!-----------------------------------------------------------------
!
! Purpose
! =======
!
! LA_GTSVX uses the LU factorization to compute the solution to a system
! of linear equations A * X = B or A**T * X = B, where A is a tridiagonal
! matrix of order N 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 LU decomposition is used to factor the matrix A
!    as A = L * U, where L is a product of permutation and unit lower
!    bidiagonal matrices and U is upper triangular with nonzeros in
!    only the main diagonal and first two superdiagonals.
! 2. If some U(i,i)=0, so that U 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_GTSVX(DL, D, DU, B, X, DLF, DF, DUF, DU2, &
!                      IPIV, FACT, TRANS, FERR, BERR, RCOND, INFO)
!    <type>(<wp>), INTENT(IN) :: DL(:), D(:), DU(:), <rhs>
!    <type>(<wp>), INTENT(OUT), OPTIONAL :: <sol>
!    <type>(<wp>), INTENT(INOUT), OPTIONAL :: DLF(:), DF(:), DUF(:), DU2(:)
!    INTEGER, INTENT(INOUT), OPTIONAL :: IPIV(:)
!    CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: FACT, TRANS
!    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
!
! =====================
!
! DL   (input) either REAL or COMPLEX array, shape (:),
!      SIZE(DL) == SIZE(D)-1. The subdiagonal elements of A.
!      INFO = -1 if SIZE(DL) /= SIZE(D) - 1.
!
! D    (input) either REAL or COMPLEX array, shape (:).
!      The diagonal elements of A.
!      INFO = -2 if SIZE(D) < 0
!
! DU   (input) either REAL or COMPLEX array, shape (:),
!      SIZE(DU) == SIZE(D)-1. The superdiagonal elements of A.
!      INFO = -3 if SIZE(DU) /= 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 eithermatrix or vector B.
!      INFO = -4 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 or INFO = n+1, the solution matrix X. n = SIZE(D)
!      INFO = -5 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 (:)
!
! DLF  Optional (input/output) either REAL or COMPLEX array, shape (:),
!      SIZE(DLF) == SIZE(DL). If FACT = 'F', then DLF is an input
!      argument and on entry contains the multipliers that define
!      the matrix L from the LU factorization of A.
!      If FACT = 'N', then DLF is an output argument and on exit
!      contains the multipliers that define the matrix L from
!      the LU factorization of A.
!      INFO = -6 if SIZE(DLF) /= SIZE(DL) or
!                if all DLF, DF, DUF and DU2 together are not present.
!
! DF   Optional (input/output) either REAL or COMPLEX array, shape (:),
!      SIZE(DF) == SIZE(D).
!      If FACT = 'F', then DF is an input argument and on entry
!      contains the diagonal elements of the upper triangular
!      matrix U from the LU factorization of A.
!      If FACT = 'N', then DF is an output argument and on exit
!      contains the n diagonal elements of the upper triangular
!      matrix U from the LU factorization of A.
!      INFO = -7 if SIZE(DF) /= SIZE(D)
!
! DUF  Optional (input/output) either REAL or COMPLEX array, shape (:),
!      SIZE(DUF) == SIZE(DU).
!      If FACT = 'F', then DUF is an input argument and on entry
!      contains the elements of the first superdiagonal of U.
!      If FACT = 'N', then DUF is an output argument and on exit
!      contains the elements of the first superdiagonal of U.
!      INFO = -8 if SIZE(DLF) /= SIZE(DL)
!
! DU2  Optional (input/output) either REAL or COMPLEX array, shape (:),
!      SIZE(DU2) == SIZE(D)-2.
!      If FACT = 'F', then DU2 is an input argument and on entry
!      contains the elements of the second superdiagonal of U.
!      If FACT = 'N', then DU2 is an output argument and on exit
!      contains the elements of the second superdiagonal of U.
!      INFO = -9 if SIZE(DU2) /= SIZE(D) - 2.
!
! IPIV Optional (input/output) INTEGER array, shape (:), SIZE(IPIV) == SIZE(D).
!      If FACT = 'F', then IPIV is an input argument and on entry
!      contains the pivot indices from the LU factorization of A.
!      If FACT = 'N', then IPIV is an output argument and on exit
!      contains the pivot indices from the LU factorization of A;
!      row i of the matrix was interchanged with row IPIV(i).
!      IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
!      a row interchange was not required.
!      INFO = -10 if SIZE(IPIV) /= SIZE(D).
!
! FACT Optional (input) CHARACTER*1
!      Specifies whether or not the factored form of A has been
!      supplied on entry.
!      = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored
!              form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
!              will not be modified.
!      = 'N':  The matrix will be copied to DLF, DF, and DUF
!              and factored.
!      If FACT is not present FACT = 'N' is assumed.
!      INFO = -11 if ( FACT /= 'F' nor 'N' ) or
!                 if FACT ='F' and either DF or IPIV is not present.
!
! TRANS Optional (input) CHARACTER*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 = Transpose)
!      If TRANS is not present TRANS = 'N' is assumed.
!      INFO = -12 if TRANS /= 'N' nor 'T' nor 'C'
!
!
! 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 = -13 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 FERR 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 = -14 if SIZE(BERR) /= 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 Optional (output) INTEGER scalar.
!      = 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) is exactly zero.  The factorization
!                   has not been completed unless i = N, but the
!                   factor U is exactly singular, so the solution
!                   and error bounds could not be 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(D,1) is a rank of the matrix A.
!--------------------------------------

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