SUBROUTINE SLARRV( N, VL, VU, D, L, PIVMIN, $ ISPLIT, M, DOL, DOU, MINRGP, $ RTOL1, RTOL2, W, WERR, WGAP, $ IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, $ WORK, IWORK, INFO ) * * -- LAPACK auxiliary routine (version 3.1.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER DOL, DOU, INFO, LDZ, M, N REAL MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU * .. * .. Array Arguments .. INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ), $ ISUPPZ( * ), IWORK( * ) REAL D( * ), GERS( * ), L( * ), W( * ), WERR( * ), $ WGAP( * ), WORK( * ) REAL Z( LDZ, * ) * .. * * Purpose * ======= * * SLARRV computes the eigenvectors of the tridiagonal matrix * T = L D L^T given L, D and APPROXIMATIONS to the eigenvalues of L D L^T. * The input eigenvalues should have been computed by SLARRE. * * Arguments * ========= * * N (input) INTEGER * The order of the matrix. N >= 0. * * VL (input) REAL * VU (input) REAL * Lower and upper bounds of the interval that contains the desired * eigenvalues. VL < VU. Needed to compute gaps on the left or right * end of the extremal eigenvalues in the desired RANGE. * * D (input/output) REAL array, dimension (N) * On entry, the N diagonal elements of the diagonal matrix D. * On exit, D may be overwritten. * * L (input/output) REAL array, dimension (N) * On entry, the (N-1) subdiagonal elements of the unit * bidiagonal matrix L are in elements 1 to N-1 of L * (if the matrix is not splitted.) At the end of each block * is stored the corresponding shift as given by SLARRE. * On exit, L is overwritten. * * PIVMIN (in) DOUBLE PRECISION * The minimum pivot allowed in the Sturm sequence. * * ISPLIT (input) INTEGER array, dimension (N) * The splitting points, at which T breaks up into blocks. * The first block consists of rows/columns 1 to * ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 * through ISPLIT( 2 ), etc. * * M (input) INTEGER * The total number of input eigenvalues. 0 <= M <= N. * * DOL (input) INTEGER * DOU (input) INTEGER * If the user wants to compute only selected eigenvectors from all * the eigenvalues supplied, he can specify an index range DOL:DOU. * Or else the setting DOL=1, DOU=M should be applied. * Note that DOL and DOU refer to the order in which the eigenvalues * are stored in W. * If the user wants to compute only selected eigenpairs, then * the columns DOL-1 to DOU+1 of the eigenvector space Z contain the * computed eigenvectors. All other columns of Z are set to zero. * * MINRGP (input) REAL * * RTOL1 (input) REAL * RTOL2 (input) REAL * Parameters for bisection. * An interval [LEFT,RIGHT] has converged if * RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) * * W (input/output) REAL array, dimension (N) * The first M elements of W contain the APPROXIMATE eigenvalues for * which eigenvectors are to be computed. The eigenvalues * should be grouped by split-off block and ordered from * smallest to largest within the block ( The output array * W from SLARRE is expected here ). Furthermore, they are with * respect to the shift of the corresponding root representation * for their block. On exit, W holds the eigenvalues of the * UNshifted matrix. * * WERR (input/output) REAL array, dimension (N) * The first M elements contain the