sposvxx - compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices
SUBROUTINE SPOSVXX(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) CHARACTER*1 EQUED, FACT, UPLO INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS REAL RCOND, RPVGRW INTEGER IWORK(*) REAL A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*) REAL S(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) SUBROUTINE SPOSVXX_64(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) CHARACTER*1 EQUED, FACT, UPLO INTEGER*8 INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS REAL RCOND, RPVGRW INTEGER*8 IWORK(*) REAL A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*) REAL S(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*), ERR_BNDS_COMP(NRHS,*) F95 INTERFACE SUBROUTINE POSVXX(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) REAL, DIMENSION(:,:) :: A, AF, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP INTEGER :: N, NRHS, LDA, LDAF, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO CHARACTER(LEN=1) :: FACT, UPLO, EQUED INTEGER, DIMENSION(:) :: IWORK REAL, DIMENSION(:) :: S, BERR, PARAMS, WORK REAL :: RCOND, RPVGRW SUBROUTINE POSVXX_64(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) REAL, DIMENSION(:,:) :: A, AF, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP INTEGER(8) :: N, NRHS, LDA, LDAF, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO CHARACTER(LEN=1) :: FACT, UPLO, EQUED INTEGER(8), DIMENSION(:) :: IWORK REAL, DIMENSION(:) :: S, BERR, PARAMS, WORK REAL :: RCOND, RPVGRW C INTERFACE #include <sunperf.h> void sposvxx (char fact, char uplo, int n, int nrhs, float *a, int lda, float *af, int ldaf, char *equed, float *s, float *b, int ldb, float *x, int ldx, float *rcond, float *rpvgrw, float *berr, int n_err_bnds, float *err_bnds_norm, float *err_bnds_comp, int nparams, float *params, int *info); void sposvxx_64 (char fact, char uplo, long n, long nrhs, float *a, long lda, float *af, long ldaf, char *equed, float *s, float *b, long ldb, float *x, long ldx, float *rcond, float *rpv- grw, 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 sposvxx(3P)
NAME
sposvxx - compute the solution to a real system of linear equations A*X
= B, where A is an N-by-N symmetric positive definite matrix and X and
B are N-by-NRHS matrices
SYNOPSIS
SUBROUTINE SPOSVXX(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B,
LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
CHARACTER*1 EQUED, FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS
REAL RCOND, RPVGRW
INTEGER IWORK(*)
REAL A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
REAL S(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*),
ERR_BNDS_COMP(NRHS,*)
SUBROUTINE SPOSVXX_64(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S,
B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
INFO)
CHARACTER*1 EQUED, FACT, UPLO
INTEGER*8 INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS
REAL RCOND, RPVGRW
INTEGER*8 IWORK(*)
REAL A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
REAL S(*), PARAMS(*), BERR(*), ERR_BNDS_NORM(NRHS,*),
ERR_BNDS_COMP(NRHS,*)
F95 INTERFACE
SUBROUTINE POSVXX(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B,
LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
REAL, DIMENSION(:,:) :: A, AF, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP
INTEGER :: N, NRHS, LDA, LDAF, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO
CHARACTER(LEN=1) :: FACT, UPLO, EQUED
INTEGER, DIMENSION(:) :: IWORK
REAL, DIMENSION(:) :: S, BERR, PARAMS, WORK
REAL :: RCOND, RPVGRW
SUBROUTINE POSVXX_64(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S,
B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
INFO)
REAL, DIMENSION(:,:) :: A, AF, B, X, ERR_BNDS_NORM, ERR_BNDS_COMP
INTEGER(8) :: N, NRHS, LDA, LDAF, LDB, LDX, N_ERR_BNDS, NPARAMS, INFO
CHARACTER(LEN=1) :: FACT, UPLO, EQUED
INTEGER(8), DIMENSION(:) :: IWORK
REAL, DIMENSION(:) :: S, BERR, PARAMS, WORK
REAL :: RCOND, RPVGRW
C INTERFACE
#include <sunperf.h>
void sposvxx (char fact, char uplo, int n, int nrhs, float *a, int lda,
float *af, int ldaf, char *equed, float *s, float *b, int
ldb, float *x, int ldx, float *rcond, float *rpvgrw, float
*berr, int n_err_bnds, float *err_bnds_norm, float
*err_bnds_comp, int nparams, float *params, int *info);
void sposvxx_64 (char fact, char uplo, long n, long nrhs, float *a,
long lda, float *af, long ldaf, char *equed, float *s, float
*b, long ldb, float *x, long ldx, float *rcond, float *rpv-
grw, float *berr, long n_err_bnds, float *err_bnds_norm,
float *err_bnds_comp, long nparams, float *params, long
*info);
PURPOSE
sposvxx uses the Cholesky factorization A = U**T*U or A = L*L**T to
compute the solution to a real system of linear equations A * X = B,
where A is an N-by-N symmetric positive definite matrix and X and B are
N-by-NRHS matrices.
