SUBROUTINE ZLARRV( 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
DOUBLE PRECISION MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ),
$ ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION D( * ), GERS( * ), L( * ), W( * ), WERR( * ),
$ WGAP( * ), WORK( * )
COMPLEX*16 Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* ZLARRV 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 DLARRE.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* VL (input) DOUBLE PRECISION
* VU (input) DOUBLE PRECISION
* 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) DOUBLE PRECISION array, dimension (N)
* On entry, the N diagonal elements of the diagonal matrix D.
* On exit, D may be overwritten.
*
* L (input/output) DOUBLE PRECISION 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 DLARRE.
* 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) DOUBLE PRECISION
*
* RTOL1 (input) DOUBLE PRECISION
* RTOL2 (input) DOUBLE PRECISION
* Parameters for bisection.
* An interval [LEFT,RIGHT] has converged if
* RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
*
* W (input/output) DOUBLE PRECISION 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 DLARRE 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) DOUBLE PRECISION array, dimension (N)
* The first M elements contain the semiwidth of the uncertainty
* interval of the corresponding eigenvalue in W
*
* WGAP (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) COMPLEX*16 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) DOUBLE PRECISION array, dimension (12*N)
*
* IWORK (workspace) INTEGER array, dimension (7*N)
*
* INFO (output) INTEGER
* = 0: successful exit
*
* > 0: A problem occured in ZLARRV.
* < 0: One of the called subroutines signaled an internal problem.
* Needs inspection of the corresponding parameter IINFO
* for further information.
*
* =-1: Problem in DLARRB when refining a child's eigenvalues.
* =-2: Problem in DLARRF 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 DLARRB 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 )
COMPLEX*16 CZERO
PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ) )
DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, HALF
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, THREE = 3.0D0,
$ FOUR = 4.0D0, HALF = 0.5D0)
* ..
* .. 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
INTEGER INDIN1, INDIN2
DOUBLE PRECISION 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 ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLARRB, DLARRF, ZDSCAL, ZLAR1V,
$ ZLASET
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN
INTRINSIC DCMPLX
* ..
* .. Executable Statements ..
* ..
* The first N entries of WORK are reserved for the eigenvalues
INDLD = N+1
INDLLD= 2*N+1
INDIN1 = 3*N + 1
INDIN2 = 4*N + 1
INDWRK = 5*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 ZLASET( 'Full', N, ZUSEDW, CZERO, CZERO,
$ Z(1,ZUSEDL), LDZ )
EPS = DLAMCH( '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 ) = DCMPLX( ONE, ZERO )
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 DCOPY( 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
DO 45 K = 1, IN - 1
D( IBEGIN+K-1 ) = DBLE( Z( IBEGIN+K-1,
$ J ) )
L( IBEGIN+K-1 ) = DBLE( Z( IBEGIN+K-1,
$ J+1 ) )
45 CONTINUE
D( IEND ) = DBLE( Z( IEND, J ) )
SIGMA = DBLE( Z( IEND, J+1 ) )
* Set the corresponding entries in Z to zero
CALL ZLASET( 'Full', IN, 2, CZERO, CZERO,
$ 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 DLARRB( 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 DLARRB.
* 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 DLARRB( 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 DLARRF needs LWORK = 2*N
CALL DLARRF( IN, D( IBEGIN ), L( IBEGIN ),
$ WORK(INDLD+IBEGIN-1),
$ NEWFST, NEWLST, WORK(WBEGIN),
$ WGAP(WBEGIN), WERR(WBEGIN),
$ SPDIAM, LGAP, RGAP, PIVMIN, TAU,
$ WORK( INDIN1 ), WORK( INDIN2 ),
$ WORK( INDWRK ), IINFO )
* In the complex case, DLARRF cannot write
* the new RRR directly into Z and needs an intermediate
* workspace
DO 56 K = 1, IN-1
Z( IBEGIN+K-1, NEWFTT ) =
$ DCMPLX( WORK( INDIN1+K-1 ), ZERO )
Z( IBEGIN+K-1, NEWFTT+1 ) =
$ DCMPLX( WORK( INDIN2+K-1 ), ZERO )
56 CONTINUE
Z( IEND, NEWFTT ) =
$ DCMPLX( WORK( INDIN1+IN-1 ), ZERO )
IF( IINFO.EQ.0 ) THEN
* a new RRR for the cluster was found by DLARRF
* update shift and store it
SSIGMA = SIGMA + TAU
Z( IEND, NEWFTT+1 ) = DCMPLX( SSIGMA, ZERO )
* 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(DBLE(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 DLARRB( 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 ZLAR1V( 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 ZLAR1V( 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 ZDSCAL( 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 ZLARRV
*
END