cla_gbrfsx_extended - ear equations for general banded matrices by performing extra-precise iterative refinement and provide error bounds and backward error esti- mates for the solution
SUBROUTINE CLA_GBRFSX_EXTENDED (PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 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) INTEGER INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS, PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH LOGICAL COLEQU, IGNORE_CWISE REAL RTHRESH, DZ_UB INTEGER IPIV(*) COMPLEX AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), Y(LDY,*), RES(*),DY(*), Y_TAIL(*) REAL C(*), AYB(*), RCOND, BERR_OUT(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) SUBROUTINE CLA_GBRFSX_EXTENDED_64 (PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 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) INTEGER*8 INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS, PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH LOGICAL*8 COLEQU, IGNORE_CWISE REAL RTHRESH, DZ_UB INTEGER*8 IPIV(*) COMPLEX AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), Y(LDY,*), RES(*),DY(*), Y_TAIL(*) REAL C(*), AYB(*), RCOND, BERR_OUT(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) F95 INTERFACE SUBROUTINE LA_GBRFSX_EXTENDED (PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 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) REAL, DIMENSION(:,:) :: ERR_BNDS_NORM, ERR_BNDS_COMP INTEGER :: PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDY, N_NORMS, ITHRESH, INFO LOGICAL :: COLEQU, IGNORE_CWISE INTEGER, DIMENSION(:) :: IPIV REAL, DIMENSION(:) :: C, BERR_OUT, AYB REAL :: RCOND, RTHRESH, DZ_UB COMPLEX, DIMENSION(:) :: RES, DY, Y_TAIL COMPLEX, DIMENSION(:,:) :: AB, AFB, B, Y SUBROUTINE LA_GBRFSX_EXTENDED_64 ( PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 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) REAL, DIMENSION(:,:) :: ERR_BNDS_NORM, ERR_BNDS_COMP INTEGER(8) :: PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDY, N_NORMS, ITHRESH, INFO LOGICAL(8) :: COLEQU, IGNORE_CWISE INTEGER(8), DIMENSION(:) :: IPIV REAL, DIMENSION(:) :: C, BERR_OUT, AYB REAL :: RCOND, RTHRESH, DZ_UB COMPLEX, DIMENSION(:) :: RES, DY, Y_TAIL COMPLEX, DIMENSION(:,:) :: AB, AFB, B, Y C INTERFACE #include <sunperf.h> void cla_gbrfsx_extended (int prec_type, int trans_type, int n, int kl, int ku, int nrhs, floatcomplex *ab, int ldab, floatcomplex *afb, int ldafb, int *ipiv, int colequ, float *c, floatcom- plex *b, int ldb, floatcomplex *y, int ldy, float *berr_out, int n_norms, float *err_bnds_norm, float *err_bnds_comp, float rcond, int ithresh, float rthresh, float dz_ub, int ignore_cwise, int *info); void cla_gbrfsx_extended_64 (long prec_type, long trans_type, long n, long kl, long ku, long nrhs, floatcomplex *ab, long ldab, floatcomplex *afb, long ldafb, long *ipiv, long colequ, float *c, floatcomplex *b, long ldb, floatcomplex *y, long ldy, float *berr_out, long n_norms, float *err_bnds_norm, float *err_bnds_comp, float rcond, long ithresh, float rthresh, float dz_ub, long ignore_cwise, long *info);
Oracle Solaris Studio Performance Library cla_gbrfsx_extended(3P) NAME cla_gbrfsx_extended - improve the computed solution to a system of lin- ear equations for general banded matrices by performing extra-precise iterative refinement and provide error bounds and backward error esti- mates for the solution SYNOPSIS SUBROUTINE CLA_GBRFSX_EXTENDED (PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 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) INTEGER INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS, PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH LOGICAL COLEQU, IGNORE_CWISE REAL RTHRESH, DZ_UB INTEGER IPIV(*) COMPLEX AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), Y(LDY,*), RES(*),DY(*), Y_TAIL(*) REAL C(*), AYB(*), RCOND, BERR_OUT(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) SUBROUTINE CLA_GBRFSX_EXTENDED_64 (PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 