*> \brief \b SLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLARRE + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2, * RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M, * W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN, * WORK, IWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER RANGE * INTEGER IL, INFO, IU, M, N, NSPLIT * REAL PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU * .. * .. Array Arguments .. * INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ), * $ INDEXW( * ) * REAL D( * ), E( * ), E2( * ), GERS( * ), * $ W( * ),WERR( * ), WGAP( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> To find the desired eigenvalues of a given real symmetric *> tridiagonal matrix T, SLARRE sets any "small" off-diagonal *> elements to zero, and for each unreduced block T_i, it finds *> (a) a suitable shift at one end of the block's spectrum, *> (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and *> (c) eigenvalues of each L_i D_i L_i^T. *> The representations and eigenvalues found are then used by *> SSTEMR to compute the eigenvectors of T. *> The accuracy varies depending on whether bisection is used to *> find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to *> conpute all and then discard any unwanted one. *> As an added benefit, SLARRE also outputs the n *> Gerschgorin intervals for the matrices L_i D_i L_i^T. *> \endverbatim * * Arguments: * ========== * *> \param[in] RANGE *> \verbatim *> RANGE is CHARACTER*1 *> = 'A': ("All") all eigenvalues will be found. *> = 'V': ("Value") all eigenvalues in the half-open interval *> (VL, VU] will be found. *> = 'I': ("Index") the IL-th through IU-th eigenvalues (of the *> entire matrix) will be found. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix. N > 0. *> \endverbatim *> *> \param[in,out] VL *> \verbatim *> VL is REAL *> If RANGE='V', the lower bound for the eigenvalues. *> Eigenvalues less than or equal to VL, or greater than VU, *> will not be returned. VL < VU. *> If RANGE='I' or ='A', SLARRE computes bounds on the desired *> part of the spectrum. *> \endverbatim *> *> \param[in,out] VU *> \verbatim *> VU is REAL *> If RANGE='V', the upper bound for the eigenvalues. *> Eigenvalues less than or equal to VL, or greater than VU, *> will not be returned. VL < VU. *> If RANGE='I' or ='A', SLARRE computes bounds on the desired *> part of the spectrum. *> \endverbatim *> *> \param[in] IL *> \verbatim *> IL is INTEGER *> If RANGE='I', the index of the *> smallest eigenvalue to be returned. *> 1 <= IL <= IU <= N. *> \endverbatim *> *> \param[in] IU *> \verbatim *> IU is INTEGER *> If RANGE='I', the index of the *> largest eigenvalue to be returned. *> 1 <= IL <= IU <= N. *> \endverbatim *> *> \param[in,out] D *> \verbatim *> D is REAL array, dimension (N) *> On entry, the N diagonal elements of the tridiagonal *> matrix T. *> On exit, the N diagonal elements of the diagonal *> matrices D_i. *> \endverbatim *> *> \param[in,out] E *> \verbatim *> E is REAL array, dimension (N) *> On entry, the first (N-1) entries contain the subdiagonal *> elements of the tridiagonal matrix T; E(N) need not be set. *> On exit, E contains the subdiagonal elements of the unit *> bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), *> 1 <= I <= NSPLIT, contain the base points sigma_i on output. *> \endverbatim *> *> \param[in,out] E2 *> \verbatim *> E2 is REAL array, dimension (N) *> On entry, the first (N-1) entries contain the SQUARES of the *> subdiagonal elements of the tridiagonal matrix T; *> E2(N) need not be set. *> On exit, the entries E2( ISPLIT( I ) ), *> 1 <= I <= NSPLIT, have been set to zero *> \endverbatim *> *> \param[in] RTOL1 *> \verbatim *> RTOL1 is REAL *> \endverbatim *> *> \param[in] RTOL2 *> \verbatim *> RTOL2 is REAL *> Parameters for bisection. *> An interval [LEFT,RIGHT] has converged if *> RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) *> \endverbatim *> *> \param[in] SPLTOL *> \verbatim *> SPLTOL is REAL *> The