semiwidth of the uncertainty * interval of the corresponding eigenvalue in W * * WGAP (input/output) REAL array, dimension (N) * The separation from the right neighbor eigenvalue in W. * * IBLOCK (input) INTEGER array, dimension (N) * The indices of the blocks (submatrices) associated with the * corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue * W(i) belongs to the first block from the top, =2 if W(i) * belongs to the second block, etc. * * INDEXW (input) INTEGER array, dimension (N) * The indices of the eigenvalues within each block (submatrix); * for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the * i-th eigenvalue W(i) is the 10-th eigenvalue in the second block. * * GERS (input) REAL array, dimension (2*N) * The N Gerschgorin intervals (the i-th Gerschgorin interval * is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should * be computed from the original UNshifted matrix. * * Z (output) REAL array, dimension (LDZ, max(1,M) ) * If INFO = 0, the first M columns of Z contain the * orthonormal eigenvectors of the matrix T * corresponding to the input eigenvalues, with the i-th * column of Z holding the eigenvector associated with W(i). * Note: the user must ensure that at least max(1,M) columns are * supplied in the array Z. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', LDZ >= max(1,N). * * ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) ) * The support of the eigenvectors in Z, i.e., the indices * indicating the nonzero elements in Z. The I-th eigenvector * is nonzero only in elements ISUPPZ( 2*I-1 ) through * ISUPPZ( 2*I ). * * WORK (workspace) REAL array, dimension (12*N) * * IWORK (workspace) INTEGER array, dimension (7*N) * * INFO (output) INTEGER * = 0: successful exit * * > 0: A problem occured in SLARRV. * < 0: One of the called subroutines signaled an internal problem. * Needs inspection of the corresponding parameter IINFO * for further information. * * =-1: Problem in SLARRB when refining a child's eigenvalues. * =-2: Problem in SLARRF when computing the RRR of a child. * When a child is inside a tight cluster, it can be difficult * to find an RRR. A partial remedy from the user's point of * view is to make the parameter MINRGP smaller and recompile. * However, as the orthogonality of the computed vectors is * proportional to 1/MINRGP, the user should be aware that * he might be trading in precision when he decreases MINRGP. * =-3: Problem in SLARRB when refining a single eigenvalue * after the Rayleigh correction was rejected. * = 5: The Rayleigh Quotient Iteration failed to converge to * full accuracy in MAXITR steps. * * Further Details * =============== * * Based on contributions by * Beresford Parlett, University of California, Berkeley, USA * Jim Demmel, University of California, Berkeley, USA * Inderjit Dhillon, University of Texas, Austin, USA * Osni Marques, LBNL/NERSC, USA * Christof Voemel, University of California, Berkeley, USA * * ===================================================================== * * .. Parameters .. INTEGER MAXITR PARAMETER ( MAXITR = 10 ) REAL ZERO, ONE, TWO, THREE, FOUR, HALF PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, $ TWO = 2.0E0, THREE = 3.0E0, $ FOUR = 4.0E0, HALF = 0.5E0) * .. * .. Local