dgbrfsx - improve the computed solution to a system of linear equations and provide error bounds and backward error estimates for the solution
SUBROUTINE DGBRFSX(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) CHARACTER*1 TRANS, EQUED INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, NPARAMS, N_ERR_BNDS DOUBLE PRECISION RCOND INTEGER IPIV(*), IWORK(*) DOUBLE PRECISION AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), X(LDX,*), WORK(*) DOUBLE PRECISION R(*), C(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) SUBROUTINE DGBRFSX_64(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) CHARACTER*1 TRANS, EQUED INTEGER*8 INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, NPARAMS, N_ERR_BNDS DOUBLE PRECISION RCOND INTEGER*8 IPIV(*), IWORK(*) DOUBLE PRECISION AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), X(LDX,*), WORK(*) DOUBLE PRECISION R(*), C(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) F95 INTERFACE SUBROUTINE GBRFSX(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) INTEGER :: N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO CHARACTER(LEN=1) :: TRANS, EQUED INTEGER, DIMENSION(:) :: IPIV, IWORK REAL(8), DIMENSION(:,:) :: AB, AFB, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP REAL(8), DIMENSION(:) :: R, C, BERR, PARAMS, WORK REAL(8) :: RCOND SUBROUTINE GBRFSX_64(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) INTEGER(8) :: N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO CHARACTER(LEN=1) :: TRANS, EQUED INTEGER(8), DIMENSION(:) :: IPIV, IWORK REAL(8), DIMENSION(:,:) :: AB, AFB, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP REAL(8), DIMENSION(:) :: R, C, BERR, PARAMS, WORK REAL(8) :: RCOND C INTERFACE #include <sunperf.h> void dgbrfsx (char trans, char equed, int n, int kl, int ku, int nrhs, double *ab, int ldab, double *afb, int ldafb, int *ipiv, dou- ble *r, double *c, double *b, int ldb, double *x, int ldx, double *rcond, double *berr, int n_err_bnds, double *err_bnds_norm, double *err_bnds_comp, int nparams, double *params, int *info); void dgbrfsx_64 (char trans, char equed, long n, long kl, long ku, long nrhs, double *ab, long ldab, double *afb, long ldafb, long *ipiv, double *r, double *c, double *b, long ldb, double *x, long ldx, double *rcond, double *berr, long n_err_bnds, dou- ble *err_bnds_norm, double *err_bnds_comp, long nparams, dou- ble *params, long *info);
Oracle Solaris Studio Performance Library dgbrfsx(3P) NAME dgbrfsx - improve the computed solution to a system of linear equations and provide error bounds and backward error estimates for the solution SYNOPSIS SUBROUTINE DGBRFSX(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) CHARACTER*1 TRANS, EQUED INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, NPARAMS, N_ERR_BNDS DOUBLE PRECISION RCOND INTEGER IPIV(*), IWORK(*) DOUBLE PRECISION AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), X(LDX,*), WORK(*) DOUBLE PRECISION R(*), C(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) SUBROUTINE DGBRFSX_64(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) CHARACTER*1 TRANS, EQUED INTEGER*8 INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, NPARAMS, N_ERR_BNDS DOUBLE PRECISION RCOND INTEGER*8 IPIV(*), IWORK(*) DOUBLE PRECISION AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), X(LDX,*), WORK(*) DOUBLE PRECISION R(*), C(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) F95 INTERFACE SUBROUTINE GBRFSX(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) INTEGER :: N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO CHARACTER(LEN=1) :: TRANS, EQUED INTEGER, DIMENSION(:) :: IPIV, IWORK REAL(8), DIMENSION(:,:) :: AB, AFB, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP REAL(8), DIMENSION(:) :: R, C, BERR, PARAMS, WORK REAL(8) :: RCOND SUBROUTINE GBRFSX_64(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) INTEGER(8) :: N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO CHARACTER(LEN=1) :: TRANS, EQUED INTEGER(8), DIMENSION(:) :: IPIV, IWORK REAL(8), DIMENSION(:,:) :: AB, AFB, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP REAL(8), DIMENSION(:) :: R, C, BERR, PARAMS, WORK REAL(8) :: RCOND C INTERFACE #include <sunperf.h> void dgbrfsx (char trans, char equed, int n, int kl, int ku, int nrhs, double *ab, int ldab, double *afb, int ldafb, int *ipiv, dou- ble *r, double *c, double *b, int ldb, double *x, int ldx, double *rcond, double *berr, int n_err_bnds, double *err_bnds_norm, double *err_bnds_comp, int nparams, double *params, int *info); void dgbrfsx_64 (char trans, char equed, long n, long kl, long ku, long nrhs, double *ab, long ldab, double *afb, long ldafb, long *ipiv, double *r, double *c, double *b, long ldb, double *x, long ldx, double *rcond, double *berr, long n_err_bnds, dou- ble *err_bnds_norm, double *err_bnds_comp, long nparams, dou- ble *params, long *info); PURPOSE dgbrfsx improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solu- tion. 