```      SUBROUTINE ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
\$                   IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
\$                   INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
*
*     .. Scalar Arguments ..
CHARACTER          TRANS
INTEGER            INFO, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
INTEGER            IPIV( * )
DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16         B( LDB, * ), D( * ), DF( * ), DL( * ),
\$                   DLF( * ), DU( * ), DU2( * ), DUF( * ),
\$                   WORK( * ), X( LDX, * )
*     ..
*
*  Purpose
*  =======
*
*  ZGTRFS improves the computed solution to a system of linear
*  equations when the coefficient matrix is tridiagonal, 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)
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  DL      (input) COMPLEX*16 array, dimension (N-1)
*          The (n-1) subdiagonal elements of A.
*
*  D       (input) COMPLEX*16 array, dimension (N)
*          The diagonal elements of A.
*
*  DU      (input) COMPLEX*16 array, dimension (N-1)
*          The (n-1) superdiagonal elements of A.
*
*  DLF     (input) COMPLEX*16 array, dimension (N-1)
*          The (n-1) multipliers that define the matrix L from the
*          LU factorization of A as computed by ZGTTRF.
*
*  DF      (input) COMPLEX*16 array, dimension (N)
*          The n diagonal elements of the upper triangular matrix U from
*          the LU factorization of A.
*
*  DUF     (input) COMPLEX*16 array, dimension (N-1)
*          The (n-1) elements of the first superdiagonal of U.
*
*  DU2     (input) COMPLEX*16 array, dimension (N-2)
*          The (n-2) elements of the second superdiagonal of U.
*
*  IPIV    (input) INTEGER array, dimension (N)
*          The pivot indices; for 1 <= i <= n, 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.
*
*  B       (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
*          On entry, the solution matrix X, as computed by ZGTTRS.
*          On exit, the improved solution matrix X.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X.  LDX >= max(1,N).
*
*  FERR    (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) COMPLEX*16 array, dimension (2*N)
*
*  RWORK   (workspace) DOUBLE PRECISION 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 )
DOUBLE PRECISION   ZERO, ONE
PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION   TWO
PARAMETER          ( TWO = 2.0D+0 )
DOUBLE PRECISION   THREE
PARAMETER          ( THREE = 3.0D+0 )
*     ..
*     .. Local Scalars ..
LOGICAL            NOTRAN
CHARACTER          TRANSN, TRANST
INTEGER            COUNT, I, J, KASE, NZ
DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
COMPLEX*16         ZDUM
*     ..
*     .. Local Arrays ..
INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
EXTERNAL           XERBLA, ZAXPY, ZCOPY, ZGTTRS, ZLACN2, ZLAGTM
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          ABS, DBLE, DCMPLX, DIMAG, MAX
*     ..
*     .. External Functions ..
LOGICAL            LSAME
DOUBLE PRECISION   DLAMCH
EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Statement Functions ..
DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
*     ..
*     .. 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( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -13
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -15
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGTRFS', -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
TRANSN = 'N'
TRANST = 'C'
ELSE
TRANSN = 'C'
TRANST = 'N'
END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = 4
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
DO 110 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 ZCOPY( N, B( 1, J ), 1, WORK, 1 )
CALL ZLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE,
\$                WORK, N )
*
*        Compute abs(op(A))*abs(x) + abs(b) for use in the backward
*        error bound.
*
IF( NOTRAN ) THEN
IF( N.EQ.1 ) THEN
RWORK( 1 ) = CABS1( B( 1, J ) ) +
\$                      CABS1( D( 1 ) )*CABS1( X( 1, J ) )
ELSE
RWORK( 1 ) = CABS1( B( 1, J ) ) +
\$                      CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
\$                      CABS1( DU( 1 ) )*CABS1( X( 2, J ) )
DO 30 I = 2, N - 1
RWORK( I ) = CABS1( B( I, J ) ) +
\$                         CABS1( DL( I-1 ) )*CABS1( X( I-1, J ) ) +
\$                         CABS1( D( I ) )*CABS1( X( I, J ) ) +
\$                         CABS1( DU( I ) )*CABS1( X( I+1, J ) )
30          CONTINUE
RWORK( N ) = CABS1( B( N, J ) ) +
\$                      CABS1( DL( N-1 ) )*CABS1( X( N-1, J ) ) +
\$                      CABS1( D( N ) )*CABS1( X( N, J ) )
END IF
ELSE
IF( N.EQ.1 ) THEN
RWORK( 1 ) = CABS1( B( 1, J ) ) +
\$                      CABS1( D( 1 ) )*CABS1( X( 1, J ) )
ELSE
RWORK( 1 ) = CABS1( B( 1, J ) ) +
\$                      CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
\$                      CABS1( DL( 1 ) )*CABS1( X( 2, J ) )
DO 40 I = 2, N - 1
RWORK( I ) = CABS1( B( I, J ) ) +
\$                         CABS1( DU( I-1 ) )*CABS1( X( I-1, J ) ) +
\$                         CABS1( D( I ) )*CABS1( X( I, J ) ) +
\$                         CABS1( DL( I ) )*CABS1( X( I+1, J ) )
40          CONTINUE
RWORK( N ) = CABS1( B( N, J ) ) +
\$                      CABS1( DU( N-1 ) )*CABS1( X( N-1, J ) ) +
\$                      CABS1( D( N ) )*CABS1( X( N, J ) )
END IF
END IF
*
*        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.
*
S = ZERO
DO 50 I = 1, N
IF( RWORK( I ).GT.SAFE2 ) THEN
S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
ELSE
S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
\$             ( RWORK( I )+SAFE1 ) )
END IF
50    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 ZGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, WORK, N,
\$                   INFO )
CALL ZAXPY( N, DCMPLX( ONE ), WORK, 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 ZLACN2 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 60 I = 1, N
IF( RWORK( I ).GT.SAFE2 ) THEN
RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
ELSE
RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
\$                      SAFE1
END IF
60    CONTINUE
*
KASE = 0
70    CONTINUE
CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(op(A)**H).
*
CALL ZGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
\$                      N, INFO )
DO 80 I = 1, N
WORK( I ) = RWORK( I )*WORK( I )
80          CONTINUE
ELSE
*
*              Multiply by inv(op(A))*diag(W).
*
DO 90 I = 1, N
WORK( I ) = RWORK( I )*WORK( I )
90          CONTINUE
CALL ZGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
\$                      N, INFO )
END IF
GO TO 70
END IF
*
*        Normalize error.
*
LSTRES = ZERO
DO 100 I = 1, N
LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
100    CONTINUE
IF( LSTRES.NE.ZERO )
\$      FERR( J ) = FERR( J ) / LSTRES
*
110 CONTINUE
*
RETURN
*
*     End of ZGTRFS
*
END

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