subroutine cla_gerfsx_extended (PREC_TYPE, TRANS_TYPE, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERRS_N, ERRS_C, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
CLA_GERFSX_EXTENDED
CLA_GERFSX_EXTENDED
Purpose:
CLA_GERFSX_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 CGERFSX to perform iterative refinement. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. See comments for ERRS_N and ERRS_C for details of the error bounds. Note that this subroutine is only resonsible for setting the second fields of ERRS_N and ERRS_C.
Parameters:
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
TRANS_TYPE
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
N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER The number of right-hand-sides, i.e., the number of columns of the matrix B.
A
A is COMPLEX array, dimension (LDA,N) On entry, the N-by-N matrix A.
LDA
LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX array, dimension (LDAF,N) The factors L and U from the factorization A = P*L*U as computed by CGETRF.
LDAF
LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N) The pivot indices from the factorization A = P*L*U as computed by CGETRF; row i of the matrix was interchanged with row IPIV(i).
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.
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.
B
B is COMPLEX array, dimension (LDB,NRHS) The right-hand-side matrix B.
LDB
LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
Y
Y is COMPLEX array, dimension (LDY,NRHS) On entry, the solution matrix X, as computed by CGETRS. On exit, the improved solution matrix Y.
LDY
LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT
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
N_NORMS is INTEGER Determines which error bounds to return (see ERRS_N and ERRS_C). If N_NORMS >= 1 return normwise error bounds. If N_NORMS >= 2 return componentwise error bounds.
ERRS_N
ERRS_N 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 ERRS_N(i,:) corresponds to the ith right-hand side. The second index in ERRS_N(:,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.
ERRS_C
ERRS_C 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 ERRS_C is not accessed. If N_ERR_BNDS .LT. 3, then at most the first (:,N_ERR_BNDS) entries are returned. The first index in ERRS_C(i,:) corresponds to the ith right-hand side. The second index in ERRS_C(:,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
RES is COMPLEX array, dimension (N) Workspace to hold the intermediate residual.
AYB
AYB is REAL array, dimension (N) Workspace.
DY
DY is COMPLEX array, dimension (N) Workspace to hold the intermediate solution.
Y_TAIL
Y_TAIL is COMPLEX array, dimension (N) Workspace to hold the trailing bits of the intermediate solution.
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.
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 ERRS_N and ERRS_C may no longer be trustworthy.
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 ill-conditioned matrices. See LAWN 165 for more details.
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.
IGNORE_CWISE
IGNORE_CWISE is LOGICAL If .TRUE. then ignore componentwise convergence. Default value is .FALSE..
INFO
INFO is INTEGER = 0: Successful exit. < 0: if INFO = -i, the ith argument to CGETRS had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Definition at line 399 of file cla_gerfsx_extended.f.
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