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SUBROUTINE CLARRV( N, D, L, ISPLIT, M, W, IBLOCK, GERSCH, TOL, Z,
$ LDZ, ISUPPZ, WORK, IWORK, INFO )
*
* -- LAPACK auxiliary routine (instru to count ops, version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
INTEGER INFO, LDZ, M, N
REAL TOL
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), ISPLIT( * ), ISUPPZ( * ),
$ IWORK( * )
REAL D( * ), GERSCH( * ), L( * ), W( * ), WORK( * )
COMPLEX Z( LDZ, * )
* ..
* Common block to return operation count ..
* .. Common blocks ..
COMMON / LATIME / OPS, ITCNT
* ..
* .. Scalars in Common ..
REAL ITCNT, OPS
* ..
*
* Purpose
* =======
*
* CLARRV computes the eigenvectors of the tridiagonal matrix
* T = L D L^T given L, D and the eigenvalues of L D L^T.
* The input eigenvalues should have high relative accuracy with
* respect to the entries of L and D. The desired accuracy of the
* output can be specified by the input parameter TOL.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* 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-1)
* On entry, the (n-1) subdiagonal elements of the unit
* bidiagonal matrix L in elements 1 to N-1 of L. L(N) need
* not be set. On exit, L is overwritten.
*
* ISPLIT (input) INTEGER array, dimension (N)
* The splitting points, at which T breaks up into submatrices.
* The first submatrix consists of rows/columns 1 to
* ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
* through ISPLIT( 2 ), etc.
*
* TOL (input) REAL
* The absolute error tolerance for the
* eigenvalues/eigenvectors.
* Errors in the input eigenvalues must be bounded by TOL.
* The eigenvectors output have residual norms
* bounded by TOL, and the dot products between different
* eigenvectors are bounded by TOL. TOL must be at least
* N*EPS*|T|, where EPS is the machine precision and |T| is
* the 1-norm of the tridiagonal matrix.
*
* M (input) INTEGER
* The total number of eigenvalues found. 0 <= M <= N.
* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*
* W (input) REAL array, dimension (N)
* The first M elements of W contain the 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 ).
* Errors in W must be bounded by TOL (see above).
*
* IBLOCK (input) INTEGER array, dimension (N)
* The submatrix indices associated with the corresponding
* eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
* the first submatrix from the top, =2 if W(i) belongs to
* the second submatrix, etc.
*
* Z (output) COMPLEX array, dimension (LDZ, max(1,M) )
* If JOBZ = 'V', then if INFO = 0, the first M columns of Z
* contain the orthonormal eigenvectors of the matrix T
* corresponding to the selected eigenvalues, with the i-th
* column of Z holding the eigenvector associated with W(i).
* If JOBZ = 'N', then Z is not referenced.
* Note: the user must ensure that at least max(1,M) columns are
* supplied in the array Z; if RANGE = 'V', the exact value of M
* is not known in advance and an upper bound must be used.
*
* 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 (13*N)
*
* IWORK (workspace) INTEGER array, dimension (6*N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = 1, internal error in SLARRB
* if INFO = 2, internal error in CSTEIN
*
* Further Details
* ===============
*
* Based on contributions by
* Inderjit Dhillon, IBM Almaden, USA
* Osni Marques, LBNL/NERSC, USA
* Ken Stanley, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
* .. Parameters ..
INTEGER MGSSIZ
PARAMETER ( MGSSIZ = 20 )
REAL ZERO, ONE, FOUR
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, FOUR = 4.0E0 )
COMPLEX CZERO
PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ) )
* ..
* .. Local Scalars ..
LOGICAL MGSCLS
INTEGER I, IBEGIN, IEND, IINDC1, IINDC2, IINDR, IINDWK,
$ IINFO, IM, IN, INDERR, INDGAP, INDIN1, INDIN2,
$ INDLD, INDLLD, INDWRK, ITER, ITMP1, ITMP2, J,
$ JBLK, K, KTOT, LSBDPT, MAXITR, NCLUS, NDEPTH,
$ NDONE, NEWCLS, NEWFRS, NEWFTT, NEWLST, NEWSIZ,
$ NSPLIT, OLDCLS, OLDFST, OLDIEN, OLDLST, OLDNCL,
$ P, Q, TEMP( 1 )
REAL EPS, GAP, LAMBDA, MGSTOL, MINGMA, MINRGP,
$ NRMINV, RELGAP, RELTOL, RESID, RQCORR, SIGMA,
$ TMP1, ZTZ
* ..
