LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

Functions/Subroutines  
REAL function  sla_porcond (UPLO, N, A, LDA, AF, LDAF, CMODE, C, INFO, WORK, IWORK) 
SLA_PORCOND estimates the Skeel condition number for a symmetric positivedefinite matrix.  
subroutine  sla_porfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, 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_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positivedefinite matrices by performing extraprecise iterative refinement and provides error bounds and backward error estimates for the solution.  
REAL function  sla_porpvgrw (UPLO, NCOLS, A, LDA, AF, LDAF, WORK) 
SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positivedefinite matrix.  
subroutine  spocon (UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO) 
SPOCON  
subroutine  spoequ (N, A, LDA, S, SCOND, AMAX, INFO) 
SPOEQU  
subroutine  spoequb (N, A, LDA, S, SCOND, AMAX, INFO) 
SPOEQUB  
subroutine  sporfs (UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO) 
SPORFS  
subroutine  sporfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) 
SPORFSX  
subroutine  spotf2 (UPLO, N, A, LDA, INFO) 
SPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).  
subroutine  spotrf (UPLO, N, A, LDA, INFO) 
SPOTRF  
subroutine  spotri (UPLO, N, A, LDA, INFO) 
SPOTRI  
subroutine  spotrs (UPLO, N, NRHS, A, LDA, B, LDB, INFO) 
SPOTRS 
This is the group of real computational functions for PO matrices
REAL function sla_porcond  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldaf, * )  AF,  
integer  LDAF,  
integer  CMODE,  
real, dimension( * )  C,  
integer  INFO,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK  
) 
SLA_PORCOND estimates the Skeel condition number for a symmetric positivedefinite matrix.
Download SLA_PORCOND + dependencies [TGZ] [ZIP] [TXT]SLA_PORCOND 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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. 
[in]  LDAF  LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). 
[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 140 of file sla_porcond.f.
subroutine sla_porfsx_extended  (  integer  PREC_TYPE, 
character  UPLO,  
integer  N,  
integer  NRHS,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldaf, * )  AF,  
integer  LDAF,  
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_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positivedefinite matrices by performing extraprecise iterative refinement and provides error bounds and backward error estimates for the solution.
Download SLA_PORFSX_EXTENDED + dependencies [TGZ] [ZIP] [TXT]SLA_PORFSX_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 SPORFSX 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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. 
[in]  LDAF  LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). 
[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 SPOTRS. 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 SPOTRS had an illegal value 
Definition at line 382 of file sla_porfsx_extended.f.
REAL function sla_porpvgrw  (  character*1  UPLO, 
integer  NCOLS,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldaf, * )  AF,  
integer  LDAF,  
real, dimension( * )  WORK  
) 
SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positivedefinite matrix.
Download SLA_PORPVGRW + dependencies [TGZ] [ZIP] [TXT]SLA_PORPVGRW 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]  NCOLS  NCOLS is INTEGER The number of columns of the matrix A. NCOLS >= 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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. 
[in]  LDAF  LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). 
[in]  WORK  WORK is REAL array, dimension (2*N) 
Definition at line 105 of file sla_porpvgrw.f.
subroutine spocon  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real  ANORM,  
real  RCOND,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
SPOCON
Download SPOCON + dependencies [TGZ] [ZIP] [TXT]SPOCON estimates the reciprocal of the condition number (in the 1norm) of a real symmetric positive definite matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPOTRF. 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 = '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]  A  A is REAL array, dimension (LDA,N) The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  ANORM  ANORM is REAL The 1norm (or infinitynorm) of the symmetric 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 (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 
Definition at line 121 of file spocon.f.
subroutine spoequ  (  integer  N, 
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( * )  S,  
real  SCOND,  
real  AMAX,  
integer  INFO  
) 
SPOEQU
Download SPOEQU + dependencies [TGZ] [ZIP] [TXT]SPOEQU computes row and column scalings intended to equilibrate a symmetric positive definite 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]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  A  A is REAL array, dimension (LDA,N) The NbyN symmetric positive definite 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]  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 113 of file spoequ.f.
subroutine spoequb  (  integer  N, 
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( * )  S,  
real  SCOND,  
real  AMAX,  
integer  INFO  
) 
SPOEQUB
Download SPOEQUB + dependencies [TGZ] [ZIP] [TXT]SPOEQU computes row and column scalings intended to equilibrate a symmetric positive definite 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]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  A  A is REAL array, dimension (LDA,N) The NbyN symmetric positive definite 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]  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 113 of file spoequb.f.
subroutine sporfs  (  character  UPLO, 
integer  N,  
integer  NRHS,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldaf, * )  AF,  
integer  LDAF,  
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  
) 
SPORFS
Download SPORFS + dependencies [TGZ] [ZIP] [TXT]SPORFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, 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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. 
[in]  LDAF  LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). 
[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 SPOTRS. 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 183 of file sporfs.f.
subroutine sporfsx  (  character  UPLO, 
character  EQUED,  
integer  N,  
integer  NRHS,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldaf, * )  AF,  
integer  LDAF,  
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  
) 
SPORFSX
Download SPORFSX + dependencies [TGZ] [ZIP] [TXT]SPORFSX improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, 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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. 
[in]  LDAF  LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N). 
[in,out]  S  S is REAL array, dimension (N) The row 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 392 of file sporfsx.f.
subroutine spotf2  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer  INFO  
) 
SPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
Download SPOTF2 + dependencies [TGZ] [ZIP] [TXT]SPOTF2 computes the Cholesky factorization of a real symmetric positive definite matrix A. The factorization has the form A = U**T * U , if UPLO = 'U', or A = L * L**T, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular. 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 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 factor U or L from the Cholesky factorization A = U**T *U or A = L*L**T. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = k, the kth argument had an illegal value > 0: if INFO = k, the leading minor of order k is not positive definite, and the factorization could not be completed. 
Definition at line 110 of file spotf2.f.
subroutine spotrf  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer  INFO  
) 
SPOTRF
Download SPOTRF + dependencies [TGZ] [ZIP] [TXT]SPOTRF computes the Cholesky factorization of a real symmetric positive definite matrix A. The factorization has the form A = U**T * U, if UPLO = 'U', or A = L * L**T, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular. This is the block 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, if INFO = 0, the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,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 leading minor of order i is not positive definite, and the factorization could not be completed. 
Definition at line 108 of file spotrf.f.
subroutine spotri  (  character  UPLO, 
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer  INFO  
) 
SPOTRI
Download SPOTRI + dependencies [TGZ] [ZIP] [TXT]SPOTRI computes the inverse of a real symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPOTRF.
[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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. On exit, the upper or lower triangle of the (symmetric) inverse of A, overwriting the input factor U or L. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,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 (i,i) element of the factor U or L is zero, and the inverse could not be computed. 
Definition at line 96 of file spotri.f.
subroutine spotrs  (  character  UPLO, 
integer  N,  
integer  NRHS,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldb, * )  B,  
integer  LDB,  
integer  INFO  
) 
SPOTRS
Download SPOTRS + dependencies [TGZ] [ZIP] [TXT]SPOTRS solves a system of linear equations A*X = B with a symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPOTRF.
[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 matrix B. NRHS >= 0. 
[in]  A  A is REAL array, dimension (LDA,N) The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[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 111 of file spotrs.f.