          SUBROUTINE SGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
         $                   IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
         $                   INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     Modified to call SLACN2 in place of SLACON, 7 Feb 03, SJH.
    *
    *     .. Scalar Arguments ..
          CHARACTER          TRANS
          INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
    *     ..
    *     .. Array Arguments ..
          INTEGER            IPIV( * ), IWORK( * )
          REAL               AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
         $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SGBRFS improves the computed solution to a system of linear
    *  equations when the coefficient matrix is banded, and provides
    *  error bounds and backward error estimates for the solution.
    *
    *  Arguments
    *  =========
    *
    *  TRANS   (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)
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  KL      (input) INTEGER
    *          The number of subdiagonals within the band of A.  KL >= 0.
    *
    *  KU      (input) INTEGER
    *          The number of superdiagonals within the band of A.  KU >= 0.
    *
    *  NRHS    (input) INTEGER
    *          The number of right hand sides, i.e., the number of columns
    *          of the matrices B and X.  NRHS >= 0.
    *
    *  AB      (input) REAL array, dimension (LDAB,N)
    *          The original band matrix A, stored 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).
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array AB.  LDAB >= KL+KU+1.
    *
    *  AFB     (input) REAL array, dimension (LDAFB,N)
    *          Details of the LU factorization of the band matrix A, as
    *          computed by SGBTRF.  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.
    *
    *  LDAFB   (input) INTEGER
    *          The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.
    *
    *  IPIV    (input) INTEGER array, dimension (N)
    *          The pivot indices from SGBTRF; for 1<=i<=N, row i of the
    *          matrix was interchanged with row IPIV(i).
    *
    *  B       (input) REAL array, dimension (LDB,NRHS)
    *          The right hand side matrix B.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B.  LDB >= max(1,N).
    *
    *  X       (input/output) REAL array, dimension (LDX,NRHS)
    *          On entry, the solution matrix X, as computed by SGBTRS.
    *          On exit, the improved solution matrix X.
    *
    *  LDX     (input) INTEGER
    *          The leading dimension of the array X.  LDX >= max(1,N).
    *
    *  FERR    (output) REAL array, dimension (NRHS)
    *          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.
    *
    *  BERR    (output) REAL array, dimension (NRHS)
    *          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).
    *
    *  WORK    (workspace) REAL array, dimension (3*N)
    *
    *  IWORK   (workspace) INTEGER array, dimension (N)
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *
    *  Internal Parameters
    *  ===================
    *
    *  ITMAX is the maximum number of steps of iterative refinement.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          INTEGER            ITMAX
          PARAMETER          ( ITMAX = 5 )
          REAL               ZERO
          PARAMETER          ( ZERO = 0.0E+0 )
          REAL               ONE
          PARAMETER          ( ONE = 1.0E+0 )
          REAL               TWO
          PARAMETER          ( TWO = 2.0E+0 )
          REAL               THREE
          PARAMETER          ( THREE = 3.0E+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            NOTRAN
          CHARACTER          TRANST
          INTEGER            COUNT, I, J, K, KASE, KK, NZ
          REAL               EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
    *     ..
    *     .. Local Arrays ..
          INTEGER            ISAVE( 3 )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SAXPY, SCOPY, SGBMV, SGBTRS, SLACN2, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX, MIN
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          REAL               SLAMCH
          EXTERNAL           LSAME, SLAMCH
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
          NOTRAN = LSAME( TRANS, 'N' )
          IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
         $    LSAME( TRANS, 'C' ) ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          ELSE IF( KL.LT.0 ) THEN
             INFO = -3
          ELSE IF( KU.LT.0 ) THEN
             INFO = -4
          ELSE IF( NRHS.LT.0 ) THEN
             INFO = -5
          ELSE IF( LDAB.LT.KL+KU+1 ) THEN
             INFO = -7
          ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
             INFO = -9
          ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
             INFO = -12
          ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
             INFO = -14
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'SGBRFS', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
             DO 10 J = 1, NRHS
                FERR( J ) = ZERO
                BERR( J ) = ZERO
       10    CONTINUE
             RETURN
          END IF
    *
          IF( NOTRAN ) THEN
             TRANST = 'T'
          ELSE
             TRANST = 'N'
          END IF
    *
    *     NZ = maximum number of nonzero elements in each row of A, plus 1
    *
          NZ = MIN( KL+KU+2, N+1 )
          EPS = SLAMCH( 'Epsilon' )
          SAFMIN = SLAMCH( 'Safe minimum' )
          SAFE1 = NZ*SAFMIN
          SAFE2 = SAFE1 / EPS
    *
    *     Do for each right hand side
    *
          DO 140 J = 1, NRHS
    *
             COUNT = 1
             LSTRES = THREE
       20    CONTINUE
    *
    *        Loop until stopping criterion is satisfied.