Scalars .. LOGICAL ESKIP, NEEDBS, STP2II, TRYRQC, USEDBS, USEDRQ INTEGER DONE, I, IBEGIN, IDONE, IEND, II, IINDC1, $ IINDC2, IINDR, IINDWK, IINFO, IM, IN, INDEIG, $ INDLD, INDLLD, INDWRK, ISUPMN, ISUPMX, ITER, $ ITMP1, J, JBLK, K, MINIWSIZE, MINWSIZE, NCLUS, $ NDEPTH, NEGCNT, NEWCLS, NEWFST, NEWFTT, NEWLST, $ NEWSIZ, OFFSET, OLDCLS, OLDFST, OLDIEN, OLDLST, $ OLDNCL, P, PARITY, Q, WBEGIN, WEND, WINDEX, $ WINDMN, WINDPL, ZFROM, ZTO, ZUSEDL, ZUSEDU, $ ZUSEDW REAL BSTRES, BSTW, EPS, FUDGE, GAP, GAPTOL, GL, GU, $ LAMBDA, LEFT, LGAP, MINGMA, NRMINV, RESID, $ RGAP, RIGHT, RQCORR, RQTOL, SAVGAP, SGNDEF, $ SIGMA, SPDIAM, SSIGMA, TAU, TMP, TOL, ZTZ * .. * .. External Functions .. REAL SLAMCH EXTERNAL SLAMCH * .. * .. External Subroutines .. EXTERNAL SCOPY, SLAR1V, SLARRB, SLARRF, SLASET, $ SSCAL * .. * .. Intrinsic Functions .. INTRINSIC ABS, REAL, MAX, MIN * .. * .. Executable Statements .. * .. * The first N entries of WORK are reserved for the eigenvalues INDLD = N+1 INDLLD= 2*N+1 INDWRK= 3*N+1 MINWSIZE = 12 * N DO 5 I= 1,MINWSIZE WORK( I ) = ZERO 5 CONTINUE * IWORK(IINDR+1:IINDR+N) hold the twist indices R for the * factorization used to compute the FP vector IINDR = 0 * IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current * layer and the one above. IINDC1 = N IINDC2 = 2*N IINDWK = 3*N + 1 MINIWSIZE = 7 * N DO 10 I= 1,MINIWSIZE IWORK( I ) = 0 10 CONTINUE ZUSEDL = 1 IF(DOL.GT.1) THEN * Set lower bound for use of Z ZUSEDL = DOL-1 ENDIF ZUSEDU = M IF(DOU.LT.M) THEN * Set lower bound for use of Z ZUSEDU = DOU+1 ENDIF * The width of the part of Z that is used ZUSEDW = ZUSEDU - ZUSEDL + 1 CALL SLASET( 'Full', N, ZUSEDW, ZERO, ZERO, $ Z(1,ZUSEDL), LDZ ) EPS = SLAMCH( 'Precision' ) RQTOL = TWO * EPS * * Set expert flags for standard code. TRYRQC = .TRUE. IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN ELSE * Only selected eigenpairs are computed. Since the other evalues * are not refined by RQ iteration, bisection has to compute to full * accuracy. RTOL1 = FOUR * EPS RTOL2 = FOUR * EPS ENDIF * The entries WBEGIN:WEND in W, WERR, WGAP correspond to the * desired eigenvalues. The support of the nonzero eigenvector * entries is contained in the interval IBEGIN:IEND. * Remark that if k eigenpairs are desired, then the eigenvectors * are stored in k contiguous columns of Z. * DONE is the number of eigenvectors already computed DONE = 0 IBEGIN = 1 WBEGIN = 1 DO 170 JBLK = 1, IBLOCK( M ) IEND = ISPLIT( JBLK ) SIGMA = L( IEND ) * Find the eigenvectors of the submatrix indexed IBEGIN * through IEND. WEND = WBEGIN - 1 15 CONTINUE IF( WEND.LT.M ) THEN IF( IBLOCK( WEND+1 ).EQ.JBLK ) THEN WEND = WEND + 1 GO TO 15 END IF END IF IF( WEND.LT.WBEGIN ) THEN IBEGIN = IEND + 1 GO TO 170 ELSEIF( (WEND.LT.DOL).OR.(WBEGIN.GT.DOU) ) THEN IBEGIN = IEND + 1 WBEGIN = WEND + 1 GO TO 170 END IF * Find local spectral diameter of the block GL = GERS( 2*IBEGIN-1 ) GU = GERS( 2*IBEGIN ) DO 20 I = IBEGIN+1 , IEND GL = MIN( GERS( 2*I-1 ), GL ) GU = MAX( GERS( 2*I ), GU ) 20 CONTINUE SPDIAM = GU - GL * OLDIEN is the last index of the previous block OLDIEN = IBEGIN - 1 * Calculate the size of the current block IN = IEND - IBEGIN + 1 * The number of eigenvalues in the current block IM = WEND - WBEGIN + 1 * This is for a 