```      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

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