!-----------------------------------------------------------------
!
! Purpose
! =======
!
! LA_PPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
! compute the solution to a real system of linear equations
!    A * X = B,
! where A is either a symmetric positive definite or hermitian matrix
! stored in packed format and X and B 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 = 'E', real scaling factors are computed to equilibrate
!    the system:
!       diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
!    Whether or not the system will be equilibrated depends on the
!    scaling of the matrix A, but if equilibration is used, A is
!    overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
! 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
!    factor the matrix A (after equilibration if FACT = 'E') as
!       A = U**T* U,  if UPLO = 'U', or
!       A = L * L**T,  if UPLO = 'L',
!    where U is an upper triangular matrix and L is a lower triangular
!    matrix.
! 3. If the leading i-by-i principal minor is not positive definite,
!    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.
! 4. The system of equations is solved for X using the factored form
!    of A.
! 5. Iterative refinement is applied to improve the computed solution
!    matrix and calculate error bounds and backward error estimates
!    for it.
! 6. If equilibration was used, the matrix X is premultiplied by
!    diag(S) so that it solves the original system before
!    equilibration.
!
! Arguments
! =========
!
!  SUBROUTINE LA_PPSVX( AP, B, X, UPLO, AFP, FACT, EQUED, &
!                      S, FERR, BERR, RCOND, INFO )
!    <type>(<wp>), INTENT(INOUT) :: AP(:), <rhs>
!    <type>(<wp>), INTENT(OUT) :: <sol>
!    CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: UPLO
!    <type>(<wp>), INTENT(INOUT), OPTIONAL :: AFP(:)
!    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
!    <type> ::= REAL | COMPLEX
!    <wp>   ::= KIND(1.0) | KIND(1.0D0)
!    <rhs>  ::= B(:,:) | B(:)
!    <sol>  ::= X(:,:) | X(:)
!    <err>  ::= FERR(:), BERR(:) | FERR, BERR
!
! =====================
!
! AP   (input/output) either REAL or COMPLEX square array, shape (:),
!      SIZE(A) == n*(n+1)/2, where n is a rank of the matrix A.
!      On entry, the upper or lower triangle of the symmetric matrix
!      A, packed columnwise in a linear array, except if FACT = 'F'
!      and EQUED = 'Y', then A must contain the equilibrated matrix
!      diag(S)*A*diag(S).  The j-th column of A is stored in the
!      array AP as follows:
!      if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
!      if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
!      See below for further details.  A is not modified if
!      FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
!      On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
!      diag(S)*A*diag(S).
!      INFO = -1 if SIZE(AP) < 0 or SIZE(AP) /= n*(n+1)/2.
!
! B    (input/output) either REAL or COMPLEX rectangular array,
!      shape either (:,:) or (:), size(B,1) or size(B) == n, where
!      n*(n+1)/2 ==  SIZE(A).
!      On entry, the right hand side matrix B.
!      On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
!      B is overwritten by diag(S) * B.
!      INFO = -2 if SIZE(B,1) /= n or SIZE(B,2) < 0
!                    and if shape of B is (:,:) or
!                if SIZE(B) /= n and if shape of B is (:)
!
! X    (output) either REAL or COMPLEX rectangular array,
!      shape either (:,:) or (:), size(X,1) or size(X) == size(B,1)
!      and SIZE(X,2) == SIZE(B,2).
!      If INFO = 0 or INFO = n+1, the solution matrix/vector X to
!      the original system of equations.  Note that if EQUED = 'Y',
!      A and B are modified on exit, and the solution to the
!      equilibrated system is inv(diag(S))*X. SIZE(A) == n*(n+1)/2.
!      INFO = -3 if SIZE(X,1) /= n or SIZE(X,2) /= SIZE(B,2)
!                             and if shape of X is (:,:) or
!                if SIZE(X) /= n 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'.
!
! AFP  Optional (input/output) either REAL or COMPLEX array, shape (:),
!      SIZE(AFP) == n*(n+1)/2, where n is a rank of the matrix A.
!      If FACT = 'F', then AFP is an input argument and on entry
!      contains the triangular factor U or L from the Cholesky
!      factorization A = U'*U or A = L*L', in the same storage
!      format as A.  If EQUED .ne. 'N', then AFP is the factored
!      form of the equilibrated matrix A.
!      If FACT = 'N', then AFP is an output argument and on exit
!      returns the triangular factor U or L from the Cholesky
!      factorization A = U'*U or A = L*L' of the original matrix A.
!      If FACT = 'E', then AFP is an output argument and on exit
!      returns the triangular factor U or L from the Cholesky
!      factorization A = U'*U or A = L*L' of the equilibrated
!      matrix A (see the description of AP for the form of the
!      equilibrated matrix).
!      INFO = -5 if SIZE(AF) /= SIZE(A)
!
! FACT Optional (input) CHARACTER*1
!      Specifies whether or not 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.
!      = 'F':  On entry, AFP contains the factored form of A.
!              If EQUED = 'Y', the matrix A has been equilibrated
!              with scaling factors given by S.  AP and AFP will not
!              be modified.
!      = 'N':  The matrix A will be copied to AFP and factored.
!      = 'E':  The matrix A will be equilibrated if necessary, then
!              copied to AFP and factored.
!      If FACT is not present FACT = 'N' is assumed.
!      INFO = -6 if ( FACT /= 'F' nor 'N' nor 'E' ) or
!                 if FACT ='F' and AF is not present
!
! EQUED Optional (input or output) CHARACTER*1
!      Specifies the form of equilibration that was done.
!      = 'N':  No equilibration (always true if FACT = 'N').
!      = 'Y':  Equilibration was done, i.e., A has been replaced by
!              diag(S) * A * diag(S).
!      EQUED is an input argument if FACT = 'F'; otherwise, it is an
!      output argument.
!      If EQUED is not present EQUED = 'N' is assumed.
!      INFO = -7 if EQUED /= 'N' nor 'Y'
!
! S    Optional (input/output) REAL array, shape (:), SIZE(S) == n,
!      where n*(n+1)/2 == SIZE(A).
!      The scale factors for A; not accessed if EQUED = 'N'.  S is
!      an input argument if FACT = 'F'; otherwise, S is an output
!      argument.  If FACT = 'F' and EQUED = 'Y', each element of S
!      must be positive.
!      INFO = -8 if SIZE(S) /= n or
!                if S <= 0 and S in an input argument.
!
! FERR Optional (output) eithe REAL array, shape (:) or REAL scalar,
!      SIZE(FERR) == SIZE(B,2) if FERR has a shape (:).
!      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 = -9 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 has a shape (:).
!      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 = -9 if SIZE(BERR) /= SIZE(B,2) and if BERR is a vector.
!
! RCOND   (output) REAL
!      The estimate of the reciprocal condition number of the matrix
!      A after equilibration (if done).  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 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 A is
!                   not positive definite, so the factorization
!                   could not be completed, and the solution has not
!                   been computed. RCOND = 0 is returned.
!            = n+1: U 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.
!
!  Further Details
!  ===============
!
!  The packed storage scheme is illustrated by the following example
!  when N = 4, UPLO = 'U':
!
!  Two-dimensional storage of the Hermitian matrix A:
!
!     a11 a12 a13 a14
!         a22 a23 a24
!             a33 a34     (aij = conjg(aji))
!                 a44
!
!  Packed storage of the upper triangle of A:
!
!  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
!--------------------------------------

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