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, R and C below. In this case, the solution and error bounds returned are for the original unequilibrated system. ARGUMENTS TRANS (input) TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose = Transpose) EQUED (input) 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 = 'R': Row equilibration, i.e., A has been premultiplied by diag(R). = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C). = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R)*A*diag(C). The right hand side B has been changed accordingly. N (input) N is INTEGER 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 matrices B and X. NRHS >= 0. AB (input) AB is DOUBLE PRECISION array, dimension (LDAB,N) The original band matrix A, stored in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows: AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). LDAB (input) LDAB is INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. AFB (input) AFB is DOUBLE PRECISION array, dimension (LDAFB,N). Details of the LU factorization of the band matrix A, as com- puted by DGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. LDAFB (input) LDAFB is INTEGER The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. IPIV (input) IPIV is INTEGER array, dimension (N) The pivot indices from DGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i). R (input/output) R is DOUBLE PRECISION array, dimension (N) The row scale factors for A. If EQUED = 'R' or 'B', A is mul- tiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not accessed. R is an input argument if FACT = 'F'; other- wise, R is an output argument. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive. If R is output, each element of R is a power of the radix. If R is input, each element of R 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. C (input/output) C is DOUBLE PRECISION array, dimension (N) The column scale factors for A. If EQUED = 'C' or 'B', A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input argument if FACT = 'F'; oth- erwise, C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. If C is output, each element of C is a power of the radix. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. B (input) B is DOUBLE PRECISION 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). X (input/output) X is DOUBLE PRECISION array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by DGETRS. On exit, the improved solution matrix X. LDX (input) LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). RCOND (output) RCOND is DOUBLE PRECISION 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 sin- gular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill- conditioned. BERR (output) BERR is DOUBLE PRECISION array, dimension (NRHS) Componentwise relative backward error. This is the component- wise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). N_ERR_BNDS (input) N_ERR_BNDS is INTEGER Number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below. ERR_BNDS_NORM (output) ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 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) * dlamch('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) * dlamch('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) * dlamch('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 appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. See Lapack Working Note 165 for further details and extra cautions. ERR_BNDS_COMP (output) ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 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_ERR_BNDS .LT. 3, then at most the first (:,N_ERR_BNDS) entries are returned. The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side. The second index in ERR_BNDS_COMP(:,err) contains the 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) * dlamch('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) * dlamch('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) * dlamch('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 appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the cur- rent 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 approxi- mately 1. See Lapack Working Note 165 for further details and extra cautions. NPARAMS (input) NPARAMS is INTEGER Specifies the number of parameters set in PARAMS. If .LE. 0, the PARAMS array is never referenced and default values are used. PARAMS (input/output) PARAMS is DOUBLE PRECISION array, dimension (NPARAMS) Speci- fies algorithm parameters. If an entry is .LT. 0.0, then that entry will be filled with default value used for that parame- ter. 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: 1.0D+0 = 0.0 : No refinement is performed, and no error bounds are computed. = 1.0 : Use the double-precision refinement algorithm, possi- bly with doubled-single computations if the compilation envi- ronment 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 approxi- mate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimina- tion, 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 rel- ative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence) WORK (output) WORK is DOUBLE PRECISION array, dimension (4*N) IWORK (output) IWORK is INTEGER array, dimension (N) INFO (output) INFO is INTEGER = 0: Successful exit. The solution to every right-hand side is guaranteed. < 0: If INFO = -i, the i-th argument had an illegal value. > 0 and <= N: U(INFO,INFO) is exactly zero. The factoriza- tion has been completed, but the factor U is exactly singu- lar, 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 small- est 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. 7 Nov 2015 dgbrfsx(3P)