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.