numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
| Name | Size | Mode | |
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| lapack/SRC/zgsvj0.f | 36378B | -rw-r--r-- |
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*> \brief <b> ZGSVJ0 pre-processor for the routine zgesvj. </b> * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZGSVJ0 + dependencies *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgsvj0.f"> *> [TGZ]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgsvj0.f"> *> [ZIP]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgsvj0.f"> *> [TXT]</a> *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS, * SFMIN, TOL, NSWEEP, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP * DOUBLE PRECISION EPS, SFMIN, TOL * CHARACTER*1 JOBV * .. * .. Array Arguments .. * COMPLEX*16 A( LDA, * ), D( N ), V( LDV, * ), WORK( LWORK ) * DOUBLE PRECISION SVA( N ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGSVJ0 is called from ZGESVJ as a pre-processor and that is its main *> purpose. It applies Jacobi rotations in the same way as ZGESVJ does, but *> it does not check convergence (stopping criterion). Few tuning *> parameters (marked by [TP]) are available for the implementer. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBV *> \verbatim *> JOBV is CHARACTER*1 *> Specifies whether the output from this procedure is used *> to compute the matrix V: *> = 'V': the product of the Jacobi rotations is accumulated *> by postmultiplying the N-by-N array V. *> (See the description of V.) *> = 'A': the product of the Jacobi rotations is accumulated *> by postmultiplying the MV-by-N array V. *> (See the descriptions of MV and V.) *> = 'N': the Jacobi rotations are not accumulated. *> \endverbatim *> *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the input matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the input matrix A. *> M >= N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA,N) *> On entry, M-by-N matrix A, such that A*diag(D) represents *> the input matrix. *> On exit, *> A_onexit * diag(D_onexit) represents the input matrix A*diag(D) *> post-multiplied by a sequence of Jacobi rotations, where the *> rotation threshold and the total number of sweeps are given in *> TOL and NSWEEP, respectively. *> (See the descriptions of D, TOL and NSWEEP.) *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[in,out] D *> \verbatim *> D is COMPLEX*16 array, dimension (N) *> The array D accumulates the scaling factors from the complex scaled *> Jacobi rotations. *> On entry, A*diag(D) represents the input matrix. *> On exit, A_onexit*diag(D_onexit) represents the input matrix *> post-multiplied by a sequence of Jacobi rotations, where the *> rotation threshold and the total number of sweeps are given in *> TOL and NSWEEP, respectively. *> (See the descriptions of A, TOL and NSWEEP.) *> \endverbatim *> *> \param[in,out] SVA *> \verbatim *> SVA is DOUBLE PRECISION array, dimension (N) *> On