sgbrfsx - improve the computed solution to a system of linear equations and provide error bounds and backward error estimates for the solution
SUBROUTINE SGBRFSX(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) CHARACTER*1 TRANS, EQUED INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, NPARAMS, N_ERR_BNDS REAL RCOND INTEGER IPIV(*),IWORK(*) REAL AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), X(LDX,*), WORK(*) REAL R(*), C(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) SUBROUTINE SGBRFSX_64(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) CHARACTER*1 TRANS, EQUED INTEGER*8 INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, NPARAMS, N_ERR_BNDS REAL RCOND INTEGER*8 IPIV(*), IWORK(*) REAL AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), X(LDX,*), WORK(*) REAL R(*), C(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) F95 INTERFACE SUBROUTINE GBRFSX(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) REAL, DIMENSION(:,:) :: AB, AFB, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP INTEGER :: N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO CHARACTER(LEN=1) :: TRANS, EQUED INTEGER, DIMENSION(:) :: IPIV, IWORK REAL, DIMENSION(:) :: R, C, BERR, PARAMS, WORK REAL :: RCOND SUBROUTINE GBRFSX_64(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) REAL, DIMENSION(:,:) :: AB, AFB, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP INTEGER(8) :: N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO CHARACTER(LEN=1) :: TRANS, EQUED INTEGER(8), DIMENSION(:) :: IPIV, IWORK REAL, DIMENSION(:) :: R, C, BERR, PARAMS, WORK REAL :: RCOND C INTERFACE #include <sunperf.h> void sgbrfsx (char trans, char equed, int n, int kl, int ku, int nrhs, float *ab, int ldab, float *afb, int ldafb, int *ipiv, float *r, float *c, float *b, int ldb, float *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 sgbrfsx_64 (char trans, char equed, long n, long kl, long ku, long nrhs, float *ab, long ldab, float *afb, long ldafb, long *ipiv, float *r, float *c, float *b, long ldb, float *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 sgbrfsx(3P)
NAME
sgbrfsx - improve the computed solution to a system of linear equations
and provide error bounds and backward error estimates for the solution
SYNOPSIS
SUBROUTINE SGBRFSX(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
INFO)
CHARACTER*1 TRANS, EQUED
INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, NPARAMS,
N_ERR_BNDS
REAL RCOND
INTEGER IPIV(*),IWORK(*)
REAL AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), X(LDX,*), WORK(*)
REAL R(*), C(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*),
ERR_BNDS_COMP(NRHS,*)
SUBROUTINE SGBRFSX_64(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB,
LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
INFO)
CHARACTER*1 TRANS, EQUED
INTEGER*8 INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, NPARAMS,
N_ERR_BNDS
REAL RCOND
INTEGER*8 IPIV(*), IWORK(*)
REAL AB(LDAB,*), AFB(LDAFB,*), B(LDB,*), X(LDX,*), WORK(*)
REAL R(*), C(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*),
ERR_BNDS_COMP(NRHS,*)
F95 INTERFACE
SUBROUTINE GBRFSX(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
INFO)
REAL, DIMENSION(:,:) :: AB, AFB, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP
INTEGER :: N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDX, N_ERR_BNDS, NPARAMS,
INFO
CHARACTER(LEN=1) :: TRANS, EQUED
INTEGER, DIMENSION(:) :: IPIV, IWORK
REAL, DIMENSION(:) :: R, C, BERR, PARAMS, WORK
REAL :: RCOND
SUBROUTINE GBRFSX_64(TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB,
LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
INFO)
REAL, DIMENSION(:,:) :: AB, AFB, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP
INTEGER(8) :: N, KL, KU, NRHS, LDAB, LDAFB, LDB, LDX, N_ERR_BNDS,
NPARAMS, INFO
CHARACTER(LEN=1) :: TRANS, EQUED
INTEGER(8), DIMENSION(:) :: IPIV, IWORK
REAL, DIMENSION(:) :: R, C, BERR, PARAMS, WORK
REAL :: RCOND
C INTERFACE
#include <sunperf.h>
void sgbrfsx (char trans, char equed, int n, int kl, int ku, int nrhs,
float *ab, int ldab, float *afb, int ldafb, int *ipiv, float
*r, float *c, float *b, int ldb, float *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 sgbrfsx_64 (char trans, char equed, long n, long kl, long ku, long
nrhs, float *ab, long ldab, float *afb, long ldafb, long
*ipiv, float *r, float *c, float *b, long ldb, float *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
sgbrfsx improves the computed solution to a system of linear equations
and provides error bounds and backward error estimates for the solu-
tion. 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, R and C
below. In this case, the solution and error bounds returned are for the
original unequilibrated system.
ARGUMENTS
TRANS (input)
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
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
= 'R': Row equilibration, i.e., A has been premultiplied by
diag(R).
= 'C': Column equilibration, i.e., A has been postmultiplied
by diag(C).
= 'B': Both row and column equilibration, i.e., A has been
replaced by diag(R)*A*diag(C).
The right hand side B has been changed accordingly.
N (input)
N is INTEGER
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 matrices B and X. NRHS >= 0.
AB (input)
AB is DOUBLE PRECISION array, dimension (LDAB,N)
The original band matrix A, stored in rows 1 to KL+KU+1. The
j-th column of A is stored in the j-th column of the array AB
as follows:
AB(ku+1+i-j,j) = A(i,j)
for max(1,j-ku)<=i<=min(n,j+kl).
LDAB (input)
LDAB is INTEGER
The leading dimension of the array AB.
LDAB >= KL+KU+1.
AFB (input)
AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as com-
puted by DGBTRF. U is stored as an upper triangular band
matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
the multipliers used during the factorization are stored in
rows KL+KU+2 to 2*KL+KU+1.
LDAFB (input)
LDAFB is INTEGER
The leading dimension of the array AFB.
LDAFB >= 2*KL*KU+1.
IPIV (input)
IPIV is INTEGER array, dimension (N)
The pivot indices from SGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
R (input/output)
R is REAL array, dimension (N)
The row scale factors for A. If EQUED = 'R' or 'B', A is mul-
tiplied on the left by diag(R); if EQUED = 'N' or 'C', R is
not accessed. R is an input argument if FACT = 'F'; other-
wise, R is an output argument. If FACT = 'F' and EQUED = 'R'
or 'B', each element of R must be positive.
If R is output, each element of R is a power of the radix.
If R is input, each element of R 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.
C (input/output)
C is REAL array, dimension (N)
The column scale factors for A. If EQUED = 'C' or 'B', A is
multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
is not accessed. C is an input argument if FACT = 'F'; oth-
erwise, C is an output argument. If FACT = 'F' and EQUED =
'C' or 'B', each element of C must be positive.
If C is output, each element of C is a power of the radix.
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 (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).
X (input/output)
X is REAL 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 REAL array, dimension (4*N)
IWORK (output)
IWORK is INTEGER array, dimension (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 sgbrfsx(3P)