dlatdf.f

          SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
         $                   JPIV )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            IJOB, LDZ, N
          DOUBLE PRECISION   RDSCAL, RDSUM
    *     ..
    *     .. Array Arguments ..
          INTEGER            IPIV( * ), JPIV( * )
          DOUBLE PRECISION   RHS( * ), Z( LDZ, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLATDF uses the LU factorization of the n-by-n matrix Z computed by
    *  DGETC2 and computes a contribution to the reciprocal Dif-estimate
    *  by solving Z * x = b for x, and choosing the r.h.s. b such that
    *  the norm of x is as large as possible. On entry RHS = b holds the
    *  contribution from earlier solved sub-systems, and on return RHS = x.
    *
    *  The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
    *  where P and Q are permutation matrices. L is lower triangular with
    *  unit diagonal elements and U is upper triangular.
    *
    *  Arguments
    *  =========
    *
    *  IJOB    (input) INTEGER
    *          IJOB = 2: First compute an approximative null-vector e
    *              of Z using DGECON, e is normalized and solve for
    *              Zx = +-e - f with the sign giving the greater value
    *              of 2-norm(x). About 5 times as expensive as Default.
    *          IJOB .ne. 2: Local look ahead strategy where all entries of
    *              the r.h.s. b is choosen as either +1 or -1 (Default).
    *
    *  N       (input) INTEGER
    *          The number of columns of the matrix Z.
    *
    *  Z       (input) DOUBLE PRECISION array, dimension (LDZ, N)
    *          On entry, the LU part of the factorization of the n-by-n
    *          matrix Z computed by DGETC2:  Z = P * L * U * Q
    *
    *  LDZ     (input) INTEGER
    *          The leading dimension of the array Z.  LDA >= max(1, N).
    *
    *  RHS     (input/output) DOUBLE PRECISION array, dimension N.
    *          On entry, RHS contains contributions from other subsystems.
    *          On exit, RHS contains the solution of the subsystem with
    *          entries acoording to the value of IJOB (see above).
    *
    *  RDSUM   (input/output) DOUBLE PRECISION
    *          On entry, the sum of squares of computed contributions to
    *          the Dif-estimate under computation by DTGSYL, where the
    *          scaling factor RDSCAL (see below) has been factored out.
    *          On exit, the corresponding sum of squares updated with the
    *          contributions from the current sub-system.
    *          If TRANS = 'T' RDSUM is not touched.
    *          NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
    *
    *  RDSCAL  (input/output) DOUBLE PRECISION
    *          On entry, scaling factor used to prevent overflow in RDSUM.
    *          On exit, RDSCAL is updated w.r.t. the current contributions
    *          in RDSUM.
    *          If TRANS = 'T', RDSCAL is not touched.
    *          NOTE: RDSCAL only makes sense when DTGSY2 is called by
    *                DTGSYL.
    *
    *  IPIV    (input) INTEGER array, dimension (N).
    *          The pivot indices; for 1 <= i <= N, row i of the
    *          matrix has been interchanged with row IPIV(i).
    *
    *  JPIV    (input) INTEGER array, dimension (N).
    *          The pivot indices; for 1 <= j <= N, column j of the
    *          matrix has been interchanged with column JPIV(j).
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
    *     Umea University, S-901 87 Umea, Sweden.
    *
    *  This routine is a further developed implementation of algorithm
    *  BSOLVE in [1] using complete pivoting in the LU factorization.
    *
    *  [1] Bo Kagstrom and Lars Westin,
    *      Generalized Schur Methods with Condition Estimators for
    *      Solving the Generalized Sylvester Equation, IEEE Transactions
    *      on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
    *
    *  [2] Peter Poromaa,
    *      On Efficient and Robust Estimators for the Separation
    *      between two Regular Matrix Pairs with Applications in
    *      Condition Estimation. Report IMINF-95.05, Departement of
    *      Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          INTEGER            MAXDIM
          PARAMETER          ( MAXDIM = 8 )
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, INFO, J, K
          DOUBLE PRECISION   BM, BP, PMONE, SMINU, SPLUS, TEMP
    *     ..
