*> \brief \b ZLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZLAR1V + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD, * PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, * R, ISUPPZ, NRMINV, RESID, RQCORR, WORK ) * * .. Scalar Arguments .. * LOGICAL WANTNC * INTEGER B1, BN, N, NEGCNT, R * DOUBLE PRECISION GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID, * $ RQCORR, ZTZ * .. * .. Array Arguments .. * INTEGER ISUPPZ( * ) * DOUBLE PRECISION D( * ), L( * ), LD( * ), LLD( * ), * $ WORK( * ) * COMPLEX*16 Z( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZLAR1V computes the (scaled) r-th column of the inverse of *> the sumbmatrix in rows B1 through BN of the tridiagonal matrix *> L D L**T - sigma I. When sigma is close to an eigenvalue, the *> computed vector is an accurate eigenvector. Usually, r corresponds *> to the index where the eigenvector is largest in magnitude. *> The following steps accomplish this computation : *> (a) Stationary qd transform, L D L**T - sigma I = L(+) D(+) L(+)**T, *> (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T, *> (c) Computation of the diagonal elements of the inverse of *> L D L**T - sigma I by combining the above transforms, and choosing *> r as the index where the diagonal of the inverse is (one of the) *> largest in magnitude. *> (d) Computation of the (scaled) r-th column of the inverse using the *> twisted factorization obtained by combining the top part of the *> the stationary and the bottom part of the progressive transform. *> \endverbatim * * Arguments: * ========== * *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix L D L**T. *> \endverbatim *> *> \param[in] B1 *> \verbatim *> B1 is INTEGER *> First index of the submatrix of L D L**T. *> \endverbatim *> *> \param[in] BN *> \verbatim *> BN is INTEGER *> Last index of the submatrix of L D L**T. *> \endverbatim *> *> \param[in] LAMBDA *> \verbatim *> LAMBDA is DOUBLE PRECISION *> The shift. In order to compute an accurate eigenvector, *> LAMBDA should be a good approximation to an eigenvalue *> of L D L**T. *> \endverbatim *> *> \param[in] L *> \verbatim *> L is DOUBLE PRECISION array, dimension (N-1) *> The (n-1) subdiagonal elements of the unit bidiagonal matrix *> L, in elements 1 to N-1. *> \endverbatim *> *> \param[in] D *> \verbatim *> D is DOUBLE PRECISION array, dimension (N) *> The n diagonal elements of the diagonal matrix D. *> \endverbatim *> *> \param[in] LD *> \verbatim *> LD is DOUBLE PRECISION array, dimension (N-1) *> The n-1 elements L(i)*D(i). *> \endverbatim *> *> \param[in] LLD *> \verbatim *> LLD is DOUBLE PRECISION array, dimension (N-1) *> The n-1 elements L(i)*L(i)*D(i). *> \endverbatim *> *> \param[in] PIVMIN *> \verbatim *> PIVMIN is DOUBLE PRECISION *> The minimum pivot in the Sturm sequence. *> \endverbatim *> *> \param[in] GAPTOL *> \verbatim *> GAPTOL is DOUBLE PRECISION *> Tolerance that indicates when eigenvector entries are negligible *> w.r.t. their contribution to the residual. *> \endverbatim *> *> \param[in,out] Z *> \verbatim *> Z is COMPLEX*16 array, dimension (N) *> On input, all entries of Z must be set to 0. *> On output, Z contains the (scaled) r-th column of the *> inverse. The scaling is such that Z(R) equals 1. *> \endverbatim *> *> \param[in] WANTNC *> \verbatim *> WANTNC is LOGICAL *> Specifies whether NEGCNT has to be computed. *> \endverbatim *> *> \param[out] NEGCNT *> \verbatim *> NEGCNT is INTEGER *> If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin *> in the matrix factorization L D L**T, and NEGCNT = -1 otherwise. *> \endverbatim *> *> \param[out] ZTZ *> \verbatim *> ZTZ is DOUBLE PRECISION *> The square of the 2-norm of Z. *> \endverbatim *> *> \param[out] MINGMA *> \verbatim *> MINGMA is DOUBLE PRECISION *> The reciprocal of the largest (in magnitude) diagonal *> element of the inverse of L D L**T - sigma I. *> \endverbatim *> *> \param[in,out] R *> \verbatim *> R is INTEGER *> The twist index for the twisted factorization used to *> compute Z. *> On input, 0 <= R <= N. If R is input as 0, R is set to *> the index where (L D L**T - sigma I)^{-1} is largest *> in magnitude. If 1 <= R <= N, R is unchanged. *> On output, R contains the twist index used to compute Z. *> Ideally, R designates the position of the maximum entry in the *> eigenvector. *> \endverbatim *> *> \param[out] ISUPPZ *> \verbatim *> ISUPPZ is INTEGER array, dimension (2) *> The support of the vector in Z, i.e., the vector Z is *> nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ). *> \endverbatim *> *> \param[out] NRMINV *> \verbatim *> NRMINV is DOUBLE PRECISION *> NRMINV = 1/SQRT( ZTZ ) *> \endverbatim *> *> \param[out] RESID *> \verbatim *> RESID is DOUBLE PRECISION *> The residual of the FP vector. *> RESID = ABS( MINGMA )/SQRT( ZTZ ) *> \endverbatim *> *> \param[out] RQCORR *> \verbatim *> RQCORR is DOUBLE PRECISION *> The Rayleigh Quotient correction to LAMBDA. *> RQCORR = MINGMA*TMP *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (4*N) *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup complex16OTHERauxiliary * *> \par Contributors: * ================== *> *> Beresford Parlett, University of California, Berkeley, USA \n *> Jim Demmel, University of California, Berkeley, USA \n *> Inderjit Dhillon, University of Texas, Austin, USA \n *> Osni Marques, LBNL/NERSC, USA \n *> Christof Voemel, University of California, Berkeley, USA * * ===================================================================== SUBROUTINE ZLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD, $ PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, $ R, ISUPPZ, NRMINV, RESID, RQCORR, WORK ) * * -- 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..-- * December 2016 * * .. Scalar Arguments .. LOGICAL WANTNC INTEGER B1, BN, N, NEGCNT, R DOUBLE PRECISION GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID, $ RQCORR, ZTZ * .. * .. Array Arguments .. INTEGER ISUPPZ( * ) DOUBLE PRECISION D( * ), L( * ), LD( * ), LLD( * ), $ WORK( * ) COMPLEX*16 Z( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) COMPLEX*16 CONE PARAMETER ( CONE = ( 1.0D0, 0.0D0 ) ) * .. * .. Local Scalars .. LOGICAL SAWNAN1, SAWNAN2 INTEGER I, INDLPL, INDP, INDS, INDUMN, NEG1, NEG2, R1, $ R2 DOUBLE PRECISION DMINUS, DPLUS, EPS, S, TMP * .. * .. External Functions .. LOGICAL DISNAN DOUBLE PRECISION DLAMCH EXTERNAL DISNAN, DLAMCH * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE * .. * .. Executable Statements .. * EPS = DLAMCH( 'Precision' ) IF( R.EQ.0 ) THEN R1 = B1 R2 = BN ELSE R1 = R R2 = R END IF * Storage for LPLUS INDLPL = 0 * Storage for UMINUS INDUMN = N INDS = 2*N + 1 INDP = 3*N + 1 IF( B1.EQ.1 ) THEN WORK( INDS ) = ZERO ELSE WORK( INDS+B1-1 ) = LLD( B1-1 ) END IF * * Compute the stationary transform (using the differential form) * until the index R2. * SAWNAN1 = .FALSE. NEG1 = 0 S = WORK( INDS+B1-1 ) - LAMBDA DO 50 I = B1, R1 - 1 DPLUS = D( I ) + S WORK( INDLPL+I ) = LD( I ) / DPLUS IF(DPLUS.LT.ZERO) NEG1 = NEG1 + 1 WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I ) S = WORK( INDS+I ) - LAMBDA 50 CONTINUE SAWNAN1 = DISNAN( S ) IF( SAWNAN1 ) GOTO 60 DO 51 I = R1, R2 - 1 DPLUS = D( I ) + S WORK( INDLPL+I ) = LD( I ) / DPLUS WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I ) S = WORK( INDS+I ) - LAMBDA 51 CONTINUE SAWNAN1 = DISNAN( S ) * 60 CONTINUE IF( SAWNAN1 ) THEN * Runs a slower version of the above loop if a NaN is detected NEG1 = 0 S = WORK( INDS+B1-1 ) - LAMBDA DO 70 I = B1, R1 - 1 DPLUS = D( I ) + S IF(ABS(DPLUS).LT.PIVMIN) DPLUS = -PIVMIN WORK( INDLPL+I ) = LD( I ) / DPLUS IF(DPLUS.LT.ZERO) NEG1 = NEG1 + 1 WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I ) IF( WORK( INDLPL+I ).EQ.ZERO ) $ WORK( INDS+I ) = LLD( I ) S = WORK( INDS+I ) - LAMBDA 70 CONTINUE DO 71 I = R1, R2 - 1 DPLUS = D( I ) + S IF(ABS(DPLUS).LT.PIVMIN) DPLUS = -PIVMIN WORK( INDLPL+I ) = LD( I ) / DPLUS WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I ) IF( WORK( INDLPL+I ).EQ.ZERO ) $ WORK( INDS+I ) = LLD( I ) S = WORK( INDS+I ) - LAMBDA 71 CONTINUE END IF * * Compute the progressive transform (using the differential form) * until the index R1 * SAWNAN2 = .FALSE. NEG2 = 0 WORK( INDP+BN-1 ) = D( BN ) - LAMBDA DO 80 I = BN - 1, R1, -1 DMINUS = LLD( I ) + WORK( INDP+I ) TMP = D( I ) / DMINUS IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1 WORK( INDUMN+I ) = L( I )*TMP WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA 80 CONTINUE TMP = WORK( INDP+R1-1 ) SAWNAN2 = DISNAN( TMP ) IF( SAWNAN2 ) THEN * Runs a slower version of the above loop if a NaN is detected NEG2 = 0 DO 100 I = BN-1, R1, -1 DMINUS = LLD( I ) + WORK( INDP+I ) IF(ABS(DMINUS).LT.PIVMIN) DMINUS = -PIVMIN TMP = D( I ) / DMINUS IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1 WORK( INDUMN+I ) = L( I )*TMP WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA IF( TMP.EQ.ZERO ) $ WORK( INDP+I-1 ) = D( I ) - LAMBDA 100 CONTINUE END IF * * Find the index (from R1 to R2) of the largest (in magnitude) * diagonal element of the inverse * MINGMA = WORK( INDS+R1-1 ) + WORK( INDP+R1-1 ) IF( MINGMA.LT.ZERO ) NEG1 = NEG1 + 1 IF( WANTNC ) THEN NEGCNT = NEG1 + NEG2 ELSE NEGCNT = -1 ENDIF IF( ABS(MINGMA).EQ.ZERO ) $ MINGMA = EPS*WORK( INDS+R1-1 ) R = R1 DO 110 I = R1, R2 - 1 TMP = WORK( INDS+I ) + WORK( INDP+I ) IF( TMP.EQ.ZERO ) $ TMP = EPS*WORK( INDS+I ) IF( ABS( TMP ).LE.ABS( MINGMA ) ) THEN MINGMA = TMP R = I + 1 END IF 110 CONTINUE * * Compute the FP vector: solve N^T v = e_r * ISUPPZ( 1 ) = B1 ISUPPZ( 2 ) = BN Z( R ) = CONE ZTZ = ONE * * Compute the FP vector upwards from R * IF( .NOT.SAWNAN1 .AND. .NOT.SAWNAN2 ) THEN DO 210 I = R-1, B1, -1 Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) ) IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL ) $ THEN Z( I ) = ZERO ISUPPZ( 1 ) = I + 1 GOTO 220 ENDIF ZTZ = ZTZ + DBLE( Z( I )*Z( I ) ) 210 CONTINUE 220 CONTINUE ELSE * Run slower loop if NaN occurred. DO 230 I = R - 1, B1, -1 IF( Z( I+1 ).EQ.ZERO ) THEN Z( I ) = -( LD( I+1 ) / LD( I ) )*Z( I+2 ) ELSE Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) ) END IF IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL ) $ THEN Z( I ) = ZERO ISUPPZ( 1 ) = I + 1 GO TO 240 END IF ZTZ = ZTZ + DBLE( Z( I )*Z( I ) ) 230 CONTINUE 240 CONTINUE ENDIF * Compute the FP vector downwards from R in blocks of size BLKSIZ IF( .NOT.SAWNAN1 .AND. .NOT.SAWNAN2 ) THEN DO 250 I = R, BN-1 Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) ) IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL ) $ THEN Z( I+1 ) = ZERO ISUPPZ( 2 ) = I GO TO 260 END IF ZTZ = ZTZ + DBLE( Z( I+1 )*Z( I+1 ) ) 250 CONTINUE 260 CONTINUE ELSE * Run slower loop if NaN occurred. DO 270 I = R, BN - 1 IF( Z( I ).EQ.ZERO ) THEN Z( I+1 ) = -( LD( I-1 ) / LD( I ) )*Z( I-1 ) ELSE Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) ) END IF IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL ) $ THEN Z( I+1 ) = ZERO ISUPPZ( 2 ) = I GO TO 280 END IF ZTZ = ZTZ + DBLE( Z( I+1 )*Z( I+1 ) ) 270 CONTINUE 280 CONTINUE END IF * * Compute quantities for convergence test * TMP = ONE / ZTZ NRMINV = SQRT( TMP ) RESID = ABS( MINGMA )*NRMINV RQCORR = MINGMA*TMP * * RETURN * * End of ZLAR1V * END