
 
 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 )
            <type>(<wp>), INTENT(INOUT) :: A(:,:), <rhs>
            <type>(<wp>), INTENT(OUT) :: <sol>
            CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: UPLO
            <type>(<wp>), INTENT(INOUT), OPTIONAL :: AF(:,:)
            CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: FACT
            CHARACTER(LEN=1), INTENT(INOUT), OPTIONAL :: EQUED
            REAL(<wp>), INTENT(INOUT), OPTIONAL :: S(:)
            REAL(<wp>), INTENT(OUT), OPTIONAL :: <err>
            REAL(<wp>), INTENT(OUT), OPTIONAL :: RCOND
            INTEGER, INTENT(OUT), OPTIONAL :: INFO
       where
            <type> ::= REAL | COMPLEX
            <wp>   ::= KIND(1.0) | KIND(1.0D0)
            <rhs>  ::= B(:,:) | B(:)
            <sol>  ::= X(:,:) | X(:)
            <err>  ::= 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.

