zgtrfs.f

          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