!-----------------------------------------------------------------
!
! Purpose
! =======
!
! LA_GBSVX uses the LU factorization to compute the solution to a system
! of linear equations A * X = B, A**T * X = B, or A**H * X = B, where
! A is a band 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 by this subroutine:
! 1. If FACT = 'E', real scaling factors are computed to equilibrate
!    the system:
!       TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
!       TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
!       TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
!    or diag(C)*B (if TRANS = 'T' or 'C').
! 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
!    matrix A (after equilibration if FACT = 'E') as
!       A = L * U,
!    where L is a product of permutation and unit lower triangular
!    matrices with KL subdiagonals, and U is upper triangular with
!    KL+KU superdiagonals.
! 3. 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.
! 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
!    that it solves the original system before equilibration.
!
! Arguments
! =========
!
!  SUBROUTINE LA_GBSVX(AB, B, X, KL, ABF, IPIV, FACT, TRANS, &
!                      EQUED, R, C, FERR, BERR, RCOND, RPVGRW, INFO)
!    <type>(<wp>), INTENT(INOUT) :: AB(:,:), <rhs>
!    <type>(<wp>), INTENT(OUT) :: <sol>
!    INTEGER, INTENT(IN), OPTIONAL :: KL
!    <type>(<wp>), INTENT(INOUT), OPTIONAL :: ABF(:,:)
!    INTEGER, INTENT(INOUT), OPTIONAL :: IPIV(:)
!    CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: TRANS, FACT
!    CHARACTER(LEN=1), INTENT(INOUT), OPTIONAL :: EQUED
!    REAL(<wp>), INTENT(INOUT), OPTIONAL :: C(:), R(:)
!    REAL(<wp>), INTENT(OUT), OPTIONAL :: <err>, RCOND, RPVGRW
!    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
!
! =====================
!
! AB   (input/output) either REAL or COMPLEX rectangular array,
!      shape (:,:).
!      On entry, the matrix A in band storage, in rows 1 to KL+ku+1.
!      The j-th column of A is stored in the j-th column of the
!      array AB as follows:
!      AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl)
!      Description KL see below, while ku = SIZE(A,1)-2*KL-1 and
!      SIZE(A,2) is the order of the banded matrix (n).
!      If FACT = 'F' and EQUED is not 'N', then A must have been
!      equilibrated by the scaling factors in R and/or C.  AB is not
!      modified if FACT = 'F' or 'N', or if FACT = 'E' and
!      EQUED = 'N' on exit.
!      On exit, if EQUED .ne. 'N', A is scaled as follows:
!      EQUED = 'R':  A := diag(R) * A
!      EQUED = 'C':  A := A * diag(C)
!      EQUED = 'B':  A := diag(R) * A * diag(C).
!      INFO = -1 if SIZE(AB,1) or SIZE(AB,2) < 0
!
! B    (input/output) either REAL or COMPLEX rectangular array,
!      shape either (:,:) or (:), size(B,1) or size(B) == size(A,2).
!      On entry, the right hand side matrix B.
!      On exit,
!      if EQUED = 'N', B is not modified;
!      if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
!      diag(R)*B;
!      if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
!      overwritten by diag(C)*B.
!      INFO = -2 if SIZE(B,1) /= SIZE(AB,2) or SIZE(B,2) < 0
!                    and if shape of B is (:,:) or
!                if SIZE(B) /= SIZE(AB,2) 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,2).
!      If INFO = 0 or INFO = n+1, the solution matrix X to the original
!      system of equations.  Note that AB and B are modified on exit if
!      EQUED .ne. 'N', and the solution to the equilibrated system is
!      inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
!      inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
!      INFO = -3 if SIZE(X,1) /= SIZE(AB,2) or SIZE(X,2) /= SIZE(B,2)
!                    and if shape of X is (:,:) or
!                if SIZE(X) /= SIZE(AB,2) and if shape of X is (:)
!
! KL   Optional (input) INTEGER
!      If KL is present it indicates the number of subdiagonals
!      within the band of A.  KL >= 0. The number of superdiagonals is
!      calculated ku = SIZE(AB,1) - KL - 1.
!      If KL is not present the system assumes equal numbers of subdiagonals
!      and superdiagonals, KL = ku = ( SIZE(AB,1)-1 )/2.
!      INFO = -4 if KL < 0 or KL > SIZE(AB,1)-1
!
! ABF  Optional (input/output) either REAL or COMPLEX square array,
!      shape (:,:), SIZE(ABF,1) == SIZE(AB,1)+KL, SIZE(ABF,2) == SIZE(A,2).
!      If FACT = 'F', then AFB is an input argument and on entry
!      contains details of the LU factorization of the band matrix
!      A, as computed by DGBTRF.  U is stored as an upper triangular
!      band matrix with KL+ku superdiagonals in rows 1 to KL+ku+1,
!      and the multipliers used during the factorization are stored
!      in rows KL+ku+2 to 2*KL+ku+1.  If EQUED .ne. 'N', then AFB is
!      the factored form of the equilibrated matrix A.