1x1 block IF( IBEGIN.EQ.IEND ) THEN DONE = DONE+1 Z( IBEGIN, WBEGIN ) = ONE ISUPPZ( 2*WBEGIN-1 ) = IBEGIN ISUPPZ( 2*WBEGIN ) = IBEGIN W( WBEGIN ) = W( WBEGIN ) + SIGMA WORK( WBEGIN ) = W( WBEGIN ) IBEGIN = IEND + 1 WBEGIN = WBEGIN + 1 GO TO 170 END IF * The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND) * Note that these can be approximations, in this case, the corresp. * entries of WERR give the size of the uncertainty interval. * The eigenvalue approximations will be refined when necessary as * high relative accuracy is required for the computation of the * corresponding eigenvectors. CALL SCOPY( IM, W( WBEGIN ), 1, & WORK( WBEGIN ), 1 ) * We store in W the eigenvalue approximations w.r.t. the original * matrix T. DO 30 I=1,IM W(WBEGIN+I-1) = W(WBEGIN+I-1)+SIGMA 30 CONTINUE * NDEPTH is the current depth of the representation tree NDEPTH = 0 * PARITY is either 1 or 0 PARITY = 1 * NCLUS is the number of clusters for the next level of the * representation tree, we start with NCLUS = 1 for the root NCLUS = 1 IWORK( IINDC1+1 ) = 1 IWORK( IINDC1+2 ) = IM * IDONE is the number of eigenvectors already computed in the current * block IDONE = 0 * loop while( IDONE.LT.IM ) * generate the representation tree for the current block and * compute the eigenvectors 40 CONTINUE IF( IDONE.LT.IM ) THEN * This is a crude protection against infinitely deep trees IF( NDEPTH.GT.M ) THEN INFO = -2 RETURN ENDIF * breadth first processing of the current level of the representation * tree: OLDNCL = number of clusters on current level OLDNCL = NCLUS * reset NCLUS to count the number of child clusters NCLUS = 0 * PARITY = 1 - PARITY IF( PARITY.EQ.0 ) THEN OLDCLS = IINDC1 NEWCLS = IINDC2 ELSE OLDCLS = IINDC2 NEWCLS = IINDC1 END IF * Process the clusters on the current level DO 150 I = 1, OLDNCL J = OLDCLS + 2*I * OLDFST, OLDLST = first, last index of current cluster. * cluster indices start with 1 and are relative * to WBEGIN when accessing W, WGAP, WERR, Z OLDFST = IWORK( J-1 ) OLDLST = IWORK( J ) IF( NDEPTH.GT.0 ) THEN * Retrieve relatively robust representation (RRR) of cluster * that has been computed at the previous level * The RRR is stored in Z and overwritten once the eigenvectors * have been computed or when the cluster is refined IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN * Get representation from location of the leftmost evalue * of the cluster J = WBEGIN + OLDFST - 1 ELSE IF(WBEGIN+OLDFST-1.LT.DOL) THEN * Get representation from the left end of Z array J = DOL - 1 ELSEIF(WBEGIN+OLDFST-1.GT.DOU) THEN * Get representation from the right end of Z array J = DOU ELSE J = WBEGIN + OLDFST - 1 ENDIF ENDIF CALL SCOPY( IN, Z( IBEGIN, J ), 1, D( IBEGIN ), 1 ) CALL SCOPY( IN-1, Z( IBEGIN, J+1 ), 1, L( IBEGIN ), $ 1 ) SIGMA = Z( IEND, J+1 ) * Set the corresponding entries in Z to zero CALL SLASET( 'Full', IN, 2, ZERO, ZERO, $ Z( IBEGIN, J), LDZ ) END IF * Compute DL and DLL of current RRR DO 50 J = IBEGIN, IEND-1 TMP = D( J )*L( J ) WORK( INDLD-1+J ) = TMP WORK( INDLLD-1+J ) = TMP*L( J ) 50 CONTINUE IF( NDEPTH.GT.0 ) THEN * P and Q are index of the first and last eigenvalue to compute * within the current block P = INDEXW( WBEGIN-1+OLDFST ) Q = INDEXW( WBEGIN-1+OLDLST ) * Offset for the arrays