    *
    *        Compute residual R = B - op(A) * X,
    *        where op(A) = A, A**T, or A**H, depending on TRANS.
    *
             CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
             CALL SGBMV( TRANS, N, N, KL, KU, -ONE, AB, LDAB, X( 1, J ), 1,
         $               ONE, WORK( N+1 ), 1 )
    *
    *        Compute componentwise relative backward error from formula
    *
    *        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
    *
    *        where abs(Z) is the componentwise absolute value of the matrix
    *        or vector Z.  If the i-th component of the denominator is less
    *        than SAFE2, then SAFE1 is added to the i-th components of the
    *        numerator and denominator before dividing.
    *
             DO 30 I = 1, N
                WORK( I ) = ABS( B( I, J ) )
       30    CONTINUE
    *
    *        Compute abs(op(A))*abs(X) + abs(B).
    *
             IF( NOTRAN ) THEN
                DO 50 K = 1, N
                   KK = KU + 1 - K
                   XK = ABS( X( K, J ) )
                   DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL )
                      WORK( I ) = WORK( I ) + ABS( AB( KK+I, K ) )*XK
       40          CONTINUE
       50       CONTINUE
             ELSE
                DO 70 K = 1, N
                   S = ZERO
                   KK = KU + 1 - K
                   DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL )
                      S = S + ABS( AB( KK+I, K ) )*ABS( X( I, J ) )
       60          CONTINUE
                   WORK( K ) = WORK( K ) + S
       70       CONTINUE
             END IF
             S = ZERO
             DO 80 I = 1, N
                IF( WORK( I ).GT.SAFE2 ) THEN
                   S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
                ELSE
                   S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
         $             ( WORK( I )+SAFE1 ) )
                END IF
       80    CONTINUE
             BERR( J ) = S
    *
    *        Test stopping criterion. Continue iterating if
    *           1) The residual BERR(J) is larger than machine epsilon, and
    *           2) BERR(J) decreased by at least a factor of 2 during the
    *              last iteration, and
    *           3) At most ITMAX iterations tried.
    *
             IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
         $       COUNT.LE.ITMAX ) THEN
    *
    *           Update solution and try again.
    *
                CALL SGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
         $                   WORK( N+1 ), N, INFO )
                CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
                LSTRES = BERR( J )
                COUNT = COUNT + 1
                GO TO 20
             END IF
    *
    *        Bound error from formula
    *
    *        norm(X - XTRUE) / norm(X) .le. FERR =
    *        norm( abs(inv(op(A)))*
    *           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
    *
    *        where
    *          norm(Z) is the magnitude of the largest component of Z
    *          inv(op(A)) is the inverse of op(A)
    *          abs(Z) is the componentwise absolute value of the matrix or
    *             vector Z
    *          NZ is the maximum number of nonzeros in any row of A, plus 1
    *          EPS is machine epsilon
    *
    *        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
    *        is incremented by SAFE1 if the i-th component of
    *        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
    *
    *        Use SLACN2 to estimate the infinity-norm of the matrix
    *           inv(op(A)) * diag(W),
    *        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
    *
             DO 90 I = 1, N
                IF( WORK( I ).GT.SAFE2 ) THEN
                   WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
                ELSE
                   WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
                END IF
       90    CONTINUE
    *
             KASE = 0
      100    CONTINUE
             CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
         $                KASE, ISAVE )
             IF( KASE.NE.0 ) THEN
                IF( KASE.EQ.1 ) THEN
    *
    *              Multiply by diag(W)*inv(op(A)**T).
    *
                   CALL SGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV,
         $                      WORK( N+1 ), N, INFO )
                   DO 110 I = 1, N
                      WORK( N+I ) = WORK( N+I )*WORK( I )
      110          CONTINUE
                ELSE
    *
    *              Multiply by inv(op(A))*diag(W).
    *
                   DO 120 I = 1, N
                      WORK( N+I ) = WORK( N+I )*WORK( I )
      120          CONTINUE
                   CALL SGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
         $                      WORK( N+1 ), N, INFO )
                END IF
                GO TO 100
             END IF
    *
    *        Normalize error.
    *
             LSTRES = ZERO
             DO 130 I = 1, N
                LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
      130    CONTINUE
             IF( LSTRES.NE.ZERO )
         $      FERR( J ) = FERR( J ) / LSTRES
    *
      140 CONTINUE
    *
          RETURN
    *
    *     End of SGBRFS
    *
          END