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cla_gbrfsx_extended-f(3)

cla_gbrfsx_extended.f –

Synopsis

Functions/Subroutines

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)
CLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Function/Subroutine Documentation

subroutine cla_gbrfsx_extended (integerPREC_TYPE, integerTRANS_TYPE, integerN, integerKL, integerKU, integerNRHS, complex, dimension( ldab, * )AB, integerLDAB, complex, dimension( ldafb, * )AFB, integerLDAFB, integer, dimension( * )IPIV, logicalCOLEQU, real, dimension( * )C, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldy, * )Y, integerLDY, real, dimension( * )BERR_OUT, integerN_NORMS, real, dimension( nrhs, * )ERR_BNDS_NORM, real, dimension( nrhs, * )ERR_BNDS_COMP, complex, dimension( * )RES, real, dimension(*)AYB, complex, dimension( * )DY, complex, dimension( * )Y_TAIL, realRCOND, integerITHRESH, realRTHRESH, realDZ_UB, logicalIGNORE_CWISE, integerINFO)

CLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

 CLA_GBRFSX_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 CGBRFSX 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.

 

Parameters:

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

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.

KL

 KL is INTEGER
 The number of subdiagonals within the band of A. KL >= 0.

KU

 KU is INTEGER
 The number of superdiagonals within the band of A. KU >= 0

NRHS

 NRHS is INTEGER
 The number of right-hand-sides, i.e., the number of columns of the
 matrix B.

AB

 AB is COMPLEX array, dimension (LDAB,N)
 On entry, the N-by-N matrix AB.

LDAB

 LDAB is INTEGER
 The leading dimension of the array AB. LDAB >= max(1,N).

AFB

 AFB is COMPLEX array, dimension (LDAF,N)
 The factors L and U from the factorization
 A = P*L*U as computed by CGBTRF.

LDAFB

 LDAFB 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 CGBTRF; 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 CGBTRS.
 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 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

 ERR_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_COMP

 ERR_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 .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 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
 ERR_BNDS_NORM and ERR_BNDS_COMP 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 CGBTRS had an illegal
 value

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

Definition at line 407 of file cla_gbrfsx_extended.f.

Author

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