*> \brief \b CLATDF 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 CLATDF + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, * JPIV ) * * .. Scalar Arguments .. * INTEGER IJOB, LDZ, N * REAL RDSCAL, RDSUM * .. * .. Array Arguments .. * INTEGER IPIV( * ), JPIV( * ) * COMPLEX RHS( * ), Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CLATDF 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 CGETC2. On entry RHS = f holds the *> contribution from earlier solved sub-systems, and on return RHS = x. *> *> The factorization of Z returned by CGETC2 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 CGECON, 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 array, dimension (LDZ, N) *> On entry, the LU part of the factorization of the n-by-n *> matrix Z computed by CGETC2: 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 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 REAL *> On entry, the sum of squares of computed contributions to *> the Dif-estimate under computation by CTGSYL, 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 CTGSY2 is called by CTGSYL. *> \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 CTGSY2 is called by *> CTGSYL. *> \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 complexOTHERauxiliary * *> \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. *> *> [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 CLATDF( 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( * ) COMPLEX RHS( * ), Z( LDZ, * ) * .. * * ===================================================================== * * .. Parameters .. INTEGER MAXDIM PARAMETER ( MAXDIM = 2 ) REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) COMPLEX CONE PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. INTEGER I, INFO, J, K REAL RTEMP, SCALE, SMINU, SPLUS COMPLEX BM, BP, PMONE, TEMP * .. * .. Local Arrays .. REAL RWORK( MAXDIM ) COMPLEX WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM ) * .. * .. External Subroutines .. EXTERNAL CAXPY, CCOPY, CGECON, CGESC2, CLASSQ, CLASWP, $ CSCAL * .. * .. External Functions .. REAL SCASUM COMPLEX CDOTC EXTERNAL SCASUM, CDOTC * .. * .. Intrinsic Functions .. INTRINSIC ABS, REAL, SQRT * .. * .. Executable Statements .. * IF( IJOB.NE.2 ) THEN * * Apply permutations IPIV to RHS * CALL CLASWP( 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 + REAL( CDOTC( N-J, Z( J+1, J ), 1, Z( J+1, $ J ), 1 ) ) SMINU = REAL( CDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) ) SPLUS = SPLUS*REAL( 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 CAXPY( 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 transfered to U * and not to L. U(N, N) is an approximation to sigma_min(LU). * CALL CCOPY( 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 CCOPY( N, WORK, 1, RHS, 1 ) * * Apply the permutations JPIV to the computed solution (RHS) * CALL CLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 ) * * Compute the sum of squares * CALL CLASSQ( N, RHS, 1, RDSCAL, RDSUM ) RETURN END IF * * ENTRY IJOB = 2 * * Compute approximate nullvector XM of Z * CALL CGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO ) CALL CCOPY( N, WORK( N+1 ), 1, XM, 1 ) * * Compute RHS * CALL CLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 ) TEMP = CONE / SQRT( CDOTC( N, XM, 1, XM, 1 ) ) CALL CSCAL( N, TEMP, XM, 1 ) CALL CCOPY( N, XM, 1, XP, 1 ) CALL CAXPY( N, CONE, RHS, 1, XP, 1 ) CALL CAXPY( N, -CONE, XM, 1, RHS, 1 ) CALL CGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE ) CALL CGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE ) IF( SCASUM( N, XP, 1 ).GT.SCASUM( N, RHS, 1 ) ) $ CALL CCOPY( N, XP, 1, RHS, 1 ) * * Compute the sum of squares * CALL CLASSQ( N, RHS, 1, RDSCAL, RDSUM ) RETURN * * End of CLATDF * END