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)