entry, SVA contains the Euclidean norms of the columns of *> the matrix A*diag(D). *> On exit, SVA contains the Euclidean norms of the columns of *> the matrix A_onexit*diag(D_onexit). *> \endverbatim *> *> \param[in] MV *> \verbatim *> MV is INTEGER *> If JOBV = 'A', then MV rows of V are post-multiplied by a *> sequence of Jacobi rotations. *> If JOBV = 'N', then MV is not referenced. *> \endverbatim *> *> \param[in,out] V *> \verbatim *> V is COMPLEX*16 array, dimension (LDV,N) *> If JOBV = 'V' then N rows of V are post-multiplied by a *> sequence of Jacobi rotations. *> If JOBV = 'A' then MV rows of V are post-multiplied by a *> sequence of Jacobi rotations. *> If JOBV = 'N', then V is not referenced. *> \endverbatim *> *> \param[in] LDV *> \verbatim *> LDV is INTEGER *> The leading dimension of the array V, LDV >= 1. *> If JOBV = 'V', LDV >= N. *> If JOBV = 'A', LDV >= MV. *> \endverbatim *> *> \param[in] EPS *> \verbatim *> EPS is DOUBLE PRECISION *> EPS = DLAMCH('Epsilon') *> \endverbatim *> *> \param[in] SFMIN *> \verbatim *> SFMIN is DOUBLE PRECISION *> SFMIN = DLAMCH('Safe Minimum') *> \endverbatim *> *> \param[in] TOL *> \verbatim *> TOL is DOUBLE PRECISION *> TOL is the threshold for Jacobi rotations. For a pair *> A(:,p), A(:,q) of pivot columns, the Jacobi rotation is *> applied only if ABS(COS(angle(A(:,p),A(:,q)))) > TOL. *> \endverbatim *> *> \param[in] NSWEEP *> \verbatim *> NSWEEP is INTEGER *> NSWEEP is the number of sweeps of Jacobi rotations to be *> performed. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX*16 array, dimension (LWORK) *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> LWORK is the dimension of WORK. LWORK >= M. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit. *> < 0: if INFO = -i, then the i-th argument had an illegal value *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup gsvj0 *> *> \par Further Details: * ===================== *> *> ZGSVJ0 is used just to enable ZGESVJ to call a simplified version of *> itself to work on a submatrix of the original matrix. *> *> Contributor: * ============= *> *> Zlatko Drmac (Zagreb, Croatia) *> *> \par Bugs, Examples and Comments: * ============================ *> *> Please report all bugs and send interesting test examples and comments to *> drmac@math.hr. Thank you. * * ===================================================================== SUBROUTINE ZGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS, $ SFMIN, TOL, NSWEEP, WORK, LWORK, INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * IMPLICIT NONE * .. Scalar Arguments .. INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP DOUBLE PRECISION EPS, SFMIN, TOL CHARACTER*1 JOBV * .. * .. Array Arguments .. COMPLEX*16 A( LDA, * ), D( N ), V( LDV, * ), WORK( LWORK ) DOUBLE PRECISION SVA( N ) * .. * * ===================================================================== * * .. Local Parameters .. DOUBLE PRECISION ZERO, HALF, ONE PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0) COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = (0.0D0, 0.0D0), CONE = (1.0D0, 0.0D0) ) * .. * .. Local Scalars .. COMPLEX*16 AAPQ, OMPQ DOUBLE PRECISION AAPP, AAPP0, AAPQ1, AAQQ, APOAQ, AQOAP, BIG, $ BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS, $ ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA, $ THSIGN INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1, $ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL, $ NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND LOGICAL APPLV, ROTOK, RSVEC * .. * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, CONJG, DBLE, MIN, SIGN, SQRT * .. * .. External Functions .. DOUBLE PRECISION DZNRM2 COMPLEX*16 ZDOTC INTEGER IDAMAX LOGICAL LSAME EXTERNAL IDAMAX, LSAME, ZDOTC, DZNRM2 * .. * .. * .. External Subroutines .. * .. * from BLAS EXTERNAL ZCOPY, ZROT, ZSWAP, ZAXPY * from LAPACK EXTERNAL ZLASCL, ZLASSQ, XERBLA * .. * .. Executable Statements .. * * Test the input parameters. * APPLV = LSAME( JOBV, 'A' ) RSVEC = LSAME( JOBV, 'V' ) IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN INFO = -3 ELSE IF( LDA.LT.M ) THEN INFO = -5 ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN INFO = -8 ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR. $ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN INFO = -10 ELSE IF( TOL.LE.EPS ) THEN INFO = -13 ELSE IF( NSWEEP.LT.0 ) THEN INFO = -14 ELSE IF( LWORK.LT.M ) THEN INFO = -16 ELSE INFO = 0 END IF * * #:( IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGSVJ0', -INFO ) RETURN END IF * IF( RSVEC ) THEN MVL = N ELSE IF( APPLV ) THEN MVL = MV END IF RSVEC = RSVEC .OR. APPLV ROOTEPS = SQRT( EPS ) ROOTSFMIN = SQRT( SFMIN ) SMALL = SFMIN / EPS BIG = ONE / SFMIN ROOTBIG = ONE / ROOTSFMIN BIGTHETA = ONE / ROOTEPS ROOTTOL = SQRT( TOL ) * * .. Row-cyclic Jacobi SVD algorithm with column pivoting .. * EMPTSW = ( N*( N-1 ) ) / 2 NOTROT = 0 * * .. Row-cyclic pivot strategy with de Rijk's pivoting .. * SWBAND = 0 *[TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective * if ZGESVJ is used as a computational routine in the preconditioned * Jacobi SVD algorithm ZGEJSV. For sweeps i=1:SWBAND the procedure * works on pivots inside a band-like region around the diagonal. * The boundaries are determined dynamically, based on the number of * pivots above a threshold. * KBL = MIN( 8, N ) *[TP] KBL is a tuning parameter that defines the tile size in the * tiling of the p-q loops of pivot pairs. In general, an optimal * value of KBL depends on the matrix dimensions and on the * parameters of the computer's memory. * NBL = N / KBL IF( ( NBL*KBL ).NE.N )NBL = NBL + 1 * BLSKIP = KBL**2 *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL. * ROWSKIP = MIN( 5, KBL ) *[TP] ROWSKIP is a tuning parameter. * LKAHEAD = 1 *[TP] LKAHEAD is a tuning parameter. * * Quasi block transformations, using the lower (upper) triangular * structure of the input matrix. The quasi-block-cycling