Provided by: liblapack-doc_3.9.0-1build1_all

**NAME**

realPOcomputational

**SYNOPSIS**

Functionsreal functionsla_porcond(UPLO, N, A, LDA, AF, LDAF, CMODE, C, INFO, WORK, IWORK)SLA_PORCONDestimates the Skeel condition number for a symmetric positive-definite matrix. subroutinesla_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_EXTENDEDimproves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. real functionsla_porpvgrw(UPLO, NCOLS, A, LDA, AF, LDAF, WORK)SLA_PORPVGRWcomputes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix. subroutinespocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)SPOCONsubroutinespoequ(N, A, LDA, S, SCOND, AMAX, INFO)SPOEQUsubroutinespoequb(N, A, LDA, S, SCOND, AMAX, INFO)SPOEQUBsubroutinesporfs(UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)SPORFSsubroutinesporfsx(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)SPORFSXsubroutinespotf2(UPLO, N, A, LDA, INFO)SPOTF2computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm). subroutinespotrf(UPLO, N, A, LDA, INFO)SPOTRFrecursive subroutinespotrf2(UPLO, N, A, LDA, INFO)SPOTRF2subroutinespotri(UPLO, N, A, LDA, INFO)SPOTRIsubroutinespotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)SPOTRS

**Detailed** **Description**

This is the group of real computational functions for PO matrices

**Function** **Documentation**

realfunctionsla_porcond(characterUPLO,integerN,real,dimension(lda,*)A,integerLDA,real,dimension(ldaf,*)AF,integerLDAF,integerCMODE,real,dimension(*)C,integerINFO,real,dimension(*)WORK,integer,dimension(*)IWORK)SLA_PORCONDestimates the Skeel condition number for a symmetric positive-definite matrix.Purpose: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 infinity-norm condition number.ParametersUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.AA is REAL array, dimension (LDA,N) On entry, the N-by-N matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AFAF 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.LDAFLDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).CMODECMODE 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)CC is REAL array, dimension (N) The vector C in the formula op(A) * op2(C).INFOINFO is INTEGER = 0: Successful exit. i > 0: The ith argument is invalid.WORKWORK is REAL array, dimension (3*N). Workspace.IWORKIWORK is INTEGER array, dimension (N). Workspace.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinesla_porfsx_extended(integerPREC_TYPE,characterUPLO,integerN,integerNRHS,real,dimension(lda,*)A,integerLDA,real,dimension(ldaf,*)AF,integerLDAF,logicalCOLEQU,real,dimension(*)C,real,dimension(ldb,*)B,integerLDB,real,dimension(ldy,*)Y,integerLDY,real,dimension(*)BERR_OUT,integerN_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,realRCOND,integerITHRESH,realRTHRESH,realDZ_UB,logicalIGNORE_CWISE,integerINFO)SLA_PORFSX_EXTENDEDimproves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.Purpose:SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. This subroutine is called by 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.ParametersPREC_TYPEPREC_TYPE is INTEGER Specifies the intermediate precision to be used in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER and P = 'S': Single = 'D': Double = 'I': Indigenous = 'X' or 'E': ExtraUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.NRHSNRHS is INTEGER The number of right-hand-sides, i.e., the number of columns of the matrix B.AA is REAL array, dimension (LDA,N) On entry, the N-by-N matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AFAF 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.LDAFLDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).COLEQUCOLEQU 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.CC 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.BB is REAL array, dimension (LDB,NRHS) The right-hand-side matrix B.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).YY is REAL array, dimension (LDY,NRHS) On entry, the solution matrix X, as computed by SPOTRS. On exit, the improved solution matrix Y.LDYLDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N).BERR_OUTBERR_OUT is REAL array, dimension (NRHS) On exit, BERR_OUT(j) contains the componentwise relative backward error for right-hand-side j from the formula max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) where abs(Z) is the componentwise absolute value of the matrix or vector Z. This is computed by SLA_LIN_BERR.N_NORMSN_NORMS is INTEGER Determines which error bounds to return (see ERR_BNDS_NORM and ERR_BNDS_COMP). If N_NORMS >= 1 return normwise error bounds. If N_NORMS >= 2 return componentwise error bounds.ERR_BNDS_NORMERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows: Normwise relative error in the ith solution vector: max_j (abs(XTRUE(j,i) - X(j,i))) ------------------------------ max_j abs(X(j,i)) The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned. The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_NORM(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.ERR_BNDS_COMPERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows: Componentwise relative error in the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j ---------------------- abs(X(j,i)) The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned. The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions.RESRES is REAL array, dimension (N) Workspace to hold the intermediate residual.AYBAYB is REAL array, dimension (N) Workspace. This can be the same workspace passed for Y_TAIL.DYDY is REAL array, dimension (N) Workspace to hold the intermediate solution.Y_TAILY_TAIL is REAL array, dimension (N) Workspace to hold the trailing bits of the intermediate solution.RCONDRCOND 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.ITHRESHITHRESH 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.RTHRESHRTHRESH is REAL Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details.DZ_UBDZ_UB