!      If FACT = 'N', then AFB is an output argument and on exit
!      returns details of the LU factorization of A.
!      If FACT = 'E', then AFB is an output argument and on exit
!      returns details of the LU factorization of the equilibrated
!      matrix A (see the description of AB for the form of the
!      equilibrated matrix).
!      INFO = -5 if SIZE(ABF,1) /= SIZE(AB,1)+KL or SIZE(ABF,2) /= SIZE(AB,2)
!
! IPIV Optional (output) INTEGER array, shape (:), size(IPIV) ==
!      size(AB,2).
!      If FACT = 'F', then IPIV is an input argument and on entry
!      contains the pivot indices from the factorization A = L*U
!      as computed by DGBTRF; row i of the matrix was interchanged
!      with row IPIV(i).
!      INFO = -6 if SIZE(IPIV) /= SIZE(AB,2)
!
! 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.
!      If FACT is present then:
!         = 'F':  On entry, AFB and IPIV contain the factored form of
!                 A.  If EQUED is not 'N', the matrix A has been
!                 equilibrated with scaling factors given by R and C.
!                 AB, AFB, and IPIV are not modified.
!         = 'N':  The matrix A will be copied to AFB and factored.
!         = 'E':  The matrix A will be equilibrated if necessary, then
!                 copied to AFB and factored.
!      If FACT is not present FACT = 'N' is assumed.
!      INFO = -7 if ( FACT /= 'F' nor 'N' nor 'E' ) or
!                if FACT ='F' and either ABF or IPIV is not present.
!
! TRANS Optional (input) CHARACTER*1
!      Specifies the form of the system of equations.
!      If TRANS is present then:
!         = 'N':  A * X = B     (No transpose)
!         = 'T':  A**T * X = B  (Transpose)
!         = 'C':  A**H * X = B  (Transpose if matrix A is REAL,
!                    Conjugate transpose if matrix A is COMPLEX)
!      If TRANS is not present TRANS = 'N' is assumed.
!      INFO = -8 if TRANS /= 'N' nor 'T' nor 'C'
! 
! EQUED Optional (input or output) CHARACTER*1
!      Specifies the form of equilibration that was done.
!      If EQUED is present then:
!         = '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, i.e., A has been
!                 replaced by diag(R) * A * diag(C).
!         EQUED is an input argument if FACT = 'F'; otherwise, it is an
!         output argument.
!      If FACT is not present EQUED = 'N' is assumed.
!      INFO = -9 if ( EQUED /= 'N' nor 'R' nor 'C' nor 'B' ) or
!                if ( EQUED = 'R' or 'B' ) and R is not present or
!                if ( EQUED = 'C' or 'B' ) and C is not present.
!
! R    Optional (input or output) REAL array, shape (:),
!      SIZE(R) == SIZE(A,2). The row scale factors for A.
!      If EQUED = 'R' or 'B', A is multiplied on the left by
!      diag(R); if EQUED = 'N' or 'C', R is not accessed.  R
!      is an input argument if FACT = 'F'; otherwise, R is an
!      output argument.  If FACT = 'F' and EQUED = 'R' or 'B',
!      each element of R must be positive.
!      INFO = -10 if SIZE(R) /= SIZE(AB,2) or 
!                 if R <= 0 and R in an input argument.
!
! C    Optional (input or output) REAL array, shape (:),
!      SIZE(R) == SIZE(A,2). The column scale factors for A.
!      If EQUED = 'C' or 'B', A is multiplied on the right by
!      diag(C); if EQUED = 'N' or 'R', C is not accessed.  C is
!      an input argument if FACT = 'F'; otherwise, C is an output
!      argument.  If FACT = 'F' and EQUED = 'C' or 'B', each
!      element of C must be positive.
!      INFO = -11 if SIZE(C) /= SIZE(AB,2) or 
!                 if C <= 0 and C in an input argument.
!
! FERR Optional (output) eithe REAL array, shape (:) or REAL scalar.
!      If it is an array, SIZE(FERR) == SIZE(B,2).
!      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 = -12 if SIZE(FERR) /= SIZE(B,2) and if FERR is a vector.
!
! BERR Optional (output) eithe REAL array, shape (:) or REAL scalar.
!      If it is an array, SIZE(BERR) == SIZE(B,2).
!      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 = -13 if SIZE(BERR) /= SIZE(B,2) and if BERR is a vector.
!
! RCOND Optional (output) REAL scalar.
!      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.
!
! RPVGRW Optional (output) REAL scalar.
!      On exit, RPVGRW contains the reciprocal pivot growth
!      factor norm(A)/norm(U). The "max absolute element" norm is
!      used. 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 factorization fails with 0<INFO<=n, then
!      RPVGRW contains the reciprocal pivot growth factor for the
!      leading INFO columns of A. n = SIZE(A,2)
!
! 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 been completed, 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(AB,2) is a rank of the matrix A.
!--------------------------------------

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