* .. External Functions ..
REAL SCNRM2, SLAMCH
COMPLEX CDOTU
EXTERNAL CDOTU, SCNRM2, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL CAXPY, CLAR1V, CLASET, CSTEIN, SCOPY, SLARRB
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CMPLX, MAX, MIN, REAL, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INDERR = N + 1
INDLD = 2*N
INDLLD = 3*N
INDGAP = 4*N
INDIN1 = 5*N + 1
INDIN2 = 6*N + 1
INDWRK = 7*N + 1
*
IINDR = N
IINDC1 = 2*N
IINDC2 = 3*N
IINDWK = 4*N + 1
*
EPS = SLAMCH( 'Precision' )
*
DO 10 I = 1, 2*N
IWORK( I ) = 0
10 CONTINUE
OPS = OPS + REAL( M+1 )
DO 20 I = 1, M
WORK( INDERR+I-1 ) = EPS*ABS( W( I ) )
20 CONTINUE
CALL CLASET( 'Full', N, N, CZERO, CZERO, Z, LDZ )
MGSTOL = 5.0E0*EPS
*
NSPLIT = IBLOCK( M )
IBEGIN = 1
DO 170 JBLK = 1, NSPLIT
IEND = ISPLIT( JBLK )
*
* Find the eigenvectors of the submatrix indexed IBEGIN
* through IEND.
*
IF( IBEGIN.EQ.IEND ) THEN
Z( IBEGIN, IBEGIN ) = ONE
ISUPPZ( 2*IBEGIN-1 ) = IBEGIN
ISUPPZ( 2*IBEGIN ) = IBEGIN
IBEGIN = IEND + 1
GO TO 170
END IF
OLDIEN = IBEGIN - 1
IN = IEND - OLDIEN
OPS = OPS + REAL( 1 )
RELTOL = MIN( 1.0E-2, ONE / REAL( IN ) )
IM = IN
CALL SCOPY( IM, W( IBEGIN ), 1, WORK, 1 )
OPS = OPS + REAL( IN-1 )
DO 30 I = 1, IN - 1
WORK( INDGAP+I ) = WORK( I+1 ) - WORK( I )
30 CONTINUE
WORK( INDGAP+IN ) = MAX( ABS( WORK( IN ) ), EPS )
NDONE = 0
*
NDEPTH = 0
LSBDPT = 1
NCLUS = 1
IWORK( IINDC1+1 ) = 1
IWORK( IINDC1+2 ) = IN
*
* While( NDONE.LT.IM ) do
*
40 CONTINUE
IF( NDONE.LT.IM ) THEN
OLDNCL = NCLUS
NCLUS = 0
LSBDPT = 1 - LSBDPT
DO 150 I = 1, OLDNCL
IF( LSBDPT.EQ.0 ) THEN
OLDCLS = IINDC1
NEWCLS = IINDC2
ELSE
OLDCLS = IINDC2
NEWCLS = IINDC1
END IF
*
* If NDEPTH > 1, retrieve the relatively robust
* representation (RRR) and perform limited bisection
* (if necessary) to get approximate eigenvalues.
