Purpose
=======
LA_SYSVX computes the solution to a linear system of equations
A*X = B, where A is a real or complex symmetric matrix and X and B are
rectangular matrices or vectors.
LA_HESVX computes the solution to a linear system of equations
A*X = B, where A is a complex Hermitian matrix and X and B are
rectangular matrices or vectors.
LA_SYSVX and LA_HESVX can also optionally estimate the condition
number of A and compute error bounds.
=========
SUBROUTINE LA_SYSVX / LA HESVX( A, B, X, UPLO=uplo, AF=af, &
IPIV=ipiv, FACT=fact, FERR=ferr, BERR=berr, &
RCOND=rcond, INFO=info )
(), INTENT(IN) :: A(:,:),
(), INTENT(OUT) ::
CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: UPLO
(), INTENT(INOUT), OPTIONAL :: AF(:,:)
INTEGER, INTENT(INOUT), OPTIONAL :: IPIV(:)
CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: FACT
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) REAL or COMPLEX square array, shape (:,:).
The symmetric or Hermitian matrix A.
If UPLO = 'U', the 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
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.
B (input) REAL or COMPLEX array, shape (:,:) with size(B,1) =
size(A,1) or shape (:) with size(B) = size(A,1).
The matrix B.
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
block diagonal matrix D and the multipliers used to obtain the
factor L or U from the factorization of A, returned by a
previous call to LA_SYSVX or LA_HESVX.
If FACT = 'N', then AF is an output argument that contains the
block diagonal matrix D and the multipliers used to obtain the
factor L or U from the factorization of A.
IPIV Optional (input or output) INTEGER array, shape (:) with
size(IPIV) = size(A,1).
If FACT = 'F', then IPIV is an input argument that contains
details of the row and column interchanges and the block
structure of D.
If IPIV(k) > 0 , then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1 by 1 diagonal block.
If IPIV(k) < 0 , then there are two cases:
1. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and
D(k-1:k,k-1:k) is a 2 by 2 diagonal block.
2. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and
columns k+1 and -IPIV(k) were interchanged and
D(k:k+1,k:k+1) is a 2 by 2 diagonal block.
If FACT = 'N', then IPIV is an output argument that contains
details of the row and column interchanges and the block
structure of D; as described above.
FACT Optional (input) CHARACTER(LEN=1).
Specifies whether the factored form of the matrix A has been
supplied on entry.
= 'N': The matrix A will be copied to AF and factored.
= 'F': AF and IPIV contain the factored form of A.
Default value: 'N'.
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 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 (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: D(i,i) = 0. The factorization has been completed, but
the block diagonal matrix D is singular, so the
solution could not be computed.
= n+1: D 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.
n is the order of A.
If INFO is not present and an error occurs, then the program is
terminated with an error message.