    *     .. Local Arrays ..
          INTEGER            IWORK( MAXDIM )
          DOUBLE PRECISION   WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DAXPY, DCOPY, DGECON, DGESC2, DLASSQ, DLASWP,
         $                   DSCAL
    *     ..
    *     .. External Functions ..
          DOUBLE PRECISION   DASUM, DDOT
          EXTERNAL           DASUM, DDOT
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, SQRT
    *     ..
    *     .. Executable Statements ..
    *
          IF( IJOB.NE.2 ) THEN
    *
    *        Apply permutations IPIV to RHS
    *
             CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
    *
    *        Solve for L-part choosing RHS either to +1 or -1.
    *
             PMONE = -ONE
    *
             DO 10 J = 1, N - 1
                BP = RHS( J ) + ONE
                BM = RHS( J ) - ONE
                SPLUS = ONE
    *
    *           Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
    *           SMIN computed more efficiently than in BSOLVE [1].
    *
                SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 )
                SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 )
                SPLUS = SPLUS*RHS( J )
                IF( SPLUS.GT.SMINU ) THEN
                   RHS( J ) = BP
                ELSE IF( SMINU.GT.SPLUS ) THEN
                   RHS( J ) = BM
                ELSE
    *
    *              In this case the updating sums are equal and we can
    *              choose RHS(J) +1 or -1. The first time this happens
    *              we choose -1, thereafter +1. This is a simple way to
    *              get good estimates of matrices like Byers well-known
    *              example (see [1]). (Not done in BSOLVE.)
    *
                   RHS( J ) = RHS( J ) + PMONE
                   PMONE = ONE
                END IF
    *
    *           Compute the remaining r.h.s.
    *
                TEMP = -RHS( J )
                CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
    *
       10    CONTINUE
    *
    *        Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
    *        in BSOLVE and will hopefully give us a better estimate because
    *        any ill-conditioning of the original matrix is transfered to U
    *        and not to L. U(N, N) is an approximation to sigma_min(LU).
    *
             CALL DCOPY( N-1, RHS, 1, XP, 1 )
             XP( N ) = RHS( N ) + ONE
             RHS( N ) = RHS( N ) - ONE
             SPLUS = ZERO
             SMINU = ZERO
             DO 30 I = N, 1, -1
                TEMP = ONE / Z( I, I )
                XP( I ) = XP( I )*TEMP
                RHS( I ) = RHS( I )*TEMP
                DO 20 K = I + 1, N
                   XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP )
                   RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
       20       CONTINUE
                SPLUS = SPLUS + ABS( XP( I ) )
                SMINU = SMINU + ABS( RHS( I ) )
       30    CONTINUE
             IF( SPLUS.GT.SMINU )
         $      CALL DCOPY( N, XP, 1, RHS, 1 )
    *
    *        Apply the permutations JPIV to the computed solution (RHS)
    *
             CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
    *
    *        Compute the sum of squares
    *
             CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
    *
          ELSE
    *
    *        IJOB = 2, Compute approximate nullvector XM of Z
    *
             CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO )
             CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 )
    *
    *        Compute RHS
    *
             CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
             TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) )
             CALL DSCAL( N, TEMP, XM, 1 )
             CALL DCOPY( N, XM, 1, XP, 1 )
             CALL DAXPY( N, ONE, RHS, 1, XP, 1 )
             CALL DAXPY( N, -ONE, XM, 1, RHS, 1 )
             CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP )
             CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP )
             IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) )
         $      CALL DCOPY( N, XP, 1, RHS, 1 )
    *
    *        Compute the sum of squares
    *
             CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
    *
          END IF
    *
          RETURN
    *
    *     End of DLATDF
    *
          END