WORK, WGAP and WERR, i.e., th P-OFFSET * thru' Q-OFFSET elements of these arrays are to be used. C OFFSET = P-OLDFST OFFSET = INDEXW( WBEGIN ) - 1 * perform limited bisection (if necessary) to get approximate * eigenvalues to the precision needed. CALL SLARRB( IN, D( IBEGIN ), $ WORK(INDLLD+IBEGIN-1), $ P, Q, RTOL1, RTOL2, OFFSET, $ WORK(WBEGIN),WGAP(WBEGIN),WERR(WBEGIN), $ WORK( INDWRK ), IWORK( IINDWK ), $ PIVMIN, SPDIAM, IN, IINFO ) IF( IINFO.NE.0 ) THEN INFO = -1 RETURN ENDIF * We also recompute the extremal gaps. W holds all eigenvalues * of the unshifted matrix and must be used for computation * of WGAP, the entries of WORK might stem from RRRs with * different shifts. The gaps from WBEGIN-1+OLDFST to * WBEGIN-1+OLDLST are correctly computed in SLARRB. * However, we only allow the gaps to become greater since * this is what should happen when we decrease WERR IF( OLDFST.GT.1) THEN WGAP( WBEGIN+OLDFST-2 ) = $ MAX(WGAP(WBEGIN+OLDFST-2), $ W(WBEGIN+OLDFST-1)-WERR(WBEGIN+OLDFST-1) $ - W(WBEGIN+OLDFST-2)-WERR(WBEGIN+OLDFST-2) ) ENDIF IF( WBEGIN + OLDLST -1 .LT. WEND ) THEN WGAP( WBEGIN+OLDLST-1 ) = $ MAX(WGAP(WBEGIN+OLDLST-1), $ W(WBEGIN+OLDLST)-WERR(WBEGIN+OLDLST) $ - W(WBEGIN+OLDLST-1)-WERR(WBEGIN+OLDLST-1) ) ENDIF * Each time the eigenvalues in WORK get refined, we store * the newly found approximation with all shifts applied in W DO 53 J=OLDFST,OLDLST W(WBEGIN+J-1) = WORK(WBEGIN+J-1)+SIGMA 53 CONTINUE END IF * Process the current node. NEWFST = OLDFST DO 140 J = OLDFST, OLDLST IF( J.EQ.OLDLST ) THEN * we are at the right end of the cluster, this is also the * boundary of the child cluster NEWLST = J ELSE IF ( WGAP( WBEGIN + J -1).GE. $ MINRGP* ABS( WORK(WBEGIN + J -1) ) ) THEN * the right relative gap is big enough, the child cluster * (NEWFST,..,NEWLST) is well separated from the following NEWLST = J ELSE * inside a child cluster, the relative gap is not * big enough. GOTO 140 END IF * Compute size of child cluster found NEWSIZ = NEWLST - NEWFST + 1 * NEWFTT is the place in Z where the new RRR or the computed * eigenvector is to be stored IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN * Store representation at location of the leftmost evalue * of the cluster NEWFTT = WBEGIN + NEWFST - 1 ELSE IF(WBEGIN+NEWFST-1.LT.DOL) THEN * Store representation at the left end of Z array NEWFTT = DOL - 1 ELSEIF(WBEGIN+NEWFST-1.GT.DOU) THEN * Store representation at the right end of Z array NEWFTT = DOU ELSE NEWFTT = WBEGIN + NEWFST - 1 ENDIF ENDIF IF( NEWSIZ.GT.1) THEN * * Current child is not a singleton but a cluster. * Compute and store new representation of child. * * * Compute left and right cluster gap. * * LGAP and RGAP are not computed from WORK because * the eigenvalue approximations may stem from RRRs * different shifts. However, W hold all eigenvalues * of the unshifted matrix. Still, the entries in WGAP * have to be computed from WORK since the entries * in W might be of the same order so that gaps are not * exhibited correctly for very close eigenvalues. IF( NEWFST.EQ.1 ) THEN LGAP = MAX( ZERO, $ W(WBEGIN)-WERR(WBEGIN) - VL ) ELSE LGAP = WGAP( WBEGIN+NEWFST-2 ) ENDIF RGAP = WGAP( WBEGIN+NEWLST-1 ) * * Compute left- and rightmost eigenvalue of child * to high precision in order to