition at line 127 of file ssytrs2.f.
subroutine stgsyl  (  character  TRANS, 
integer  IJOB,  
integer  M,  
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldb, * )  B,  
integer  LDB,  
real, dimension( ldc, * )  C,  
integer  LDC,  
real, dimension( ldd, * )  D,  
integer  LDD,  
real, dimension( lde, * )  E,  
integer  LDE,  
real, dimension( ldf, * )  F,  
integer  LDF,  
real  SCALE,  
real  DIF,  
real, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
STGSYL
Download STGSYL + dependencies [TGZ] [ZIP] [TXT]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 mbyn matrices, (A, D), (B, E) and (C, F) are given matrix pairs of size mbym, nbyn and mbyn, 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 onenormbased 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 normbased 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 [12] for more information. This is a level 3 BLAS algorithm.
[in]  TRANS  TRANS is CHARACTER*1 = 'N', solve the generalized Sylvester equation (1). = 'T', solve the 'transposed' system (3). 
[in]  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 subsystems is used ). Not referenced if TRANS = 'T'. 
[in]  M  M is INTEGER The order of the matrices A and D, and the row dimension of the matrices C, F, R and L. 
[in]  N  N is INTEGER The order of the matrices B and E, and the column dimension of the matrices C, F, R and L. 
[in]  A  A is REAL array, dimension (LDA, M) The upper quasi triangular matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1, M). 
[in]  B  B is REAL array, dimension (LDB, N) The upper quasi triangular matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1, N). 
[in,out]  C  C is REAL array, dimension (LDC, N) On entry, C contains the righthandside 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 Difestimate. 
[in]  LDC  LDC is INTEGER The leading dimension of the array C. LDC >= max(1, M). 
[in]  D  D is REAL array, dimension (LDD, M) The upper triangular matrix D. 
[in]  LDD  LDD is INTEGER The leading dimension of the array D. LDD >= max(1, M). 
[in]  E  E is REAL array, dimension (LDE, N) The upper triangular matrix E. 
[in]  LDE  LDE is INTEGER The leading dimension of the array E. LDE >= max(1, N). 
[in,out]  F  F is REAL array, dimension (LDF, N) On entry, F contains the righthandside 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 Difestimate. 
[in]  LDF  LDF is INTEGER The leading dimension of the array F. LDF >= max(1, M). 
[out]  DIF  DIF is REAL On exit DIF is the reciprocal of a lower bound of the reciprocal of the Diffunction, 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. 
[out]  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. 
[out]  WORK  WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 
[in]  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. 
[out]  IWORK  IWORK is INTEGER array, dimension (M+N+6) 
[out]  INFO  INFO is INTEGER =0: successful exit <0: If INFO = i, the ith argument had an illegal value. >0: (A, D) and (B, E) have common or close eigenvalues. 
[1] B. Kagstrom and P. Poromaa, LAPACKStyle 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, S901 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):10451060, 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 745751.
Definition at line 298 of file stgsyl.f.
subroutine strsyl  (  character  TRANA, 
character  TRANB,  
integer  ISGN,  
integer  M,  
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldb, * )  B,  
integer  LDB,  
real, dimension( ldc, * )  C,  
integer  LDC,  
real  SCALE,  
integer  INFO  
) 
STRSYL
Download STRSYL + dependencies [TGZ] [ZIP] [TXT]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 MbyM and B is NbyN; the right hand side C and the solution X are MbyN; 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 1by1 and 2by2 diagonal blocks; each 2by2 diagonal block has its diagonal elements equal and its offdiagonal elements of opposite sign.
[in]  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) 
[in]  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) 
[in]  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 
[in]  M  M is INTEGER The order of the matrix A, and the number of rows in the matrices X and C. M >= 0. 
[in]  N  N is INTEGER The order of the matrix B, and the number of columns in the matrices X and C. N >= 0. 
[in]  A  A is REAL array, dimension (LDA,M) The upper quasitriangular matrix A, in Schur canonical form. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[in]  B  B is REAL array, dimension (LDB,N) The upper quasitriangular matrix B, in Schur canonical form. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in,out]  C  C is REAL array, dimension (LDC,N) On entry, the MbyN right hand side matrix C. On exit, C is overwritten by the solution matrix X. 
[in]  LDC  LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M) 
[out]  SCALE  SCALE is REAL The scale factor, scale, set <= 1 to avoid overflow in X. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value = 1: A and B have common or very close eigenvalues; perturbed values were used to solve the equation (but the matrices A and B are unchanged). 
Definition at line 164 of file strsyl.f.