sla_syrfsx_extended - ear equations for symmetric indefinite matrices by performing extra- precise iterative refinement and provide error bounds and backward error estimates for the solution
SUBROUTINE SLA_SYRFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, 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, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, N_NORMS, ITHRESH CHARACTER*1 UPLO LOGICAL COLEQU, IGNORE_CWISE REAL RTHRESH, DZ_UB INTEGER IPIV(*) REAL A(LDA,*), AF(LDAF,*), B(LDB,*), Y(LDY,*), RES(*),DY(*), Y_TAIL(*) REAL C(*), AYB(*), RCOND, BERR_OUT(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) SUBROUTINE SLA_SYRFSX_EXTENDED_64( PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, 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, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, N_NORMS, ITHRESH CHARACTER*1 UPLO LOGICAL*8 COLEQU, IGNORE_CWISE REAL RTHRESH, DZ_UB INTEGER*8 IPIV(*) REAL A(LDA,*), AF(LDAF,*), 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_SYRFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, 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(:,:) :: A, AF, B, Y, ERR_BNDS_NORM, ERR_BNDS_COMP CHARACTER(LEN=1) :: UPLO INTEGER :: PREC_TYPE, N, NRHS, LDA, LDAF, LDB, LDY, N_NORMS, ITHRESH, INFO LOGICAL :: COLEQU, IGNORE_CWISE INTEGER, DIMENSION(:) :: IPIV REAL, DIMENSION(:) :: C, BERR_OUT, RES, AYB, DY, Y_TAIL REAL :: RCOND, RTHRESH, DZ_UB SUBROUTINE LA_SYRFSX_EXTENDED_64( PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, 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(:,:) :: A, AF, B, Y, ERR_BNDS_NORM, ERR_BNDS_COMP CHARACTER(LEN=1) :: UPLO INTEGER(8) :: PREC_TYPE, N, NRHS, LDA, LDAF, LDB, LDY, N_NORMS, ITHRESH, INFO LOGICAL(8) :: COLEQU, IGNORE_CWISE INTEGER(8), DIMENSION(:) :: IPIV REAL, DIMENSION(:) :: C, BERR_OUT, RES, AYB, DY, Y_TAIL REAL :: RCOND, RTHRESH, DZ_UB C INTERFACE #include <sunperf.h> void sla_syrfsx_extended (int prec_type, char uplo, int n, int nrhs, float *a, int lda, float *af, int ldaf, int *ipiv, int colequ, float *c, float *b, int ldb, float *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 sla_syrfsx_extended_64 (long prec_type, char uplo, long n, long nrhs, float *a, long lda, float *af, long ldaf, long *ipiv, long colequ, float *c, float *b, long ldb, float *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 sla_syrfsx_extended(3P)
NAME
sla_syrfsx_extended - improve the computed solution to a system of lin-
ear equations for symmetric indefinite matrices by performing extra-
precise iterative refinement and provide error bounds and backward
error estimates for the solution
SYNOPSIS
SUBROUTINE SLA_SYRFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA, AF,
LDAF, 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, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, N_NORMS, ITHRESH
CHARACTER*1 UPLO
LOGICAL COLEQU, IGNORE_CWISE
REAL RTHRESH, DZ_UB
INTEGER IPIV(*)
REAL A(LDA,*), AF(LDAF,*), B(LDB,*), Y(LDY,*), RES(*),DY(*), Y_TAIL(*)
REAL C(*), AYB(*), RCOND, BERR_OUT(*), ERR_BNDS_NORM(NRHS,*),
ERR_BNDS_COMP(NRHS,*)
SUBROUTINE SLA_SYRFSX_EXTENDED_64( PREC_TYPE, UPLO, N, NRHS, A, LDA,
AF, LDAF, 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, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, N_NORMS,
ITHRESH
CHARACTER*1 UPLO
LOGICAL*8 COLEQU, IGNORE_CWISE
REAL RTHRESH, DZ_UB
INTEGER*8 IPIV(*)
REAL A(LDA,*), AF(LDAF,*), 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_SYRFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA, AF,
LDAF, 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(:,:) :: A, AF, B, Y, ERR_BNDS_NORM, ERR_BNDS_COMP
CHARACTER(LEN=1) :: UPLO
INTEGER :: PREC_TYPE, N, NRHS, LDA, LDAF, LDB, LDY, N_NORMS, ITHRESH,
INFO
LOGICAL :: COLEQU, IGNORE_CWISE
INTEGER, DIMENSION(:) :: IPIV
REAL, DIMENSION(:) :: C, BERR_OUT, RES, AYB, DY, Y_TAIL
REAL :: RCOND, RTHRESH, DZ_UB
SUBROUTINE LA_SYRFSX_EXTENDED_64( PREC_TYPE, UPLO, N, NRHS, A, LDA, AF,
LDAF, 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(:,:) :: A, AF, B, Y, ERR_BNDS_NORM, ERR_BNDS_COMP
CHARACTER(LEN=1) :: UPLO
INTEGER(8) :: PREC_TYPE, N, NRHS, LDA, LDAF, LDB, LDY, N_NORMS,
ITHRESH, INFO
LOGICAL(8) :: COLEQU, IGNORE_CWISE
INTEGER(8), DIMENSION(:) :: IPIV
REAL, DIMENSION(:) :: C, BERR_OUT, RES, AYB, DY, Y_TAIL
REAL :: RCOND, RTHRESH, DZ_UB
C INTERFACE
#include <sunperf.h>
void sla_syrfsx_extended (int prec_type, char uplo, int n, int nrhs,
float *a, int lda, float *af, int ldaf, int *ipiv, int
colequ, float *c, float *b, int ldb, float *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 sla_syrfsx_extended_64 (long prec_type, char uplo, long n, long
nrhs, float *a, long lda, float *af, long ldaf, long *ipiv,
long colequ, float *c, float *b, long ldb, float *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
sla_syrfsx_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 SSYRFSX 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 CHARACTER and
P = 'S': Single
= 'D': Double
= 'I': Indigenous
= 'X', 'E': Extra
UPLO (input)
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input)
N is INTEGER
The number of linear equations, i.e., the order of the matrix
A. N >= 0.
NRHS (input)
NRHS is INTEGER
The number of right-hand-sides, i.e., the number of columns
of the matrix B.
A (input)
A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA (input)
LDA is INTEGER
The leading dimension of the array A.
LDA >= max(1,N).
AF (input)
AF is REAL array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by SSYTRF.
LDAF (input)
LDAF is INTEGER
The leading dimension of the array AF.
LDAF >= max(1,N).
IPIV (input)
IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D as
determined by SSYTRF.
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 REAL 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 REAL array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by SSYTRS.
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 SLA_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 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 sub-
routine 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 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 componentwise
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 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.
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 REAL array, dimension (N)
Workspace to hold the intermediate residual.
AYB (input)
AYB is REAL array, dimension (N)
Workspace. This can be the same workspace passed for Y_TAIL.
DY (input)
DY is REAL array, dimension (N)
Workspace to hold the intermediate solution.
Y_TAIL (input)
Y_TAIL is REAL array, dimension (N)
Workspace to hold the trailing bits of the intermediate solu-
tion.
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 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.
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 fac-
torizations other than LU. If the factorization uses a tech-
nique 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 conver-
gence. 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 SLA_SYRFSX_EXTENDED
had an illegal value.
7 Nov 2015 sla_syrfsx_extended(3P)