csyrfsx - improve the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite, provide error bounds and backward error estimates for the solution
SUBROUTINE CSYRFSX(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO) CHARACTER*1 UPLO, EQUED INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS REAL RCOND INTEGER IPIV(*) COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*) REAL S(*), PARAMS(*), BERR(*), RWORK(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) SUBROUTINE CSYRFSX_64(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO) CHARACTER*1 UPLO, EQUED INTEGER*8 INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS REAL RCOND INTEGER*8 IPIV(*) COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*) REAL S(*), PARAMS(*), BERR(*), RWORK(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) F95 INTERFACE SUBROUTINE SYRFSX(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO) REAL, DIMENSION(:,:) :: ERR_BNDS_NORM, ERR_BNDS_COMP INTEGER :: N, NRHS, LDA, LDAF, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO CHARACTER(LEN=1) :: UPLO, EQUED INTEGER, DIMENSION(:) :: IPIV REAL, DIMENSION(:) :: S, BERR, PARAMS, RWORK COMPLEX, DIMENSION(:,:) :: A, AF, B, X COMPLEX, DIMENSION(:) :: WORK REAL :: RCOND SUBROUTINE SYRFSX_64(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO) REAL, DIMENSION(:,:) :: ERR_BNDS_NORM, ERR_BNDS_COMP INTEGER(8) :: N, NRHS, LDA, LDAF, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO CHARACTER(LEN=1) :: UPLO, EQUED INTEGER(8), DIMENSION(:) :: IPIV REAL, DIMENSION(:) :: S, BERR, PARAMS, RWORK COMPLEX, DIMENSION(:,:) :: A, AF, B, X COMPLEX, DIMENSION(:) :: WORK REAL :: RCOND C INTERFACE #include <sunperf.h> void csyrfsx (char uplo, char equed, int n, int nrhs, floatcomplex *a, int lda, floatcomplex *af, int ldaf, int *ipiv, float *s, floatcomplex *b, int ldb, floatcomplex *x, int ldx, float *rcond, float *berr, int n_err_bnds, float *err_bnds_norm, float *err_bnds_comp, int nparams, float *params, int *info); void csyrfsx_64 (char uplo, char equed, long n, long nrhs, floatcomplex *a, long lda, floatcomplex *af, long ldaf, long *ipiv, float *s, floatcomplex *b, long ldb, floatcomplex *x, long ldx, float *rcond, float *berr, long n_err_bnds, float *err_bnds_norm, float *err_bnds_comp, long nparams, float *params, long *info);
Oracle Solaris Studio Performance Library csyrfsx(3P)
NAME
csyrfsx - improve the computed solution to a system of linear equations
when the coefficient matrix is symmetric indefinite, provide error
bounds and backward error estimates for the solution
SYNOPSIS
SUBROUTINE CSYRFSX(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B,
LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
CHARACTER*1 UPLO, EQUED
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS
REAL RCOND
INTEGER IPIV(*)
COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
REAL S(*), PARAMS(*), BERR(*), RWORK(*), ERR_BNDS_NORM(NRHS,*),
ERR_BNDS_COMP(NRHS,*)
SUBROUTINE CSYRFSX_64(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S,
B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
CHARACTER*1 UPLO, EQUED
INTEGER*8 INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS
REAL RCOND
INTEGER*8 IPIV(*)
COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
REAL S(*), PARAMS(*), BERR(*), RWORK(*), ERR_BNDS_NORM(NRHS,*),
ERR_BNDS_COMP(NRHS,*)
F95 INTERFACE
SUBROUTINE SYRFSX(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B,
LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
REAL, DIMENSION(:,:) :: ERR_BNDS_NORM, ERR_BNDS_COMP
INTEGER :: N, NRHS, LDA, LDAF, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO
CHARACTER(LEN=1) :: UPLO, EQUED
INTEGER, DIMENSION(:) :: IPIV
REAL, DIMENSION(:) :: S, BERR, PARAMS, RWORK
COMPLEX, DIMENSION(:,:) :: A, AF, B, X
COMPLEX, DIMENSION(:) :: WORK
REAL :: RCOND
SUBROUTINE SYRFSX_64(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S,
B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
REAL, DIMENSION(:,:) :: ERR_BNDS_NORM, ERR_BNDS_COMP
INTEGER(8) :: N, NRHS, LDA, LDAF, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO
CHARACTER(LEN=1) :: UPLO, EQUED
INTEGER(8), DIMENSION(:) :: IPIV
REAL, DIMENSION(:) :: S, BERR, PARAMS, RWORK
COMPLEX, DIMENSION(:,:) :: A, AF, B, X
COMPLEX, DIMENSION(:) :: WORK
REAL :: RCOND
C INTERFACE
#include <sunperf.h>
void csyrfsx (char uplo, char equed, int n, int nrhs, floatcomplex *a,
int lda, floatcomplex *af, int ldaf, int *ipiv, float *s,
floatcomplex *b, int ldb, floatcomplex *x, int ldx, float
*rcond, float *berr, int n_err_bnds, float *err_bnds_norm,
float *err_bnds_comp, int nparams, float *params, int *info);
void csyrfsx_64 (char uplo, char equed, long n, long nrhs, floatcomplex
*a, long lda, floatcomplex *af, long ldaf, long *ipiv, float
*s, floatcomplex *b, long ldb, floatcomplex *x, long ldx,
float *rcond, float *berr, long n_err_bnds, float
*err_bnds_norm, float *err_bnds_comp, long nparams, float
*params, long *info);
PURPOSE
csyrfsx improves the computed solution to a system of linear equations
when the coefficient matrix is symmetric indefinite, and provides error
bounds and backward error estimates for the solution. 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 and S
below. In this case, the solution and error bounds returned are for the
original unequilibrated system.
ARGUMENTS
UPLO (input)
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
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
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S). The right hand side B has
been changed accordingly.
N (input)
N is INTEGER
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 matrices B and X. NRHS >= 0.
A (input)
A is COMPLEX array, dimension (LDA,N)
The symmetric matrix A. If UPLO = 'U', the leading N-by-N
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A
is not referenced. If UPLO = 'L', the leading N-by-N lower
triangular part of A contains the lower triangular part of
the matrix A, and the strictly upper triangular part of A is
not referenced.
LDA (input)
LDA is INTEGER
The leading dimension of the array A.
LDA >= max(1,N).
AF (input)
AF is COMPLEX array, dimension (LDAF,N)
The factored form of the matrix A. AF contains the block
diagonal matrix D and the multipliers used to obtain the fac-
tor U or L from the factorization A=U*D*U**T or A=L*D*L**T 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.
S (input/output)
S is REAL array, dimension (N)
The scale factors for A. If EQUED = 'Y', A is multiplied on
the left and right by diag(S). S is an input argument if FACT
= = 'Y', each element of S must be positive. If S is output,
each element of S is a power of the radix. If S is input,
each element of S 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 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).
X (input/output)
X is COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
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 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.
BERR (output)
BERR is REAL 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 REAL 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) * 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.
See Lapack Working Note 165 for further details and extra
cautions.
ERR_BNDS_COMP (output)
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 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) * 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.
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 REAL array, dimension NPARAMS
Specifies algorithm parameters. If an entry is .LT. 0.0, then
that entry will be filled with default value used for that
parameter. 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: 1.0
= 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 COMPLEX array, dimension (2*N)
RWORK (output)
RWORK is REAL array, dimension (2*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 csyrfsx(3P)