shift as close * as possible and obtain as large relative gaps * as possible * DO 55 K =1,2 IF(K.EQ.1) THEN P = INDEXW( WBEGIN-1+NEWFST ) ELSE P = INDEXW( WBEGIN-1+NEWLST ) ENDIF OFFSET = INDEXW( WBEGIN ) - 1 CALL SLARRB( IN, D(IBEGIN), $ WORK( INDLLD+IBEGIN-1 ),P,P, $ RQTOL, RQTOL, OFFSET, $ WORK(WBEGIN),WGAP(WBEGIN), $ WERR(WBEGIN),WORK( INDWRK ), $ IWORK( IINDWK ), PIVMIN, SPDIAM, $ IN, IINFO ) 55 CONTINUE * IF((WBEGIN+NEWLST-1.LT.DOL).OR. $ (WBEGIN+NEWFST-1.GT.DOU)) THEN * if the cluster contains no desired eigenvalues * skip the computation of that branch of the rep. tree * * We could skip before the refinement of the extremal * eigenvalues of the child, but then the representation * tree could be different from the one when nothing is * skipped. For this reason we skip at this place. IDONE = IDONE + NEWLST - NEWFST + 1 GOTO 139 ENDIF * * Compute RRR of child cluster. * Note that the new RRR is stored in Z * C SLARRF needs LWORK = 2*N CALL SLARRF( IN, D( IBEGIN ), L( IBEGIN ), $ WORK(INDLD+IBEGIN-1), $ NEWFST, NEWLST, WORK(WBEGIN), $ WGAP(WBEGIN), WERR(WBEGIN), $ SPDIAM, LGAP, RGAP, PIVMIN, TAU, $ Z(IBEGIN, NEWFTT),Z(IBEGIN, NEWFTT+1), $ WORK( INDWRK ), IINFO ) IF( IINFO.EQ.0 ) THEN * a new RRR for the cluster was found by SLARRF * update shift and store it SSIGMA = SIGMA + TAU Z( IEND, NEWFTT+1 ) = SSIGMA * WORK() are the midpoints and WERR() the semi-width * Note that the entries in W are unchanged. DO 116 K = NEWFST, NEWLST FUDGE = $ THREE*EPS*ABS(WORK(WBEGIN+K-1)) WORK( WBEGIN + K - 1 ) = $ WORK( WBEGIN + K - 1) - TAU FUDGE = FUDGE + $ FOUR*EPS*ABS(WORK(WBEGIN+K-1)) * Fudge errors WERR( WBEGIN + K - 1 ) = $ WERR( WBEGIN + K - 1 ) + FUDGE * Gaps are not fudged. Provided that WERR is small * when eigenvalues are close, a zero gap indicates * that a new representation is needed for resolving * the cluster. A fudge could lead to a wrong decision * of judging eigenvalues 'separated' which in * reality are not. This could have a negative impact * on the orthogonality of the computed eigenvectors. 116 CONTINUE NCLUS = NCLUS + 1 K = NEWCLS + 2*NCLUS IWORK( K-1 ) = NEWFST IWORK( K ) = NEWLST ELSE INFO = -2 RETURN ENDIF ELSE * * Compute eigenvector of singleton * ITER = 0 * TOL = FOUR * LOG(REAL(IN)) * EPS * K = NEWFST WINDEX = WBEGIN + K - 1 WINDMN = MAX(WINDEX - 1,1) WINDPL = MIN(WINDEX + 1,M) LAMBDA = WORK( WINDEX ) DONE = DONE + 1 * Check if eigenvector computation is to be skipped IF((WINDEX.LT.DOL).OR. $ (WINDEX.GT.DOU)) THEN ESKIP = .TRUE. GOTO 125 ELSE ESKIP = .FALSE. ENDIF LEFT = WORK( WINDEX ) - WERR( WINDEX ) RIGHT = WORK( WINDEX ) + WERR( WINDEX ) INDEIG = INDEXW( WINDEX ) * Note that since we compute the eigenpairs for a child, * all eigenvalue approximations are w.r.t the same shift. * In this case, the entries in WORK should be used for * computing the gaps since they exhibit even very small * differences in the eigenvalues, as opposed to the * entries in W which might "look" the same. IF( K .EQ. 1) THEN * In the case RANGE='I' and with not much initial * accuracy in LAMBDA and VL, the formula * LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA ) * can lead to an overestimation of the left gap and * thus to inadequately early RQI 'convergence'. * Prevent this by forcing