usually * invokes cubic convergence. Big part of this cycle is done inside * canonical subspaces of dimensions less than M. * * * .. Row-cyclic pivot strategy with de Rijk's pivoting .. * DO 1993 i = 1, NSWEEP * * .. go go go ... * MXAAPQ = ZERO MXSINJ = ZERO ISWROT = 0 * NOTROT = 0 PSKIPPED = 0 * * Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs * 1 <= p < q <= N. This is the first step toward a blocked implementation * of the rotations. New implementation, based on block transformations, * is under development. * DO 2000 ibr = 1, NBL * igl = ( ibr-1 )*KBL + 1 * DO 1002 ir1 = 0, MIN( LKAHEAD, NBL-ibr ) * igl = igl + ir1*KBL * DO 2001 p = igl, MIN( igl+KBL-1, N-1 ) * * .. de Rijk's pivoting * q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1 IF( p.NE.q ) THEN CALL ZSWAP( M, A( 1, p ), 1, A( 1, q ), 1 ) IF( RSVEC )CALL ZSWAP( MVL, V( 1, p ), 1, $ V( 1, q ), 1 ) TEMP1 = SVA( p ) SVA( p ) = SVA( q ) SVA( q ) = TEMP1 AAPQ = D(p) D(p) = D(q) D(q) = AAPQ END IF * IF( ir1.EQ.0 ) THEN * * Column norms are periodically updated by explicit * norm computation. * Caveat: * Unfortunately, some BLAS implementations compute SNCRM2(M,A(1,p),1) * as SQRT(S=ZDOTC(M,A(1,p),1,A(1,p),1)), which may cause the result to * overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and to * underflow for ||A(:,p)||_2 < SQRT(underflow_threshold). * Hence, DZNRM2 cannot be trusted, not even in the case when * the true norm is far from the under(over)flow boundaries. * If properly implemented DZNRM2 is available, the IF-THEN-ELSE-END IF * below should be replaced with "AAPP = DZNRM2( M, A(1,p), 1 )". * IF( ( SVA( p ).LT.ROOTBIG ) .AND. $ ( SVA( p ).GT.ROOTSFMIN ) ) THEN SVA( p ) = DZNRM2( M, A( 1, p ), 1 ) ELSE TEMP1 = ZERO AAPP = ONE CALL ZLASSQ( M, A( 1, p ), 1, TEMP1, AAPP ) SVA( p ) = TEMP1*SQRT( AAPP ) END IF AAPP = SVA( p ) ELSE AAPP = SVA( p ) END IF * IF( AAPP.GT.ZERO ) THEN * PSKIPPED = 0 * DO 2002 q = p + 1, MIN( igl+KBL-1, N ) * AAQQ = SVA( q ) * IF( AAQQ.GT.ZERO ) THEN * AAPP0 = AAPP IF( AAQQ.GE.ONE ) THEN ROTOK = ( SMALL*AAPP ).LE.AAQQ IF( AAPP.LT.( BIG / AAQQ ) ) THEN AAPQ = ( ZDOTC( M, A( 1, p ), 1, $ A( 1, q ), 1 ) / AAQQ ) / AAPP ELSE CALL ZCOPY( M, A( 1, p ), 1, $ WORK, 1 ) CALL ZLASCL( 'G', 0, 0, AAPP, ONE, $ M, 1, WORK, LDA, IERR ) AAPQ = ZDOTC( M, WORK, 1, $ A( 1, q ), 1 ) / AAQQ END IF ELSE ROTOK = AAPP.LE.( AAQQ / SMALL ) IF( AAPP.GT.( SMALL / AAQQ ) ) THEN AAPQ = ( ZDOTC( M, A( 1, p ), 1, $ A( 1, q ), 1 ) / AAPP ) / AAQQ ELSE CALL ZCOPY( M, A( 1, q ), 1, $ WORK, 1 ) CALL ZLASCL( 'G', 0, 0, AAQQ, $ ONE, M, 1, $ WORK, LDA, IERR ) AAPQ = ZDOTC( M, A( 1, p ), 1, $ WORK, 1 ) / AAPP END IF END IF * * AAPQ = AAPQ * CONJG( CWORK(p) ) * CWORK(q) AAPQ1 = -ABS(AAPQ) MXAAPQ = MAX( MXAAPQ, -AAPQ1 ) * * TO rotate or NOT to rotate, THAT is the question ... * IF( ABS( AAPQ1 ).GT.TOL ) THEN OMPQ = AAPQ / ABS(AAPQ) * * .. rotate *[RTD] ROTATED = ROTATED + ONE * IF( ir1.EQ.0 ) THEN NOTROT = 0 PSKIPPED = 0 ISWROT = ISWROT + 1 END IF * IF( ROTOK ) THEN * AQOAP = AAQQ / AAPP APOAQ = AAPP / AAQQ THETA = -HALF*ABS( AQOAP-APOAQ )/AAPQ1 * IF( ABS( THETA ).GT.BIGTHETA ) THEN * T = HALF / THETA CS = ONE CALL ZROT( M, A(1,p), 1, A(1,q), $ 1, $ CS, CONJG(OMPQ)*T ) IF ( RSVEC ) THEN CALL ZROT( MVL, V(1,p), 1, $ V(1,q), 1, CS, CONJG(OMPQ)*T ) END IF SVA( q ) = AAQQ*SQRT( MAX( ZERO, $ ONE+T*APOAQ*AAPQ1 ) ) AAPP = AAPP*SQRT( MAX( ZERO, $ ONE-T*AQOAP*AAPQ1 ) ) MXSINJ = MAX( MXSINJ, ABS( T ) ) * ELSE * * .. choose correct signum for THETA and rotate * THSIGN = -SIGN( ONE, AAPQ1 ) T = ONE / ( THETA+THSIGN* $ SQRT( ONE+THETA*THETA ) ) CS = SQRT( ONE / ( ONE+T*T ) ) SN = T*CS * MXSINJ = MAX( MXSINJ, ABS( SN ) ) SVA( q ) = AAQQ*SQRT( MAX( ZERO, $ ONE+T*APOAQ*AAPQ1 ) ) AAPP = AAPP*SQRT( MAX( ZERO, $ ONE-T*AQOAP*AAPQ1 ) ) * CALL ZROT( M, A(1,p), 1, A(1,q), $ 1, $ CS, CONJG(OMPQ)*SN ) IF ( RSVEC ) THEN CALL ZROT( MVL, V(1,p), 1, $ V(1,q), 1, CS, CONJG(OMPQ)*SN ) END IF END IF D(p) = -D(q) * OMPQ * ELSE * .. have to use modified Gram-Schmidt like transformation CALL ZCOPY( M, A( 1, p ), 1, $ WORK, 1 ) CALL ZLASCL( 'G', 0, 0, AAPP, ONE, $ M, $ 1, WORK, LDA, $ IERR ) CALL ZLASCL( 'G', 0, 0, AAQQ, ONE, $ M, $ 1, A( 1, q ), LDA, IERR ) CALL ZAXPY( M, -AAPQ, WORK, 1, $ A( 1, q ), 1 ) CALL ZLASCL( 'G', 0, 0, ONE, AAQQ, $ M, $ 1, A( 1, q ), LDA, IERR ) SVA( q ) = AAQQ*SQRT( MAX( ZERO, $ ONE-AAPQ1*AAPQ1 ) ) MXSINJ = MAX( MXSINJ, SFMIN ) END IF * END IF ROTOK THEN ... ELSE * * In the case of cancellation in updating SVA(q), SVA(p) * recompute SVA(q), SVA(p). * IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS ) $ THEN IF( ( AAQQ.LT.ROOTBIG ) .AND. $ ( AAQQ.GT.ROOTSFMIN ) ) THEN SVA( q ) = DZNRM2( M, A( 1, q ), $ 1 ) ELSE T = ZERO AAQQ = ONE CALL ZLASSQ( M, A( 1, q ), 1, T, $ AAQQ ) SVA( q ) = T*SQRT( AAQQ ) END IF END IF IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN IF( ( AAPP.LT.ROOTBIG ) .AND. $ ( AAPP.GT.ROOTSFMIN ) ) THEN AAPP = DZNRM2( M, A( 1, p ), 1 ) ELSE T = ZERO AAPP = ONE CALL ZLASSQ( M, A( 1, p ), 1, T, $ AAPP ) AAPP = T*SQRT( AAPP ) END IF SVA( p ) = AAPP END IF * ELSE * A(:,p) and A(:,q) already numerically orthogonal IF( ir1.EQ.0 )NOTROT = NOTROT + 1 *[RTD] SKIPPED = SKIPPED + 1 PSKIPPED = PSKIPPED + 1 END IF ELSE * A(:,q) is zero column IF( ir1.EQ.0 )NOTROT = NOTROT + 1 PSKIPPED = PSKIPPED + 1 END IF * IF( ( i.LE.SWBAND ) .AND. $ ( PSKIPPED.GT.ROWSKIP ) ) THEN IF( ir1.EQ.0 )AAPP = -AAPP NOTROT = 0 GO TO 2103 END IF * 2002 CONTINUE * END q-LOOP * 2103 CONTINUE * bailed out of q-loop * SVA( p ) = AAPP * ELSE SVA( p ) = AAPP IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) ) $ NOTROT = NOTROT + MIN( igl+KBL-1, N ) - p END IF * 2001 CONTINUE * end of the p-loop * end of doing the block ( ibr, ibr ) 1002 CONTINUE * end of ir1-loop * * ... go to the off diagonal blocks * igl = ( ibr-1 )*KBL + 1 * DO 2010 jbc = ibr + 1, NBL * jgl = ( jbc-1 )*KBL + 1 * * doing the block at ( ibr, jbc ) * IJBLSK = 0 DO 2100 p = igl, MIN( igl+KBL-1, N ) * AAPP = SVA( p ) IF( AAPP.GT.ZERO ) THEN * PSKIPPED = 0 * DO 2200 q = jgl, MIN( jgl+KBL-1, N ) * AAQQ = SVA( q ) IF( AAQQ.GT.ZERO ) THEN AAPP0 = AAPP * * .. M x 2 Jacobi SVD .. * * Safe Gram matrix computation * IF( AAQQ.GE.ONE ) THEN IF( AAPP.GE.AAQQ ) THEN ROTOK = ( SMALL*AAPP ).LE.AAQQ ELSE ROTOK = ( SMALL*AAQQ ).LE.AAPP END IF IF( AAPP.LT.