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) INTEGER*8 INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS, PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH LOGICAL*8 COLEQU, IGNORE_CWISE REAL RTHRESH, DZ_UB INTEGER*8 IPIV(*) COMPLEX AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), Y(LDY,*), RES(*),DY(*), Y_TAIL(*) REAL C(*), AYB(*), RCOND, BERR_OUT(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) F95 INTERFACE SUBROUTINE LA_GBRFSX_EXTENDED (PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 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) REAL, DIMENSION(:,:) :: ERR_BNDS_NORM, ERR_BNDS_COMP INTEGER :: PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDY, N_NORMS, ITHRESH, INFO LOGICAL :: COLEQU, IGNORE_CWISE INTEGER, DIMENSION(:) :: IPIV REAL, DIMENSION(:) :: C, BERR_OUT, AYB REAL :: RCOND, RTHRESH, DZ_UB COMPLEX, DIMENSION(:) :: RES, DY, Y_TAIL COMPLEX, DIMENSION(:,:) :: AB, AFB, B, Y SUBROUTINE LA_GBRFSX_EXTENDED_64 ( PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 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) REAL, DIMENSION(:,:) :: ERR_BNDS_NORM, ERR_BNDS_COMP INTEGER(8) :: PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDY, N_NORMS, ITHRESH, INFO LOGICAL(8) :: COLEQU, IGNORE_CWISE INTEGER(8), DIMENSION(:) :: IPIV REAL, DIMENSION(:) :: C, BERR_OUT, AYB REAL :: RCOND, RTHRESH, DZ_UB COMPLEX, DIMENSION(:) :: RES, DY, Y_TAIL COMPLEX, DIMENSION(:,:) :: AB, AFB, B, Y C INTERFACE #include <sunperf.h> void cla_gbrfsx_extended (int prec_type, int trans_type, int n, int kl, int ku, int nrhs, floatcomplex *ab, int ldab, floatcomplex *afb, int ldafb, int *ipiv, int colequ, float *c, floatcom- plex *b, int ldb, floatcomplex *y, int ldy, float *berr_out, int n_norms, float *err_bnds_norm, float *err_bnds_comp, float rcond, int ithresh, float rthresh, float dz_ub, int ignore_cwise, int *info); void cla_gbrfsx_extended_64 (long prec_type, long trans_type, long n, long kl, long ku, long nrhs, floatcomplex *ab, long ldab, floatcomplex *afb, long ldafb, long *ipiv, long colequ, float *c, floatcomplex *b, long ldb, floatcomplex *y, long ldy, float *berr_out, long n_norms, float *err_bnds_norm, float *err_bnds_comp, float rcond, long ithresh, float rthresh, float dz_ub, long ignore_cwise, long *info); PURPOSE cla_gbrfsx_extended improves the computed solution to a system of lin- ear equations by performing extra-precise iterative refinement and pro- vides error bounds and backward error estimates for the solution. This subroutine is called by CGBRFSX to perform iterative refinement. In addition to normwise error bound, the code provides maximum component- wise error bound if possible. See comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds. Note that this subrou- tine is only resonsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP. ARGUMENTS PREC_TYPE (input) PREC_TYPE is INTEGER Specifies the intermediate precision to be used in refine- ment. The value is defined by ILAPREC(P) where P is a CHAR- ACTER and P = 'S': Single = 'D': Double = 'I': Indigenous = 'X', 'E': Extra TRANS_TYPE (input) TRANS_TYPE is INTEGER Specifies the transposition operation on A. The value is defined by ILATRANS(T) where T is a CHARACTER and T = 'N': No transpose = 'T': Transpose = 'C': Conjugate transpose N (input) N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. KL (input) KL is INTEGER The number of subdiagonals within the band of A. KL >= 0. KU (input) KU is INTEGER The number of superdiagonals within the band of A. KU >= 0 NRHS (input) NRHS is INTEGER The number of right-hand-sides, i.e., the number of columns of the matrix B. AB (input) AB is COMPLEX array, dimension (LDAB,N) On entry, the N-by-N matrix AB. LDAB (input) LDAB is INTEGER The leading dimension of the array AB. LDAB >= max(1,N). AFB (input) AFB is COMPLEX array, dimension (LDAFB,N) The factors L and U from the factorization A=P*L*U as com- puted by CGBTRF. LDAFB (input) LDAFB is INTEGER The leading dimension of the array AFB. LDAFB >= max(1,N). IPIV (input) IPIV is INTEGER array, dimension (N) The pivot indices from the factorization A = P*L*U as com- puted by CGBTRF; row i of the matrix was interchanged with row IPIV(i). COLEQU (input) COLEQU is LOGICAL If .TRUE. then column equilibration was done to A before calling this routine. This is needed to compute the solution and error bounds correctly. C (input) 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 esti- mates. Scaling by powers of the radix does not cause round- ing errors unless the result underflows or overflows. Round- ing errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error esti- mates that may not be reliable. B (input) B is COMPLEX array, dimension (LDB,NRHS) The right-hand-side matrix B. LDB (input) LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). Y (input/output) Y is COMPLEX array, dimension (LDY,NRHS) On entry, the solution matrix X, as computed by CGBTRS. On exit, the improved solution matrix Y. LDY (input) LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N). BERR_OUT (output) BERR_OUT is REAL array, dimension (NRHS) On exit, BERR_OUT(j) contains the componentwise relative backward error for right-hand-side j from the formula max(i)(abs(RES(i))/(abs(op(A_s))*abs(Y)+abs(B_s))(i)) where abs(Z) is the componentwise absolute value of the matrix or vector Z. This is computed by CLA_LIN_BERR. N_NORMS (input) N_NORMS is INTEGER Determines which error bounds to return (see ERR_BNDS_NORM and ERR_BNDS_COMP). If N_NORMS >= 1 return normwise error bounds. If N_NORMS >= 2 return componentwise error bounds. ERR_BNDS_NORM (input/output) ERR_BNDS_NORM is REAL array, dimension (NRHS, N_NORMS) For each right-hand side, this array contains information about various error bounds and condition numbers correspond- ing to the normwise relative error, which is defined as fol- lows: 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 follow- ing 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 esti- mate is "guaranteed". These reciprocal condition numbers are 1/(norm(Z^{-1},inf)*norm(Z,inf)) for some appro- priately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions. ERR_BNDS_COMP (input/output) ERR_BNDS_COMP is REAL array, dimension (NRHS, N_NORMS) For each right-hand side, this array contains information about various error bounds and condition numbers correspond- ing 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_NORMS .LT. 3, then at most the first (:,N_NORMS) entries are returned. The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_COMP(:,err) contains the following three fields: err = 1 "Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). err = 2 "Guaranteed" error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condition numbers are 1/(norm(Z^{-1},inf)*norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions. RES (input) RES is COMPLEX array, dimension (N) Workspace to hold the intermediate residual. AYB (input) AYB is REAL array, dimension (N) Workspace. DY (input) DY is COMPLEX array, dimension (N) Workspace to hold the intermediate solution. Y_TAIL (input) Y_TAIL is COMPLEX array, dimension (N) Workspace to hold the trailing bits of the intermediate solution. RCOND (input) RCOND is REAL Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill- conditioned. ITHRESH (input) ITHRESH is INTEGER The maximum number of residual computations allowed for refinement. The default is 10. For 'aggressive' set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. RTHRESH (input) RTHRESH is REAL Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details. DZ_UB (input) DZ_UB is REAL Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we definte as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details. IGNORE_CWISE (input) IGNORE_CWISE is LOGICAL If .TRUE. then ignore componentwise convergence. Default value is .FALSE.. INFO (output) INFO is INTEGER = 0: Successful exit. < 0: if INFO = -i, the ith argument to CGBTRS had an illegal value. 7 Nov 2015 cla_gbrfsx_extended(3P)