!-----------------------------------------------------------------
!
! Purpose
! =======
!
! LA_PTSVX uses the factorization A = L*D*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 tridiagonal 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 matrix A is factored as A = L*D*L**T, where L
!    is a unit lower bidiagonal matrix and D is diagonal.  The
!    factorization can also be regarded as having the form
!    A = U**T*D*U.
! 2. 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.
! 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_PTSVX(D, E, B, X, DF, EF, FACT, FERR, BERR, &
!                      RCOND, INFO)
!    REAL(<wp>), INTENT(IN) :: D(:)
!    <type>(<wp>), INTENT(IN) :: E(:), <rhs>
!    <type>(<wp>), INTENT(OUT) :: <sol>
!    <type>(<wp>), INTENT(INOUT), OPTIONAL :: DF(:), EF(:)
!    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
!
! =====================
!
! D    (input) either REAL array, shape (:), SIZE(D) == n.
!      The n diagonal elements of the tridiagonal matrix A.
!      INFO = -1 if SIZE(D) < 0
!
! E    (input) either REAL or COMPLEX array, shape (:), SIZE(E) == SIZE(D)-1.
!      The (n-1) subdiagonal elements of the tridiagonal matrix A.
!      INFO = -2 if SIZE(E) /= SIZE(D) - 1.
!
! B    (input) either REAL or COMPLEX rectangular array,
!      shape either (:,:) or (:), size(B,1) or size(B) == size(D).
!      The right hand side matrix/vector B.
!      INFO = -3 if SIZE(B,1) /= SIZE(D) or SIZE(B,2) < 0
!                    and if shape of B is (:,:) or
!                if SIZE(B) /= SIZE(D) and if shape of B is (:)
!
! X    (output) either REAL or COMPLEX rectangular array, shape either
!      (:,:) or (:), size(B,1) or size(B) == size(D), and
!      SIZE(X,2) == SIZE(B,2) if B and X are matrices.
!      If INFO = 0 of INFO = n+1, the solution matrix X, n = SIZE(D).
!      INFO = -4 if SIZE(X,1) /= SIZE(D) or SIZE(X,2) /= SIZE(B,2)
!                    and if shape of X is (:,:) or
!                if SIZE(X) /= SIZE(D) and if shape of X is (:)
!
! DF   Optional (input/output) either REAL array, shape (:),
!      SIZE(DF) == SIZE(D).
!      If FACT = 'F', then DF is an input argument and on entry
!      contains the n diagonal elements of the diagonal matrix D
!      from the L*D*L**T factorization of A.
!      If FACT = 'N', then DF is an output argument and on exit
!      contains the n diagonal elements of the diagonal matrix D
!      from the L*D*L**T factorization of A.
!      INFO = -5 if SIZE(DF) /= SIZE(D) or
!                if both DF and EF together are not present.
!
! EF   Optional (input/output) either REAL or COMPLEX array, shape (:),
!      SIZE(EF) == SIZE(E). n = SIZE(D).
!      If FACT = 'F', then EF is an input argument and on entry
!      contains the (n-1) subdiagonal elements of the unit
!      bidiagonal factor L from the L*D*L**T factorization of A.
!      If FACT = 'N', then EF is an output argument and on exit
!      contains the (n-1) subdiagonal elements of the unit
!      bidiagonal factor L from the L*D*L**T factorization of A.
!      INFO = -6 if SIZE(EF) /= SIZE(E)
!
! FACT Optional (input) CHARACTER*1
!      Specifies whether or not the factored form of A has been
!      supplied on entry.
!      = 'F':  On entry, DF and EF contain the factored form of A.
!              D, E, DF, and EF will not be modified.
!      = 'N':  The matrix A will be copied to DF and EF and
!              factored.
!      If FACT is not present FACT = 'N' is assumed.
!      INFO = -7 if ( FACT /= 'F' nor 'N' ) or
!                if FACT ='F' and DF and EF are not present.
!
! FERR Optional (output) either REAL array, shape (:) or REAL scalar.
!      SIZE(FERR) == SIZE(B,2), if FERR is an array.
!      The 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).
!      INFO = -8 if SIZE(FERR) /= SIZE(B,2) and if FERR is a vector.
!
! BERR Optional (output) either REAL array, shape (:) 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).
!      INFO = -9 if SIZE(BERR) /= SIZE(B,2) and if BERR is a vector.
!
! RCOND (output) REAL.
!      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 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(AB,2) is a rank of the matrix A.
!--------------------------------------

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