*> \brief \b SLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE SLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
* JPIV )
*
* .. Scalar Arguments ..
* INTEGER IJOB, LDZ, N
* REAL RDSCAL, RDSUM
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), JPIV( * )
* REAL RHS( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLATDF uses the LU factorization of the n-by-n matrix Z computed by
*> SGETC2 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 SGETC2 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 SGECON, 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 REAL array, dimension (LDZ, N)
*> On entry, the LU part of the factorization of the n-by-n
*> matrix Z computed by SGETC2: 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 REAL 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).
*> \endverbatim
*>
*> \param[in,out] RDSUM
*> \verbatim
*> RDSUM is REAL
*> On entry, the sum of squares of computed contributions to
*> the Dif-estimate under computation by STGSYL, 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 STGSY2 is called by STGSYL.
*> \endverbatim
*>
*> \param[in,out] RDSCAL
*> \verbatim
*> RDSCAL is REAL
*> 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 STGSY2 is called by
*> STGSYL.
*> \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.
*
*> \date June 2016
*
*> \ingroup realOTHERauxiliary
*
*> \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:
* ================
*>
*> \verbatim
*>
*>
*> [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.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
$ JPIV )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
INTEGER IJOB, LDZ, N
REAL RDSCAL, RDSUM
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), JPIV( * )
REAL RHS( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXDIM
PARAMETER ( MAXDIM = 8 )
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J, K
REAL BM, BP, PMONE, SMINU, SPLUS, TEMP
* ..
* .. Local Arrays ..
INTEGER IWORK( MAXDIM )
REAL WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SCOPY, SGECON, SGESC2, SLASSQ, SLASWP,
$ SSCAL
* ..
* .. External Functions ..
REAL SASUM, SDOT
EXTERNAL SASUM, SDOT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IF( IJOB.NE.2 ) THEN
*
* Apply permutations IPIV to RHS
*
CALL SLASWP( 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 + SDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 )
SMINU = SDOT( 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 SAXPY( 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 SCOPY( 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 SCOPY( N, XP, 1, RHS, 1 )
*
* Apply the permutations JPIV to the computed solution (RHS)
*
CALL SLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
*
* Compute the sum of squares
*
CALL SLASSQ( N, RHS, 1, RDSCAL, RDSUM )
*
ELSE
*
* IJOB = 2, Compute approximate nullvector XM of Z
*
CALL SGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO )
CALL SCOPY( N, WORK( N+1 ), 1, XM, 1 )
*
* Compute RHS
*
CALL SLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
TEMP = ONE / SQRT( SDOT( N, XM, 1, XM, 1 ) )
CALL SSCAL( N, TEMP, XM, 1 )
CALL SCOPY( N, XM, 1, XP, 1 )
CALL SAXPY( N, ONE, RHS, 1, XP, 1 )
CALL SAXPY( N, -ONE, XM, 1, RHS, 1 )
CALL SGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP )
CALL SGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP )
IF( SASUM( N, XP, 1 ).GT.SASUM( N, RHS, 1 ) )
$ CALL SCOPY( N, XP, 1, RHS, 1 )
*
* Compute the sum of squares
*
CALL SLASSQ( N, RHS, 1, RDSCAL, RDSUM )
*
END IF
*
RETURN
*
* End of SLATDF
*
END