*
J = OLDCLS + 2*I
OLDFST = IWORK( J-1 )
OLDLST = IWORK( J )
IF( NDEPTH.GT.0 ) THEN
J = OLDIEN + OLDFST
DO 45 K = 1, IN
D( IBEGIN+K-1 ) = REAL( Z( IBEGIN+K-1,
$ OLDIEN+OLDFST ) )
L( IBEGIN+K-1 ) = REAL( Z( IBEGIN+K-1,
$ OLDIEN+OLDFST+1 ) )
45 CONTINUE
SIGMA = L( IEND )
END IF
K = IBEGIN
OPS = OPS + REAL( 2*(IN-1) )
DO 50 J = 1, IN - 1
WORK( INDLD+J ) = D( K )*L( K )
WORK( INDLLD+J ) = WORK( INDLD+J )*L( K )
K = K + 1
50 CONTINUE
IF( NDEPTH.GT.0 ) THEN
CALL SLARRB( IN, D( IBEGIN ), L( IBEGIN ),
$ WORK( INDLD+1 ), WORK( INDLLD+1 ),
$ OLDFST, OLDLST, SIGMA, RELTOL, WORK,
$ WORK( INDGAP+1 ), WORK( INDERR ),
$ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 1
RETURN
END IF
END IF
*
* Classify eigenvalues of the current representation (RRR)
* as (i) isolated, (ii) loosely clustered or (iii) tightly
* clustered
*
NEWFRS = OLDFST
DO 140 J = OLDFST, OLDLST
OPS = OPS + REAL( 1 )
IF( J.EQ.OLDLST .OR. WORK( INDGAP+J ).GE.RELTOL*
$ ABS( WORK( J ) ) ) THEN
NEWLST = J
ELSE
*
* continue (to the next loop)
*
OPS = OPS + REAL( 1 )
RELGAP = WORK( INDGAP+J ) / ABS( WORK( J ) )
IF( J.EQ.NEWFRS ) THEN
MINRGP = RELGAP
ELSE
MINRGP = MIN( MINRGP, RELGAP )
END IF
GO TO 140
END IF
NEWSIZ = NEWLST - NEWFRS + 1
MAXITR = 10
NEWFTT = OLDIEN + NEWFRS
IF( NEWSIZ.GT.1 ) THEN
MGSCLS = NEWSIZ.LE.MGSSIZ .AND. MINRGP.GE.MGSTOL
IF( .NOT.MGSCLS ) THEN
DO 55 K = 1, IN
WORK( INDIN1+K-1 ) = REAL( Z( IBEGIN+K-1,
$ NEWFTT ) )
WORK( INDIN2+K-1 ) = REAL( Z( IBEGIN+K-1,
$ NEWFTT+1 ) )
55 CONTINUE
CALL SLARRF( IN, D( IBEGIN ), L( IBEGIN ),
$ WORK( INDLD+1 ), WORK( INDLLD+1 ),
$ NEWFRS, NEWLST, WORK,
$ WORK( INDIN1 ), WORK( INDIN2 ),
$ WORK( INDWRK ), IWORK( IINDWK ),
$ INFO )
IF( INFO.EQ.0 ) THEN
NCLUS = NCLUS + 1
K = NEWCLS + 2*NCLUS
IWORK( K-1 ) = NEWFRS
IWORK( K ) = NEWLST
ELSE
INFO = 0
IF( MINRGP.GE.MGSTOL ) THEN
MGSCLS = .TRUE.
ELSE
*
* Call CSTEIN to process this tight cluster.
* This happens only if MINRGP <= MGSTOL
* and SLARRF returns INFO = 1. The latter
* means that a new RRR to "break" the
* cluster could not be found.
*
WORK( INDWRK ) = D( IBEGIN )
OPS = OPS + REAL( IN-1 )
DO 60 K = 1, IN - 1
WORK( INDWRK+K ) = D( IBEGIN+K ) +
$ WORK( INDLLD+K )
60 CONTINUE
DO 70 K = 1, NEWSIZ
IWORK( IINDWK+K-1 ) = 1
70 CONTINUE
DO 80 K = NEWFRS, NEWLST
ISUPPZ( 2*( IBEGIN+K )-3 ) = 1
ISUPPZ( 2*( IBEGIN+K )-2 ) = IN
80 CONTINUE
TEMP( 1 ) = IN
CALL CSTEIN( IN, WORK( INDWRK ),
$ WORK( INDLD+1 ), NEWSIZ,
$ WORK( NEWFRS ),
$ IWORK( IINDWK ), TEMP( 1 ),
$ Z( IBEGIN, NEWFTT ), LDZ,
$ WORK( INDWRK+IN ),
$ IWORK( IINDWK+IN ),
$ IWORK( IINDWK+2*IN ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = 2
RETURN
END IF
NDONE = NDONE + NEWSIZ
END IF
END IF
END IF
ELSE
MGSCLS = .FALSE.