is REAL Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we definte as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details.IGNORE_CWISEIGNORE_CWISE is LOGICAL If .TRUE. then ignore componentwise convergence. Default value is .FALSE..INFOINFO is INTEGER = 0: Successful exit. < 0: if INFO = -i, the ith argument to SPOTRS had an illegal valueAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016realfunctionsla_porpvgrw(character*1UPLO,integerNCOLS,real,dimension(lda,*)A,integerLDA,real,dimension(ldaf,*)AF,integerLDAF,real,dimension(*)WORK)SLA_PORPVGRWcomputes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.Purpose: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.ParametersUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NCOLSNCOLS is INTEGER The number of columns of the matrix A. NCOLS >= 0.AA is REAL array, dimension (LDA,N) On entry, the N-by-N matrix A.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AFAF 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.LDAFLDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).WORKWORK is REAL array, dimension (2*N)AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinespocon(characterUPLO,integerN,real,dimension(lda,*)A,integerLDA,realANORM,realRCOND,real,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)SPOCONPurpose:SPOCON estimates the reciprocal of the condition number (in the 1-norm) 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))).ParametersUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.AA 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).ANORMANORM is REAL The 1-norm (or infinity-norm) of the symmetric matrix A.RCONDRCOND is REAL The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.WORKWORK is REAL array, dimension (3*N)IWORKIWORK is INTEGER array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinespoequ(integerN,real,dimension(lda,*)A,integerLDA,real,dimension(*)S,realSCOND,realAMAX,integerINFO)SPOEQUPurpose:SPOEQU computes row and column scalings intended to equilibrate a symmetric positive definite matrix A and reduce its condition number (with respect to the two-norm). 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.ParametersNN is INTEGER The order of the matrix A. N >= 0.AA is REAL array, dimension (LDA,N) The N-by-N symmetric positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).SS is REAL array, dimension (N) If INFO = 0, S contains the scale factors for A.SCONDSCOND 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.AMAXAMAX 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.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the i-th diagonal element is nonpositive.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinespoequb(integerN,real,dimension(lda,*)A,integerLDA,real,dimension(*)S,realSCOND,realAMAX,integerINFO)SPOEQUBPurpose:SPOEQUB computes row and column scalings intended to equilibrate a symmetric positive definite matrix A and reduce its condition number (with respect to the two-norm). 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. This routine differs from SPOEQU by restricting the scaling factors to a power of the radix. Barring over- and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled diagonal entries are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).ParametersNN is INTEGER The order of the matrix A. N >= 0.AA is REAL array, dimension (LDA,N) The N-by-N symmetric positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).SS is REAL array, dimension (N) If INFO = 0, S contains the scale factors for A.SCONDSCOND 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.AMAXAMAX 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.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the i-th diagonal element is nonpositive.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinesporfs(characterUPLO,integerN,integerNRHS,real,dimension(lda,*)A,integerLDA,real,dimension(ldaf,*)AF,integerLDAF,real,dimension(ldb,*)B,integerLDB,real,dimension(ldx,*)X,integerLDX,real,dimension(*)FERR,real,dimension(*)BERR,real,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)SPORFSPurpose: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.ParametersUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.NRHSNRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.AA is REAL array, dimension (LDA,N) The symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AFAF 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.LDAFLDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).BB is REAL array, dimension (LDB,NRHS) The right hand side matrix B.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).XX is REAL array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by SPOTRS. On exit, the improved solution matrix X.LDXLDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).FERRFERR is REAL array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.BERRBERR 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).WORKWORK is REAL array, dimension (3*N)IWORKIWORK is INTEGER array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueInternalParameters:ITMAX is the maximum number of steps of iterative refinement.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinesporfsx(characterUPLO,characterEQUED,integerN,integerNRHS,real,dimension(lda,*)A,integerLDA,real,dimension(ldaf,*)AF,integerLDAF,real,dimension(*)S,real,dimension(ldb,*)B,integerLDB,real,dimension(ldx,*)X,integerLDX,realRCOND,real,dimension(*)BERR,integerN_ERR_BNDS,real,dimension(nrhs,*)ERR_BNDS_NORM,real,dimension(nrhs,*)ERR_BNDS_COMP,integerNPARAMS,real,dimension(*)PARAMS,real,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)SPORFSXPurpose: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.ParametersUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.EQUEDEQUED 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.NN is INTEGER The order of the matrix A. N >= 0.NRHSNRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.AA is REAL array, dimension (LDA,N) The symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).AFAF 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.LDAFLDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).SS 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.BB is REAL array, dimension (LDB,NRHS) The right hand side matrix B.