If requested, both normwise and maximum componentwise error bounds are
returned. SPOSVXX will return a solution with a tiny guaranteed error
(O(eps) where eps is the working machine precision) unless the matrix
is very ill-conditioned, in which case a warning is returned. Relevant
condition numbers also are calculated and returned.
SPOSVXX accepts user-provided factorizations and equilibration factors;
see the definitions of the FACT and EQUED options. Solving with
refinement and using a factorization from a previous SPOSVXX call will
also produce a solution with either O(eps) errors or warnings, but we
cannot make that claim for general user-provided factorizations and
equilibration factors if they differ from what SPOSVXX would itself
produce.
ARGUMENTS
FACT (input)
FACT is CHARACTER*1
Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 'F': On entry, AF contains the factored form of A. If
EQUED is not 'N', the matrix A has been equilibrated with
scaling factors given by S. A and AF are not modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AF and factored.
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 matrices B and X. NRHS >= 0.
A (input/output)
A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A, except if FACT = 'F' and
EQUED = diag(S)*A*diag(S). 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. A is not modified if FACT = 'F' or 'N', or
if FACT = 'E' and EQUED =
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
diag(S)*A*diag(S).
LDA (input)
LDA is INTEGER
The leading dimension of the array A.
LDA >= max(1,N).
AF (input/output)
AF is REAL array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry con-
tains the triangular factor U or L from the Cholesky factor-
ization A=U**T*U or A=L*L**T, in the same storage format as
A. If EQUED .ne. 'N', then AF is the factored form of the
equilibrated matrix diag(S)*A*diag(S).
If FACT = 'N', then AF is an output argument and on exit
returns the triangular factor U or L from the Cholesky fac-
torization A=U**T*U or A=L*L**T of the original matrix A.
If FACT = 'E', then AF is an output argument and on exit
returns the triangular factor U or L from the Cholesky fac-
torization A = U**T*U or A = L*L**T of the equilibrated
matrix A (see the description of A for the form of the equi-
librated matrix).
LDAF (input)
LDAF is INTEGER
The leading dimension of the array AF.
LDAF >= max(1,N).
EQUED (input/output)
EQUED is CHARACTER*1
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S).
EQUED is an input argument if FACT = 'F'; otherwise, it is an
output argument.
S (input/output)
S is REAL array, dimension (N)
The row 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 scal-
ing lead to refining with a matrix that is not equivalent to
the input matrix, producing error estimates that may not be
reliable.
B (input/output)
B is REAL array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit,
if EQUED = 'N', B is not modified;
if EQUED = 'Y', B is overwritten by diag(S)*B.
LDB (input)
LDB is INTEGER
The leading dimension of the array B.
LDB >= max(1,N).
X (output)
X is REAL array, dimension (LDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X to the original
system of equations. Note that A and B are modified on exit
if EQUED .ne. 'N', and the solution to the equilibrated sys-
tem is inv(diag(S))*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.
RPVGRW (output)
RPVGRW is REAL
Reciprocal pivot growth. On exit, this contains the recipro-
cal pivot growth factor norm(A)/norm(U). The "max absolute
element" norm is used. If this is much less than 1, then the
stability of the LU factorization of the (equilibrated)
matrix A could be poor. This also means that the solution X,
estimated condition numbers, and error bounds could be unre-
liable. If factorization fails with 0<INFO<=N, then this con-
tains the reciprocal pivot growth factor for the leading INFO
columns of A.
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 sposvxx(3P)