a small left gap. LGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT)) ELSE LGAP = WGAP(WINDMN) ENDIF IF( K .EQ. IM) THEN * In the case RANGE='I' and with not much initial * accuracy in LAMBDA and VU, the formula * can lead to an overestimation of the right gap and * thus to inadequately early RQI 'convergence'. * Prevent this by forcing a small right gap. RGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT)) ELSE RGAP = WGAP(WINDEX) ENDIF GAP = MIN( LGAP, RGAP ) IF(( K .EQ. 1).OR.(K .EQ. IM)) THEN * The eigenvector support can become wrong * because significant entries could be cut off due to a * large GAPTOL parameter in LAR1V. Prevent this. GAPTOL = ZERO ELSE GAPTOL = GAP * EPS ENDIF ISUPMN = IN ISUPMX = 1 * Update WGAP so that it holds the minimum gap * to the left or the right. This is crucial in the * case where bisection is used to ensure that the * eigenvalue is refined up to the required precision. * The correct value is restored afterwards. SAVGAP = WGAP(WINDEX) WGAP(WINDEX) = GAP * We want to use the Rayleigh Quotient Correction * as often as possible since it converges quadratically * when we are close enough to the desired eigenvalue. * However, the Rayleigh Quotient can have the wrong sign * and lead us away from the desired eigenvalue. In this * case, the best we can do is to use bisection. USEDBS = .FALSE. USEDRQ = .FALSE. * Bisection is initially turned off unless it is forced NEEDBS = .NOT.TRYRQC 120 CONTINUE * Check if bisection should be used to refine eigenvalue IF(NEEDBS) THEN * Take the bisection as new iterate USEDBS = .TRUE. ITMP1 = IWORK( IINDR+WINDEX ) OFFSET = INDEXW( WBEGIN ) - 1 CALL SLARRB( IN, D(IBEGIN), $ WORK(INDLLD+IBEGIN-1),INDEIG,INDEIG, $ ZERO, TWO*EPS, OFFSET, $ WORK(WBEGIN),WGAP(WBEGIN), $ WERR(WBEGIN),WORK( INDWRK ), $ IWORK( IINDWK ), PIVMIN, SPDIAM, $ ITMP1, IINFO ) IF( IINFO.NE.0 ) THEN INFO = -3 RETURN ENDIF LAMBDA = WORK( WINDEX ) * Reset twist index from inaccurate LAMBDA to * force computation of true MINGMA IWORK( IINDR+WINDEX ) = 0 ENDIF * Given LAMBDA, compute the eigenvector. CALL SLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ), $ L( IBEGIN ), WORK(INDLD+IBEGIN-1), $ WORK(INDLLD+IBEGIN-1), $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ), $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA, $ IWORK( IINDR+WINDEX ), ISUPPZ( 2*WINDEX-1 ), $ NRMINV, RESID, RQCORR, WORK( INDWRK ) ) IF(ITER .EQ. 0) THEN BSTRES = RESID BSTW = LAMBDA ELSEIF(RESID.LT.BSTRES) THEN BSTRES = RESID BSTW = LAMBDA ENDIF ISUPMN = MIN(ISUPMN,ISUPPZ( 2*WINDEX-1 )) ISUPMX = MAX(ISUPMX,ISUPPZ( 2*WINDEX )) ITER = ITER + 1 * sin alpha <= |resid|/gap * Note that both the residual and the gap are * proportional to the matrix, so ||T|| doesn't play * a role in the quotient * * Convergence test for Rayleigh-Quotient iteration * (omitted when Bisection has been used) * IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT. $ RQTOL*ABS( LAMBDA ) .AND. .NOT. USEDBS) $ THEN * We need to check that the RQCORR update doesn't * move the eigenvalue away from the desired one and * towards a neighbor. -> protection with bisection IF(INDEIG.LE.NEGCNT) THEN * The wanted eigenvalue lies to the left SGNDEF = -ONE ELSE * The wanted eigenvalue lies to the right SGNDEF = ONE ENDIF * We only use the RQCORR if it improves the * the iterate reasonably. IF( ( RQCORR*SGNDEF.GE.ZERO ) $ .AND.