Purpose
=======
LA_POSVX computes the solution to a linear system of equations
A*X = B, where A is real symmetric or complex Hermitian and, in either
case, positive definite, and where X and B are rectangular matrices or
vectors.
LA_POSVX can also optionally equilibrate the system if A is poorly
scaled, estimate the condition number of (the equilibrated) A, and
compute error bounds.
=========
SUBROUTINE LA_POSVX( A, B, X, UPLO=uplo, AF=af, FACT=fact, &
EQUED=equed, S=s, FERR=ferr, BERR=berr, &
RCOND=rcond, INFO=info )
(), INTENT(INOUT) :: A(:,:),
(), INTENT(OUT) ::
CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: UPLO
(), INTENT(INOUT), OPTIONAL :: AF(:,:)
CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: FACT
CHARACTER(LEN=1), INTENT(INOUT), OPTIONAL :: EQUED
REAL(), INTENT(INOUT), OPTIONAL :: S(:)
REAL(), INTENT(OUT), OPTIONAL ::
REAL(), INTENT(OUT), OPTIONAL :: RCOND
INTEGER, INTENT(OUT), OPTIONAL :: INFO
where
::= REAL | COMPLEX
::= KIND(1.0) | KIND(1.0D0)
::= B(:,:) | B(:)
::= X(:,:) | X(:)
::= FERR(:), BERR(:) | FERR, BERR
Arguments
=========
A (input/output) REAL or COMPLEX square array, shape (:,:).
On entry, the matrix A or its equilibration:
If UPLO = 'U', then the upper triangular part of A contains
the upper triangular part of (the equilibrated) A, and the
strictly lower triangular part of A is not referenced.
If UPLO = 'L', then the lower triangular part of A contains
the lower triangular part of (the equilibrated) A, and the
strictly upper triangular part of A is not referenced.
If FACT = 'F' and EQUED = 'Y', then A has been equilibrated
by the scaling factors in S during a previous call to
LA_POSVX.
On exit, if FACT = 'E', then the equilibrated version of A is
stored in A; otherwise, A is unchanged.
B (input/output) REAL or COMPLEX array, shape (:,:) with
size(B,1) = size(A,1) or shape (:) with size(B) =
size(A,1).
On entry, the matrix B.
On exit, the scaled version of B if the system has been
equilibrated; otherwise, B is unchanged.
X (output) REAL or COMPLEX array, shape (:,:) with size(X,1) =
size(A,1) and size(X,2) = size(B,2), or shape (:) with size(X)
= size(A,1).
The solution matrix X .
UPLO Optional (input) CHARACTER(LEN=1).
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
Default value: 'U'.
AF Optional (input or output) REAL or COMPLEX array, shape (:,:)
with the same size as A.
If FACT = 'F' then AF is an input argument that contains the
factor U or L from the Cholesky factorization of (the
equilibrated) A, in the same storage format as A, returned by
a previous call to LA_POSVX
If FACT /= 'F' then AF is an output argument that contains the
factor U or L from the Cholesky factorization of (the
equilibrated) A in the same storage format as A.
FACT Optional (input) CHARACTER(LEN=1).
Specifies whether the factored form of the matrix A is
supplied on entry, and, if not, whether A should be
equilibrated before it is factored.
= 'N': The matrix A will be copied to AF and factored
(no equilibration).
= 'E': The matrix A will be equilibrated, then copied to AF
and factored.
= 'F': AF contains the factored form of (the equilibrated)
A.
Default value: 'N'.
EQUED Optional (input or output) CHARACTER(LEN=1).
Specifies the form of equilibration that was done.
EQUED is an input argument if FACT = 'F', otherwise it is an
output argument:
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Equilibration, i.e., A has been premultiplied and
postmultiplied by diag(S).
Default value: 'N'.
S Optional (input or output) REAL array, shape (:) with size(S)=
size(A,1).
The scaling factors for A.
S is an input argument if FACT = 'F' and EQUED = 'Y'.
S is an output argument if FACT = 'E' and EQUED = 'Y'.
FERR Optional (output) REAL array of shape (:), with size(FERR) =
size(X,2), or REAL scalar.
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X). If XTRUE is
the true solution corresponding to X(j), FERR(j) is an
estimated upper bound for the magnitude of the largest element
in (X(j)-XTRUE) divided by the magnitude of the largest
element in X(j). The estimate is as reliable as the estimate
for RCOND, and is almost always a slight overestimate of the
true error.
BERR Optional (output) REAL array of shape (:), with size(BERR) =
size(X,2), or REAL scalar.
The componentwise 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).
RCOND Optional (output) REAL
The estimate of the reciprocal condition number of (the
equilibrated) A. If RCOND is less than the machine precision,
the matrix is singular to working precision. This condition is
indicated by a return code of INFO > 0.
INFO Optional (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, and i is
<= n: the leading minor of order i of (the equilibrated)
A is not positive definite, so the factorization
could not be completed and the solution and error
bounds could not be computed. RCOND= 0 is returned.
= n+1: U or L is nonsingular, but RCOND is less than
machine precision, so the matrix is singular to
working precision. Nevertheless, the solution and
error bounds are computed because the computed
solution can be more accurate than the value of
RCOND would suggest.
If INFO is not present and an error occurs, then the program
is terminated with an error message.