( BIG / AAQQ ) ) THEN AAPQ = ( ZDOTC( M, A( 1, p ), 1, $ A( 1, q ), 1 ) / AAQQ ) / AAPP ELSE CALL ZCOPY( M, A( 1, p ), 1, $ WORK, 1 ) CALL ZLASCL( 'G', 0, 0, AAPP, $ ONE, M, 1, $ WORK, LDA, IERR ) AAPQ = ZDOTC( M, WORK, 1, $ A( 1, q ), 1 ) / AAQQ END IF ELSE IF( AAPP.GE.AAQQ ) THEN ROTOK = AAPP.LE.( AAQQ / SMALL ) ELSE ROTOK = AAQQ.LE.( AAPP / SMALL ) END IF IF( AAPP.GT.( SMALL / AAQQ ) ) THEN AAPQ = ( ZDOTC( M, A( 1, p ), 1, $ A( 1, q ), 1 ) / MAX(AAQQ,AAPP) ) $ / MIN(AAQQ,AAPP) ELSE CALL ZCOPY( M, A( 1, q ), 1, $ WORK, 1 ) CALL ZLASCL( 'G', 0, 0, AAQQ, $ ONE, M, 1, $ WORK, LDA, IERR ) AAPQ = ZDOTC( M, A( 1, p ), 1, $ WORK, 1 ) / AAPP END IF END IF * * AAPQ = AAPQ * CONJG(CWORK(p))*CWORK(q) AAPQ1 = -ABS(AAPQ) MXAAPQ = MAX( MXAAPQ, -AAPQ1 ) * * TO rotate or NOT to rotate, THAT is the question ... * IF( ABS( AAPQ1 ).GT.TOL ) THEN OMPQ = AAPQ / ABS(AAPQ) NOTROT = 0 *[RTD] ROTATED = ROTATED + 1 PSKIPPED = 0 ISWROT = ISWROT + 1 * IF( ROTOK ) THEN * AQOAP = AAQQ / AAPP APOAQ = AAPP / AAQQ THETA = -HALF*ABS( AQOAP-APOAQ )/ AAPQ1 IF( AAQQ.GT.AAPP0 )THETA = -THETA * IF( ABS( THETA ).GT.BIGTHETA ) THEN T = HALF / THETA CS = ONE CALL ZROT( M, A(1,p), 1, A(1,q), $ 1, $ CS, CONJG(OMPQ)*T ) IF( RSVEC ) THEN CALL ZROT( MVL, V(1,p), 1, $ V(1,q), 1, CS, CONJG(OMPQ)*T ) END IF SVA( q ) = AAQQ*SQRT( MAX( ZERO, $ ONE+T*APOAQ*AAPQ1 ) ) AAPP = AAPP*SQRT( MAX( ZERO, $ ONE-T*AQOAP*AAPQ1 ) ) MXSINJ = MAX( MXSINJ, ABS( T ) ) ELSE * * .. choose correct signum for THETA and rotate * THSIGN = -SIGN( ONE, AAPQ1 ) IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN T = ONE / ( THETA+THSIGN* $ SQRT( ONE+THETA*THETA ) ) CS = SQRT( ONE / ( ONE+T*T ) ) SN = T*CS MXSINJ = MAX( MXSINJ, ABS( SN ) ) SVA( q ) = AAQQ*SQRT( MAX( ZERO, $ ONE+T*APOAQ*AAPQ1 ) ) AAPP = AAPP*SQRT( MAX( ZERO, $ ONE-T*AQOAP*AAPQ1 ) ) * CALL ZROT( M, A(1,p), 1, A(1,q), $ 1, $ CS, CONJG(OMPQ)*SN ) IF( RSVEC ) THEN CALL ZROT( MVL, V(1,p), 1, $ V(1,q), 1, CS, CONJG(OMPQ)*SN ) END IF END IF D(p) = -D(q) * OMPQ * ELSE * .. have to use modified Gram-Schmidt like transformation IF( AAPP.GT.AAQQ ) THEN CALL ZCOPY( M, A( 1, p ), 1, $ WORK, 1 ) CALL ZLASCL( 'G', 0, 0, AAPP, $ ONE, $ M, 1, WORK,LDA, $ IERR ) CALL ZLASCL( 'G', 0, 0, AAQQ, $ ONE, $ M, 1, A( 1, q ), LDA, $ IERR ) CALL ZAXPY( M, -AAPQ, WORK, $ 1, A( 1, q ), 1 ) CALL ZLASCL( 'G', 0, 0, ONE, $ AAQQ, $ M, 1, A( 1, q ), LDA, $ IERR ) SVA( q ) = AAQQ*SQRT( MAX( ZERO, $ ONE-AAPQ1*AAPQ1 ) ) MXSINJ = MAX( MXSINJ, SFMIN ) ELSE CALL ZCOPY( M, A( 1, q ), 1, $ WORK, 1 ) CALL ZLASCL( 'G', 0, 0, AAQQ, $ ONE, $ M, 1, WORK,LDA, $ IERR ) CALL ZLASCL( 'G', 0, 0, AAPP, $ ONE, $ M, 1, A( 1, p ), LDA, $ IERR ) CALL