( LAMBDA + RQCORR.LE. RIGHT) $ .AND.( LAMBDA + RQCORR.GE. LEFT) $ ) THEN USEDRQ = .TRUE. * Store new midpoint of bisection interval in WORK IF(SGNDEF.EQ.ONE) THEN * The current LAMBDA is on the left of the true * eigenvalue LEFT = LAMBDA * We prefer to assume that the error estimate * is correct. We could make the interval not * as a bracket but to be modified if the RQCORR * chooses to. In this case, the RIGHT side should * be modified as follows: * RIGHT = MAX(RIGHT, LAMBDA + RQCORR) ELSE * The current LAMBDA is on the right of the true * eigenvalue RIGHT = LAMBDA * See comment about assuming the error estimate is * correct above. * LEFT = MIN(LEFT, LAMBDA + RQCORR) ENDIF WORK( WINDEX ) = $ HALF * (RIGHT + LEFT) * Take RQCORR since it has the correct sign and * improves the iterate reasonably LAMBDA = LAMBDA + RQCORR * Update width of error interval WERR( WINDEX ) = $ HALF * (RIGHT-LEFT) ELSE NEEDBS = .TRUE. ENDIF IF(RIGHT-LEFT.LT.RQTOL*ABS(LAMBDA)) THEN * The eigenvalue is computed to bisection accuracy * compute eigenvector and stop USEDBS = .TRUE. GOTO 120 ELSEIF( ITER.LT.MAXITR ) THEN GOTO 120 ELSEIF( ITER.EQ.MAXITR ) THEN NEEDBS = .TRUE. GOTO 120 ELSE INFO = 5 RETURN END IF ELSE STP2II = .FALSE. IF(USEDRQ .AND. USEDBS .AND. $ BSTRES.LE.RESID) THEN LAMBDA = BSTW STP2II = .TRUE. ENDIF IF (STP2II) THEN * improve error angle by second step CALL SLAR1V( IN, 1, IN, LAMBDA, $ D( IBEGIN ), L( IBEGIN ), $ WORK(INDLD+IBEGIN-1), $ WORK(INDLLD+IBEGIN-1), $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ), $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA, $ IWORK( IINDR+WINDEX ), $ ISUPPZ( 2*WINDEX-1 ), $ NRMINV, RESID, RQCORR, WORK( INDWRK ) ) ENDIF WORK( WINDEX ) = LAMBDA END IF * * Compute FP-vector support w.r.t. whole matrix * ISUPPZ( 2*WINDEX-1 ) = ISUPPZ( 2*WINDEX-1 )+OLDIEN ISUPPZ( 2*WINDEX ) = ISUPPZ( 2*WINDEX )+OLDIEN ZFROM = ISUPPZ( 2*WINDEX-1 ) ZTO = ISUPPZ( 2*WINDEX ) ISUPMN = ISUPMN + OLDIEN ISUPMX = ISUPMX + OLDIEN * Ensure vector is ok if support in the RQI has changed IF(ISUPMN.LT.ZFROM) THEN DO 122 II = ISUPMN,ZFROM-1 Z( II, WINDEX ) = ZERO 122 CONTINUE ENDIF IF(ISUPMX.GT.ZTO) THEN DO 123 II = ZTO+1,ISUPMX Z( II, WINDEX ) = ZERO 123 CONTINUE ENDIF CALL SSCAL( ZTO-ZFROM+1, NRMINV, $ Z( ZFROM, WINDEX ), 1 ) 125 CONTINUE * Update W W( WINDEX ) = LAMBDA+SIGMA * Recompute the gaps on the left and right * But only allow them to become larger and not * smaller (which can only happen through "bad" * cancellation and doesn't reflect the theory * where the initial gaps are underestimated due * to WERR being too crude.) IF(.NOT.ESKIP) THEN IF( K.GT.1) THEN WGAP( WINDMN ) = MAX( WGAP(WINDMN), $ W(WINDEX)-WERR(WINDEX) $ - W(WINDMN)-WERR(WINDMN) ) ENDIF IF( WINDEX.LT.WEND ) THEN WGAP( WINDEX ) = MAX( SAVGAP, $ W( WINDPL )-WERR( WINDPL ) $ - W( WINDEX )-WERR( WINDEX) ) ENDIF ENDIF IDONE = IDONE + 1 ENDIF * here ends the code for the current child * 139 CONTINUE * Proceed to any remaining child nodes NEWFST = J + 1 140 CONTINUE 150 CONTINUE NDEPTH = NDEPTH + 1 GO TO 40 END IF IBEGIN = IEND + 1 WBEGIN = WEND + 1 170 CONTINUE * RETURN * * End of SLARRV * END