*> \brief \b ZLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZLATDF + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
* JPIV )
*
* .. Scalar Arguments ..
* INTEGER IJOB, LDZ, N
* DOUBLE PRECISION RDSCAL, RDSUM
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), JPIV( * )
* COMPLEX*16 RHS( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZLATDF computes the contribution to the reciprocal Dif-estimate
*> by solving for x in Z * x = b, where b is chosen such that the norm
*> of x is as large as possible. It is assumed that LU decomposition
*> of Z has been computed by ZGETC2. On entry RHS = f holds the
*> contribution from earlier solved sub-systems, and on return RHS = x.
*>
*> The factorization of Z returned by ZGETC2 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] IJOB
*> \verbatim
*> IJOB is INTEGER
*> IJOB = 2: First compute an approximative null-vector e
*> of Z using ZGECON, 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 chosen as either +1 or
*> -1. Default.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix Z.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is COMPLEX*16 array, dimension (LDZ, N)
*> On entry, the LU part of the factorization of the n-by-n
*> matrix Z computed by ZGETC2: Z = P * L * U * Q
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDA >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] RHS
*> \verbatim
*> RHS is COMPLEX*16 array, dimension (N).
*> On entry, RHS contains contributions from other subsystems.
*> On exit, RHS contains the solution of the subsystem with
*> entries according to the value of IJOB (see above).
*> \endverbatim
*>
*> \param[in,out] RDSUM
*> \verbatim
*> RDSUM is DOUBLE PRECISION
*> On entry, the sum of squares of computed contributions to
*> the Dif-estimate under computation by ZTGSYL, 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 ZTGSY2 is called by CTGSYL.
*> \endverbatim
*>
*> \param[in,out] RDSCAL
*> \verbatim
*> RDSCAL is 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 ZTGSY2 is called by
*> ZTGSYL.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N).
*> The pivot indices; for 1 <= i <= N, row i of the
*> matrix has been interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] JPIV
*> \verbatim
*> JPIV is INTEGER array, dimension (N).
*> The pivot indices; for 1 <= j <= N, column j of the
*> matrix has been interchanged with column JPIV(j).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16OTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> This routine is a further developed implementation of algorithm
*> BSOLVE in [1] using complete pivoting in the LU factorization.
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
* ================
*>
*> [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.
*>\n
*> [2] Peter Poromaa,
*> On Efficient and Robust Estimators for the Separation
*> between two Regular Matrix Pairs with Applications in
*> Condition Estimation. Report UMINF-95.05, Department of
*> Computing Science, Umea University, S-901 87 Umea, Sweden,
*> 1995.
*
* =====================================================================
SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
$ JPIV )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IJOB, LDZ, N
DOUBLE PRECISION RDSCAL, RDSUM
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), JPIV( * )
COMPLEX*16 RHS( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXDIM
PARAMETER ( MAXDIM = 2 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
COMPLEX*16 CONE
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J, K
DOUBLE PRECISION RTEMP, SCALE, SMINU, SPLUS
COMPLEX*16 BM, BP, PMONE, TEMP
* ..
* .. Local Arrays ..
DOUBLE PRECISION RWORK( MAXDIM )
COMPLEX*16 WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
* ..
* .. External Subroutines ..
EXTERNAL ZAXPY, ZCOPY, ZGECON, ZGESC2, ZLASSQ, ZLASWP,
$ ZSCAL
* ..
* .. External Functions ..
DOUBLE PRECISION DZASUM
COMPLEX*16 ZDOTC
EXTERNAL DZASUM, ZDOTC
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, SQRT
* ..
* .. Executable Statements ..
*
IF( IJOB.NE.2 ) THEN
*
* Apply permutations IPIV to RHS
*
CALL ZLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
*
* Solve for L-part choosing RHS either to +1 or -1.
*
PMONE = -CONE
DO 10 J = 1, N - 1
BP = RHS( J ) + CONE
BM = RHS( J ) - CONE
SPLUS = ONE
*
* Lockahead for L- part RHS(1:N-1) = +-1
* SPLUS and SMIN computed more efficiently than in BSOLVE[1].
*
SPLUS = SPLUS + DBLE( ZDOTC( N-J, Z( J+1, J ), 1, Z( J+1,
$ J ), 1 ) )
SMINU = DBLE( ZDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) )
SPLUS = SPLUS*DBLE( 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 = CONE
END IF
*
* Compute the remaining r.h.s.
*
TEMP = -RHS( J )
CALL ZAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
10 CONTINUE
*
* Solve for U- part, lockahead 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 transferred to U
* and not to L. U(N, N) is an approximation to sigma_min(LU).
*
CALL ZCOPY( N-1, RHS, 1, WORK, 1 )
WORK( N ) = RHS( N ) + CONE
RHS( N ) = RHS( N ) - CONE
SPLUS = ZERO
SMINU = ZERO
DO 30 I = N, 1, -1
TEMP = CONE / Z( I, I )
WORK( I ) = WORK( I )*TEMP
RHS( I ) = RHS( I )*TEMP
DO 20 K = I + 1, N
WORK( I ) = WORK( I ) - WORK( K )*( Z( I, K )*TEMP )
RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
20 CONTINUE
SPLUS = SPLUS + ABS( WORK( I ) )
SMINU = SMINU + ABS( RHS( I ) )
30 CONTINUE
IF( SPLUS.GT.SMINU )
$ CALL ZCOPY( N, WORK, 1, RHS, 1 )
*
* Apply the permutations JPIV to the computed solution (RHS)
*
CALL ZLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
*
* Compute the sum of squares
*
CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
RETURN
END IF
*
* ENTRY IJOB = 2
*
* Compute approximate nullvector XM of Z
*
CALL ZGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO )
CALL ZCOPY( N, WORK( N+1 ), 1, XM, 1 )
*
* Compute RHS
*
CALL ZLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
TEMP = CONE / SQRT( ZDOTC( N, XM, 1, XM, 1 ) )
CALL ZSCAL( N, TEMP, XM, 1 )
CALL ZCOPY( N, XM, 1, XP, 1 )
CALL ZAXPY( N, CONE, RHS, 1, XP, 1 )
CALL ZAXPY( N, -CONE, XM, 1, RHS, 1 )
CALL ZGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE )
CALL ZGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE )
IF( DZASUM( N, XP, 1 ).GT.DZASUM( N, RHS, 1 ) )
$ CALL ZCOPY( N, XP, 1, RHS, 1 )
*
* Compute the sum of squares
*
CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
RETURN
*
* End of ZLATDF
*
END