cgtrfs.f

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*> \brief \b CGTRFS
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE CGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
*                          IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
*                          INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            INFO, LDB, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       REAL               BERR( * ), FERR( * ), RWORK( * )
*       COMPLEX            B( LDB, * ), D( * ), DF( * ), DL( * ),
*      $                   DLF( * ), DU( * ), DU2( * ), DUF( * ),
*      $                   WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CGTRFS 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is 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)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrix B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*>          DL is COMPLEX array, dimension (N-1)
*>          The (n-1) subdiagonal elements of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is COMPLEX array, dimension (N)
*>          The diagonal elements of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*>          DU is COMPLEX array, dimension (N-1)
*>          The (n-1) superdiagonal elements of A.
*> \endverbatim
*>
*> \param[in] DLF
*> \verbatim
*>          DLF is COMPLEX array, dimension (N-1)
*>          The (n-1) multipliers that define the matrix L from the
*>          LU factorization of A as computed by CGTTRF.
*> \endverbatim
*>
*> \param[in] DF
*> \verbatim
*>          DF is COMPLEX array, dimension (N)
*>          The n diagonal elements of the upper triangular matrix U from
*>          the LU factorization of A.
*> \endverbatim
*>
*> \param[in] DUF
*> \verbatim
*>          DUF is COMPLEX array, dimension (N-1)
*>          The (n-1) elements of the first superdiagonal of U.
*> \endverbatim
*>
*> \param[in] DU2
*> \verbatim
*>          DU2 is COMPLEX array, dimension (N-2)
*>          The (n-2) elements of the second superdiagonal of U.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is 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.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is COMPLEX array, dimension (LDB,NRHS)
*>          The right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is COMPLEX array, dimension (LDX,NRHS)
*>          On entry, the solution matrix X, as computed by CGTTRS.
*>          On exit, the improved solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*>          FERR is 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.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is 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).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  ITMAX is the maximum number of steps of iterative refinement.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gtrfs
*
*  =====================================================================
      SUBROUTINE cgtrfs( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
     $                   DU2,
     $                   IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
     $                   INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            INFO, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      REAL               BERR( * ), FERR( * ), RWORK( * )
      COMPLEX            B( LDB, * ), D( * ), DF( * ), DL( * ),
     $                   dlf( * ), du( * ), du2( * ), duf( * ),
     $                   work( * ), x( ldx, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      real               zero, one
      parameter( zero = 0.0e+0, one = 1.0e+0 )
      REAL               TWO
      parameter( two = 2.0e+0 )
      REAL               THREE
      parameter( three = 3.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN
      CHARACTER          TRANSN, TRANST
      INTEGER            COUNT, I, J, KASE, NZ
      REAL               EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
      COMPLEX            ZDUM
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           caxpy, ccopy, cgttrs, clacn2, clagtm,
     $                   xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, aimag, cmplx, max, real
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               SLAMCH
      EXTERNAL           LSAME, SLAMCH
*     ..
*     .. Statement Functions ..
      REAL               CABS1
*     ..
*     .. Statement Function definitions ..
      cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( 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( 'CGTRFS', -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 = slamch( 'Epsilon' )
      safmin = slamch( 'Safe minimum' )
      safe1 = real( 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 ccopy( n, b( 1, j ), 1, work, 1 )
         CALL clagtm( 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 cgttrs( trans, n, 1, dlf, df, duf, du2, ipiv, work,
     $                   n,
     $                   info )
            CALL caxpy( n, cmplx( 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 CLACN2 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 ) ) + real( nz )*
     $                      eps*rwork( i )
            ELSE
               rwork( i ) = cabs1( work( i ) ) + real( nz )*
     $                      eps*rwork( i ) + safe1
            END IF
   60    CONTINUE
*
         kase = 0
   70    CONTINUE
         CALL clacn2( 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 cgttrs( 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 cgttrs( 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 CGTRFS
*
      END