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diff --git a/SRC/zlatdf.f b/SRC/zlatdf.f new file mode 100644 index 00000000..d637b8f1 --- /dev/null +++ b/SRC/zlatdf.f @@ -0,0 +1,241 @@ + SUBROUTINE ZLATDF( 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( * ) + COMPLEX*16 RHS( * ), Z( LDZ, * ) +* .. +* +* Purpose +* ======= +* +* 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. +* +* Arguments +* ========= +* +* IJOB (input) 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 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 ZGETC2: 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 according 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 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. +* +* 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 ZTGSY2 is called by +* ZTGSYL. +* +* 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 UMINF-95.05, Department of +* Computing Science, Umea University, S-901 87 Umea, Sweden, +* 1995. +* +* ===================================================================== +* +* .. 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 transfered 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 |