END IF
IF( NEWSIZ.EQ.1 .OR. MGSCLS ) THEN
KTOT = NEWFTT
DO 100 K = NEWFRS, NEWLST
ITER = 0
90 CONTINUE
LAMBDA = WORK( K )
CALL CLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ),
$ L( IBEGIN ), WORK( INDLD+1 ),
$ WORK( INDLLD+1 ),
$ GERSCH( 2*OLDIEN+1 ),
$ Z( IBEGIN, KTOT ), ZTZ, MINGMA,
$ IWORK( IINDR+KTOT ),
$ ISUPPZ( 2*KTOT-1 ),
$ WORK( INDWRK ) )
OPS = OPS + REAL( 4 )
TMP1 = ONE / ZTZ
NRMINV = SQRT( TMP1 )
RESID = ABS( MINGMA )*NRMINV
RQCORR = MINGMA*TMP1
IF( K.EQ.IN ) THEN
GAP = WORK( INDGAP+K-1 )
ELSE IF( K.EQ.1 ) THEN
GAP = WORK( INDGAP+K )
ELSE
GAP = MIN( WORK( INDGAP+K-1 ),
$ WORK( INDGAP+K ) )
END IF
ITER = ITER + 1
OPS = OPS + REAL( 3 )
IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT.
$ FOUR*EPS*ABS( LAMBDA ) ) THEN
OPS = OPS + REAL( 1 )
WORK( K ) = LAMBDA + RQCORR
IF( ITER.LT.MAXITR ) THEN
GO TO 90
END IF
END IF
IWORK( KTOT ) = 1
IF( NEWSIZ.EQ.1 )
$ NDONE = NDONE + 1
OPS = OPS + REAL( 2*IN )
CALL CSSCAL( IN, NRMINV, Z( IBEGIN, KTOT ), 1 )
KTOT = KTOT + 1
100 CONTINUE
IF( NEWSIZ.GT.1 ) THEN
ITMP1 = ISUPPZ( 2*NEWFTT-1 )
ITMP2 = ISUPPZ( 2*NEWFTT )
KTOT = OLDIEN + NEWLST
DO 120 P = NEWFTT + 1, KTOT
DO 110 Q = NEWFTT, P - 1
OPS = OPS + REAL( 10*IN )
TMP1 = -CDOTU( IN, Z( IBEGIN, P ), 1,
$ Z( IBEGIN, Q ), 1 )
CALL CAXPY( IN, CMPLX( TMP1, ZERO ),
$ Z( IBEGIN, Q ), 1,
$ Z( IBEGIN, P ), 1 )
110 CONTINUE
OPS = OPS + REAL( 8*IN+1 )
TMP1 = ONE / SCNRM2( IN, Z( IBEGIN, P ), 1 )
CALL CSSCAL( IN, TMP1, Z( IBEGIN, P ), 1 )
ITMP1 = MIN( ITMP1, ISUPPZ( 2*P-1 ) )
ITMP2 = MAX( ITMP2, ISUPPZ( 2*P ) )
120 CONTINUE
DO 130 P = NEWFTT, KTOT
ISUPPZ( 2*P-1 ) = ITMP1
ISUPPZ( 2*P ) = ITMP2
130 CONTINUE
NDONE = NDONE + NEWSIZ
END IF
END IF
NEWFRS = J + 1
140 CONTINUE
150 CONTINUE
NDEPTH = NDEPTH + 1
GO TO 40
END IF
J = 2*IBEGIN
DO 160 I = IBEGIN, IEND
ISUPPZ( J-1 ) = ISUPPZ( J-1 ) + OLDIEN
ISUPPZ( J ) = ISUPPZ( J ) + OLDIEN
J = J + 2
160 CONTINUE
IBEGIN = IEND + 1
170 CONTINUE
*
RETURN
*
* End of CLARRV
*
END
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