ZAXPY( M, -CONJG(AAPQ), $ WORK, 1, A( 1, p ), 1 ) CALL ZLASCL( 'G', 0, 0, ONE, $ AAPP, $ M, 1, A( 1, p ), LDA, $ IERR ) SVA( p ) = AAPP*SQRT( MAX( ZERO, $ ONE-AAPQ1*AAPQ1 ) ) MXSINJ = MAX( MXSINJ, SFMIN ) END IF END IF * END IF ROTOK THEN ... ELSE * * In the case of cancellation in updating SVA(q), SVA(p) * .. recompute SVA(q), SVA(p) IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS ) $ THEN IF( ( AAQQ.LT.ROOTBIG ) .AND. $ ( AAQQ.GT.ROOTSFMIN ) ) THEN SVA( q ) = DZNRM2( M, A( 1, q ), $ 1) ELSE T = ZERO AAQQ = ONE CALL ZLASSQ( M, A( 1, q ), 1, T, $ AAQQ ) SVA( q ) = T*SQRT( AAQQ ) END IF END IF IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN IF( ( AAPP.LT.ROOTBIG ) .AND. $ ( AAPP.GT.ROOTSFMIN ) ) THEN AAPP = DZNRM2( M, A( 1, p ), 1 ) ELSE T = ZERO AAPP = ONE CALL ZLASSQ( M, A( 1, p ), 1, T, $ AAPP ) AAPP = T*SQRT( AAPP ) END IF SVA( p ) = AAPP END IF * end of OK rotation ELSE NOTROT = NOTROT + 1 *[RTD] SKIPPED = SKIPPED + 1 PSKIPPED = PSKIPPED + 1 IJBLSK = IJBLSK + 1 END IF ELSE NOTROT = NOTROT + 1 PSKIPPED = PSKIPPED + 1 IJBLSK = IJBLSK + 1 END IF * IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) ) $ THEN SVA( p ) = AAPP NOTROT = 0 GO TO 2011 END IF IF( ( i.LE.SWBAND ) .AND. $ ( PSKIPPED.GT.ROWSKIP ) ) THEN AAPP = -AAPP NOTROT = 0 GO TO 2203 END IF * 2200 CONTINUE * end of the q-loop 2203 CONTINUE * SVA( p ) = AAPP * ELSE * IF( AAPP.EQ.ZERO )NOTROT = NOTROT + $ MIN( jgl+KBL-1, N ) - jgl + 1 IF( AAPP.LT.ZERO )NOTROT = 0 * END IF * 2100 CONTINUE * end of the p-loop 2010 CONTINUE * end of the jbc-loop 2011 CONTINUE *2011 bailed out of the jbc-loop DO 2012 p = igl, MIN( igl+KBL-1, N ) SVA( p ) = ABS( SVA( p ) ) 2012 CONTINUE *** 2000 CONTINUE *2000 :: end of the ibr-loop * * .. update SVA(N) IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) ) $ THEN SVA( N ) = DZNRM2( M, A( 1, N ), 1 ) ELSE T = ZERO AAPP = ONE CALL ZLASSQ( M, A( 1, N ), 1, T, AAPP ) SVA( N ) = T*SQRT( AAPP ) END IF * * Additional steering devices * IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR. $ ( ISWROT.LE.N ) ) )SWBAND = i * IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.SQRT( DBLE( N ) )* $ TOL ) .AND. ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN GO TO 1994 END IF * IF( NOTROT.GE.EMPTSW )GO TO 1994 * 1993 CONTINUE * end i=1:NSWEEP loop * * #:( Reaching this point means that the procedure has not converged. INFO = NSWEEP - 1 GO TO 1995 * 1994 CONTINUE * #:) Reaching this point means numerical convergence after the i-th * sweep. * INFO = 0 * #:) INFO = 0 confirms successful iterations. 1995 CONTINUE * * Sort the vector SVA() of column norms. DO 5991 p = 1, N - 1 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1 IF( p.NE.q ) THEN TEMP1 = SVA( p ) SVA( p ) = SVA( q ) SVA( q ) = TEMP1 AAPQ = D( p ) D( p ) = D( q ) D( q ) = AAPQ CALL ZSWAP( M, A( 1, p ), 1, A( 1, q ), 1 ) IF( RSVEC )CALL ZSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 ) END IF 5991 CONTINUE * RETURN * .. * .. END OF ZGSVJ0 * .. END