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