!-----------------------------------------------------------------
!
! Purpose
! =======
!
! LA_SYSVX / LA_SYSVX uses the diagonal pivoting factorization to
! compute the solution to a system of linear equations A * X = B,
! where A is a symmetric, hermitial or complex matrix and X and B
! are either rectangular matrices or vectors.
! Error bounds on the solution and a condition estimate are also
! provided.
!
! Description
! ===========
!
! The following steps are performed:
! 1. If FACT = 'N', the diagonal pivoting method is used to factor A.
!    The form of the factorization is
!       A = U * D * U**T,  if UPLO = 'U', or
!       A = L * D * L**T,  if UPLO = 'L',
!    where U (or L) is a product of permutation and unit upper (lower)
!    triangular matrices, and D is symmetric and block diagonal with
!    1-by-1 and 2-by-2 diagonal blocks.
! 2. If some D(i,i)=0, so that D is exactly singular, then the routine
!    returns with INFO = i. Otherwise, the factored form of A is used
!    to estimate the condition number of the matrix A.  If the
!    reciprocal of the condition number is less than machine precision,
!    INFO = N+1 is returned as a warning, but the routine still goes on
!    to solve for X and compute error bounds as described below.
! 3. The system of equations is solved for X using the factored form
!    of A.
! 4. Iterative refinement is applied to improve the computed solution
!    matrix and calculate error bounds and backward error estimates
!    for it.
!
! Arguments
! =========
!
!  SUBROUTINE LA_SYSVX / LA_HESVX( A, B, X, UPLO, AF, IPIV, &
!                                  FACT, FERR, BERR, RCOND, INFO )
!     <type>(<wp>), INTENT(IN) :: A(:,:), <rhs>
!     <type>(<wp>), INTENT(OUT) :: <sol>
!     CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: UPLO
!     <type>(<wp>), INTENT(INOUT), OPTIONAL :: AF(:,:)
!     INTEGER, INTENT(INOUT), OPTIONAL :: IPIV(:)
!     CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: FACT
!     REAL(<wp>), INTENT(OUT), OPTIONAL :: <err>, RCOND
!     INTEGER, INTENT(OUT), OPTIONAL :: INFO
!    <type> ::= REAL | COMPLEX
!    <wp>   ::= KIND(1.0) | KIND(1.0D0)
!    <rhs>  ::= B(:,:) | B(:)
!    <sol>  ::= X(:,:) | X(:)
!    <err>  ::= FERR(:), BERR(:) | FERR, BERR
!
! =====================
!
! A    (input) either REAL or COMPLEX square array, shape (:,:),
!      SIZE(A,1) == SIZE(A,2).
!      The symmetric matrix A.  If UPLO = 'U', the leading
!      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 leading 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.
!      INFO = -1 if SIZE(A,2) /= SIZE(A,1) or SIZE(A,1) < 0.
!
! B    (input) either REAL or COMPLEX rectangular array,
!      shape either (:,:) or (:), size(B,1) or size(B) == size(A,1).
!      The right hand side matrix B.
!      INFO = -2 if SIZE(B,1) /= SIZE(A,1) or SIZE(B,2) < 0
!                    and if shape of B is (:,:) or
!                if SIZE(B) /= SIZE(A,1) and if shape of B is (:)
!
! X    (output) either REAL or COMPLEX rectangular array,
!      shape either (:,:) or (:), size(X,1) or size(X) == size(A,1)
!      and SIZE(X,2) == SIZE(B,2).
!      If INFO = 0 or INFO = n+1, the solution matrix X, where
!      n = SIZE(A,1).
!      INFO = -3 if SIZE(X,1) /= SIZE(A,1) or SIZE(X,2) /= SIZE(B,2)
!                    and if shape of X is (:,:) or
!                if SIZE(X) /= SIZE(A,1) and if shape of X is (:)
!
! UPLO Optional (input) CHARACTER*1
!      = 'U':  Upper triangle of A is stored;
!      = 'L':  Lower triangle of A is stored.
!      If UPLO is not present UPLO = 'U' is assumed.
!      INFO -4 if UPLO /= 'U' nor 'L'.
!
! AF   Optional (input/output) either REAL or COMPLEX array, shape (:,:),
!      SIZE(AF,1) == SIZE(A) and SIZE(AF,2) == SIZE(A,2).
!      If FACT = 'F', then AF is an input argument and on entry
!      contains the block diagonal matrix D and the multipliers used
!      to obtain the factor U or L from the factorization
!      A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.
!      If FACT = 'N', then AF is an output argument and on exit
!      returns the block diagonal matrix D and the multipliers used
!      to obtain the factor U or L from the factorization
!      A = U*D*U**T or A = L*D*L**T.
!      INFO = -5 if SIZE(AF,1) /= SIZE(A,1) or SIZE(AF,2) /= SIZE(A,2).
!
! IPIV Optional (input/output) INTEGER array, shape (:), SIZE(IPIV) == SIZE(A,1).
!      If FACT = 'F', then IPIV is an input argument and on entry
!      contains details of the interchanges and the block structure
!      of D, as determined by _SYTRF.
!      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 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.  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 and on exit
!      contains details of the interchanges and the block structure
!      of D, as determined by _SYTRF.
!      INFO = -6 if SIZE(IPIV) /= SIZE(A,1).
!
! FACT Optional (input) CHARACTER*1
!      Specifies whether or not the factored form of A has been
!      supplied on entry.
!      = 'F':  On entry, AF and IPIV contain the factored form of
!              A.  AF and IPIV will not be modified.
!      = 'N':  The matrix A will be copied to AF and factored.
!      If FACT is not present FACT = 'N' is assumed.
!      INFO = -7 if ( FACT /= 'F' nor 'N' ) or
!                 if FACT ='F' and AF and IPIV are not present.
!
! FERR Optional (output) eithe REAL array, shape (:) or REAL scalar.
!      SIZE(FERR) == SIZE(B,2), if FERR is an array
!      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.
!      INFO = -8 if SIZE(FERR) /= SIZE(B,2) and if FERR is a vector.
!
! BERR Optional (output) eithe REAL array, shape (:) or REAL scalar.
!      SIZE(BERR) == SIZE(B,2), if BERR is an array.
!      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).
!      INFO = -10 if SIZE(FERR) /= SIZE(B,2) and if BERR is a vector.
!
! RCOND Optional (output) REAL
!      The estimate of the reciprocal condition number of the matrix
!      A.  If RCOND is less than the machine precision (in
!      particular, if RCOND = 0), 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) is exactly zero.  The factorization
!                   has been completed but the factor D is exactly
!                   singular, so the solution and error bounds could
!                   not be computed. RCOND = 0 is returned.
!            = n+1: D is nonsingular, but RCOND is less than machine
!                   precision, meaning that the matrix is singular
!                   to working precision.  Nevertheless, the
!                   solution and error bounds are computed because
!                   there are a number of situations where 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.
!      n = SIZE(A,1) is a rank of the matrix A.
!--------------------------------------

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