Purpose
 =======
 
    LA_GESVX computes the solution to a real or complex linear system of
 equations of the form A*X = B, A^T*X = B or A^H*X = B, where A is a 
 square matrix and X and B are rectangular matrices or vectors.
    LA_GESVX can also optionally equilibrate the system if A is poorly
 scaled, estimate the condition number of (the equilibrated) A, return 
 the pivot growth factor, and compute error bounds.
 
 =========
 
     SUBROUTINE LA_GESVX ( A, B, X, AF=af, IPIV=ipiv, FACT=fact, &
                  TRANS=trans, EQUED=equed, R=r, C=c, FERR=ferr, &
                  BERR=berr, RCOND=rcond, RPVGRW=rpvgrw, &
                  INFO=info )
          <type>(<wp>), INTENT(INOUT) :: A(:,:), <rhs>
          <type>(<wp>), INTENT(OUT) :: <sol>
          <type>(<wp>), INTENT(INOUT), OPTIONAL :: AF(:,:)
          INTEGER, INTENT(INOUT), OPTIONAL :: IPIV(:)
          CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: FACT, &
                                       TRANS
          CHARACTER(LEN=1), INTENT(INOUT), OPTIONAL :: EQUED
          REAL(<wp>), INTENT(INOUT), OPTIONAL :: R(:), C(:)
          REAL(<wp>), INTENT(OUT), OPTIONAL :: <err>, RCOND, RPVGRW
          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 FACT = 'F' and EQUED /= 'N' then A has been equilibrated 
           by the scaling factors in R and/or C during a previous call
 	    to LA_GESVX.
           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 .
 AF        Optional (input or output) REAL or COMPLEX square array, 
           shape (:,:) with the same size as A.
           If FACT = 'F' then AF is an input argument that contains the
           factors L and U of (the equilibrated) A returned by a
 	   previous call to LA_GESVX.
           If FACT /= 'F' then AF is an output argument that contains 
           the factors L and U of (the equilibrated) 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 
           the pivot indices from the factorization of (the 
	   equilibrated) A, returned by a previous call to LA_GESVX.
           If FACT /= 'F' then IPIV is an output argument that contains
           the pivot indices from the factorization of (the
 	   equilibrated) A.
 FACT      Optional (input) CHARACTER(LEN=1).
           Specifies whether 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.
            = '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 and IPIV contain the factored form of (the 
 	          equilibrated) A.
           Default value: 'N'.
 TRANS     Optional (input) CHARACTER(LEN=1).
           Specifies the form of the system of equations:
            = 'N': A*X = B (No transpose)
            = 'T': A^T*X = B (Transpose)
            = 'C': A^H*X = B (Conjugate transpose)
 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').
            = 'R': Row equilibration, i.e., A has been premultiplied by
 	          diag(R).
            = 'C': Column equilibration, i.e., A has been postmultiplied 
 	          by diag(C).
            = 'B': Both row and column equilibration.
           Default value: 'N'.
 R         Optional (input or output) REAL array, shape (:) with size(R)
           = size(A,1). The row scale factors for A.
           R is an input argument if FACT = 'F' and EQUED = 'R' or 'B'.
           R is an output argument if FACT = 'E' and EQUED = 'R' or 'B'.
 C         Optional (input or output) REAL array, shape (:) with size(C) 
           = size(A,1). The column scale factors for A.
           C is an input argument if FACT = 'F' and EQUED = 'C' or 'B'.
           C is an output argument if FACT = 'E' and EQUED = 'C' or 'B'.
 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.
 RPVGRW    Optional (output) REAL.
           The reciprocal pivot growth factor ||A||inf = ||U||inf. If
           RPVGRW 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 , condition estimator 
           RCOND, and forward error bound FERR could be unreliable. If
           the factorization fails with 0 < INFO <= size(A,1), then
           RPVGRW contains the reciprocal pivot growth factor for the 
           leading INFO columns of A.
 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: U(i,i) = 0. The factorization has been completed,
                    but the factor U is singular, so the solution could 
 		   not be computed.
               = n+1: U 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.