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).XX is REAL array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by SGETRS. On exit, the improved solution matrix X.LDXLDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).RCONDRCOND 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.BERRBERR is REAL array, dimension (NRHS) Componentwise relative backward error. This is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).N_ERR_BNDSN_ERR_BNDS is INTEGER Number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below.ERR_BNDS_NORMERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows: Normwise relative error in the ith solution vector: max_j (abs(XTRUE(j,i) - X(j,i))) ------------------------------ max_j abs(X(j,i)) The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned. The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_NORM(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. See Lapack Working Note 165 for further details and extra cautions.ERR_BNDS_COMPERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows: Componentwise relative error in the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j ---------------------- abs(X(j,i)) The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned. The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. See Lapack Working Note 165 for further details and extra cautions.NPARAMSNPARAMS is INTEGER Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used.PARAMSPARAMS is REAL array, dimension NPARAMS Specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters. PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative refinement or not. Default: 1.0 = 0.0: No refinement is performed, and no error bounds are computed. = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION. (other values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual computations allowed for refinement. Default: 10 Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence)WORKWORK is REAL array, dimension (4*N)IWORKIWORK is INTEGER array, dimension (N)INFOINFO is INTEGER = 0: Successful exit. The solution to every right-hand side is guaranteed. < 0: If INFO = -i, the i-th argument had an illegal value > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right- hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateApril 2012subroutinespotf2(characterUPLO,integerN,real,dimension(lda,*)A,integerLDA,integerINFO)SPOTF2computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).Purpose: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.ParametersUPLOUPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangularNN is INTEGER The order of the matrix A. N >= 0.AA 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th 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.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinespotrf(characterUPLO,integerN,real,dimension(lda,*)A,integerLDA,integerINFO)SPOTRFSPOTRFVARIANT: top-looking block version of the algorithm, calling Level 3 BLAS.Purpose: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.ParametersUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.AA 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th 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.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016Purpose: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 top-looking block version of the algorithm, calling Level 3 BLAS.ParametersUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.AA 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th 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.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016recursivesubroutinespotrf2(characterUPLO,integerN,real,dimension(lda,*)A,integerLDA,integerINFO)SPOTRF2Purpose:SPOTRF2 computes the Cholesky factorization of a real symmetric positive definite matrix A using the recursive algorithm. 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 recursive version of the algorithm. It divides the matrix into four submatrices: [ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2 A = [ -----|----- ] with n1 = n/2 [ A21 | A22 ] n2 = n-n1 The subroutine calls itself to factor A11. Update and scale A21 or A12, update A22 then call itself to factor A22.ParametersUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.AA 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th 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.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateNovember 2017subroutinespotri(characterUPLO,integerN,real,dimension(lda,*)A,integerLDA,integerINFO)SPOTRIPurpose: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.ParametersUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.AA 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th 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.AuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016subroutinespotrs(characterUPLO,integerN,integerNRHS,real,dimension(lda,*)A,integerLDA,real,dimension(ldb,*)B,integerLDB,integerINFO)SPOTRSPurpose: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.ParametersUPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.NRHSNRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.AA 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.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).BB is REAL array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthorUniv. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.DateDecember 2016

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