threshold for splitting. *> \endverbatim *> *> \param[out] NSPLIT *> \verbatim *> NSPLIT is INTEGER *> The number of blocks T splits into. 1 <= NSPLIT <= N. *> \endverbatim *> *> \param[out] ISPLIT *> \verbatim *> ISPLIT is 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., and the NSPLIT-th consists of rows/columns *> ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. *> \endverbatim *> *> \param[out] M *> \verbatim *> M is INTEGER *> The total number of eigenvalues (of all L_i D_i L_i^T) *> found. *> \endverbatim *> *> \param[out] W *> \verbatim *> W is REAL array, dimension (N) *> The first M elements contain the eigenvalues. The *> eigenvalues of each of the blocks, L_i D_i L_i^T, are *> sorted in ascending order ( SLARRE may use the *> remaining N-M elements as workspace). *> \endverbatim *> *> \param[out] WERR *> \verbatim *> WERR is REAL array, dimension (N) *> The error bound on the corresponding eigenvalue in W. *> \endverbatim *> *> \param[out] WGAP *> \verbatim *> WGAP is REAL array, dimension (N) *> The separation from the right neighbor eigenvalue in W. *> The gap is only with respect to the eigenvalues of the same block *> as each block has its own representation tree. *> Exception: at the right end of a block we store the left gap *> \endverbatim *> *> \param[out] IBLOCK *> \verbatim *> IBLOCK is 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. *> \endverbatim *> *> \param[out] INDEXW *> \verbatim *> INDEXW is 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 block 2 *> \endverbatim *> *> \param[out] GERS *> \verbatim *> GERS is REAL array, dimension (2*N) *> The N Gerschgorin intervals (the i-th Gerschgorin interval *> is (GERS(2*i-1), GERS(2*i)). *> \endverbatim *> *> \param[out] PIVMIN *> \verbatim *> PIVMIN is REAL *> The minimum pivot in the Sturm sequence for T. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (6*N) *> Workspace. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (5*N) *> Workspace. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> > 0: A problem occurred in SLARRE. *> < 0: One of the called subroutines signaled an internal problem. *> Needs inspection of the corresponding parameter IINFO *> for further information. *> *> =-1: Problem in SLARRD. *> = 2: No base representation could be found in MAXTRY iterations. *> Increasing MAXTRY and recompilation might be a remedy. *> =-3: Problem in SLARRB when computing the refined root *> representation for SLASQ2. *> =-4: Problem in SLARRB when preforming bisection on the *> desired part of the spectrum. *> =-5: Problem in SLASQ2. *> =-6: Problem in SLASQ2. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date June 2016 * *> \ingroup OTHERauxiliary * *> \par Further Details: * ===================== *> *> \verbatim *> *> The base representations are required to suffer very little *> element growth and consequently define all their eigenvalues to *> high relative accuracy. *> \endverbatim * *> \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 \n *> * ===================================================================== SUBROUTINE SLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2, $ RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M, $ W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN, $ WORK, IWORK, INFO ) * * -- 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..-- * June 2016 * * .. Scalar Arguments .. CHARACTER RANGE INTEGER IL, INFO, IU, M, N, NSPLIT REAL PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU * .. * .. Array Arguments .. INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ), $ INDEXW( * ) REAL D( * ), E( * ), E2( * ), GERS( * ), $ W( * ),WERR( * ), WGAP( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. REAL FAC, FOUR, FOURTH, FUDGE, HALF, HNDRD, $ MAXGROWTH, ONE, PERT, TWO, ZERO PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, $ TWO = 2.0E0, FOUR=4.0E0, $ HNDRD = 100.0E0, $ PERT = 4.0E0, $ HALF = ONE/TWO, FOURTH = ONE/FOUR, FAC= HALF, $ MAXGROWTH = 64.0E0, FUDGE = 2.0E0 ) INTEGER MAXTRY, ALLRNG, INDRNG, VALRNG PARAMETER ( MAXTRY = 6, ALLRNG = 1, INDRNG = 2, $ VALRNG = 3 ) * .. * .. Local Scalars .. LOGICAL FORCEB, NOREP, USEDQD INTEGER CNT, CNT1, CNT2, I, IBEGIN, IDUM, IEND, IINFO, $ IN, INDL, INDU, IRANGE, J, JBLK, MB, MM, $ WBEGIN, WEND REAL AVGAP, BSRTOL, CLWDTH, DMAX, DPIVOT, EABS, $ EMAX, EOLD, EPS, GL, GU, ISLEFT, ISRGHT, RTL, $ RTOL, S1, S2, SAFMIN, SGNDEF, SIGMA, SPDIAM, $ TAU, TMP, TMP1 * .. * .. Local Arrays .. INTEGER ISEED( 4 ) * .. * .. External Functions .. LOGICAL LSAME REAL SLAMCH EXTERNAL SLAMCH, LSAME * .. * .. External Subroutines .. EXTERNAL SCOPY, SLARNV, SLARRA, SLARRB, SLARRC, SLARRD, $ SLASQ2 * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN * .. * .. Executable Statements .. * INFO = 0 * * Decode RANGE * IF( LSAME( RANGE, 'A' ) ) THEN IRANGE = ALLRNG ELSE IF( LSAME( RANGE, 'V' ) ) THEN IRANGE = VALRNG ELSE IF( LSAME( RANGE, 'I' ) ) THEN IRANGE = INDRNG END IF M = 0 * Get machine constants SAFMIN = SLAMCH( 'S' ) EPS = SLAMCH( 'P' ) * Set parameters RTL = HNDRD*EPS * If one were ever to ask for less initial precision in BSRTOL, * one should keep in mind that for the subset case, the extremal * eigenvalues must be at least as accurate as the current setting * (eigenvalues in the middle need not as much accuracy) BSRTOL = SQRT(EPS)*(0.5E-3) * Treat case of 1x1 matrix for quick return IF( N.EQ.1 ) THEN IF( (IRANGE.EQ.ALLRNG).OR. $ ((IRANGE.EQ.VALRNG).AND.(D(1).GT.VL).AND.(D(1).LE.VU)).OR. $ ((IRANGE.EQ.INDRNG).AND.(IL.EQ.1).AND.(IU.EQ.1)) ) THEN M = 1 W(1) = D(1) * The computation error of the eigenvalue is zero WERR(1) = ZERO WGAP(1) = ZERO IBLOCK( 1 ) = 1 INDEXW( 1 ) = 1 GERS(1) = D( 1 ) GERS(2) = D( 1 ) ENDIF * store the shift for the initial RRR, which is zero in this case E(1) = ZERO RETURN END IF * General case: tridiagonal matrix of order > 1 * * Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter. * Compute maximum off-diagonal entry and pivmin. GL = D(1) GU = D(1) EOLD = ZERO EMAX = ZERO E(N) = ZERO DO 5 I = 1,N WERR(I) = ZERO WGAP(I) = ZERO EABS = ABS( E(I) ) IF( EABS .GE. EMAX ) THEN EMAX = EABS END IF TMP1 = EABS + EOLD GERS( 2*I-1) = D(I) - TMP1 GL = MIN( GL, GERS( 2*I - 1)) GERS( 2*I ) = D(I) + TMP1 GU = MAX( GU, GERS(2*I) ) EOLD = EABS 5 CONTINUE * The minimum pivot allowed in the Sturm sequence for T PIVMIN = SAFMIN * MAX( ONE, EMAX**2 ) * Compute spectral diameter. The Gerschgorin bounds give an * estimate that is wrong by at most a factor of SQRT(2) SPDIAM = GU - GL * Compute splitting points CALL SLARRA( N, D, E, E2, SPLTOL, SPDIAM, $ NSPLIT, ISPLIT, IINFO ) * Can force use of bisection instead of faster DQDS. * Option left in the code for future multisection work. FORCEB = .FALSE. * Initialize USEDQD, DQDS should be used for ALLRNG unless someone * explicitly wants bisection. USEDQD = (( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB)) IF( (IRANGE.EQ.ALLRNG) .AND. (.NOT. FORCEB) ) THEN * Set interval [VL,VU] that contains all eigenvalues VL = GL VU = GU ELSE * We call SLARRD to find crude approximations to the eigenvalues * in the desired range. In case IRANGE = INDRNG, we also obtain the * interval (VL,VU] that contains all the wanted eigenvalues. * An interval [LEFT,RIGHT] has converged if * RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT)) * SLARRD needs a WORK of size 4*N, IWORK of size 3*N CALL SLARRD( RANGE, 'B', N, VL, VU, IL, IU, GERS, $ BSRTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT, $ MM, W, WERR, VL, VU, IBLOCK, INDEXW, $ WORK, IWORK, IINFO ) IF( IINFO.NE.0 ) THEN INFO = -1 RETURN ENDIF * Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0 DO 14 I = MM+1,N W( I ) = ZERO WERR( I ) = ZERO IBLOCK( I ) = 0 INDEXW( I ) = 0 14 CONTINUE END IF *** * Loop over unreduced blocks IBEGIN = 1 WBEGIN = 1 DO 170 JBLK = 1, NSPLIT IEND = ISPLIT( JBLK ) IN = IEND - IBEGIN + 1 * 1 X 1 block IF( IN.EQ.1 ) THEN IF( (IRANGE.EQ.ALLRNG).OR.( (IRANGE.EQ.VALRNG).AND. $ ( D( IBEGIN ).GT.VL ).AND.( D( IBEGIN ).LE.VU ) ) $ .OR. ( (IRANGE.EQ.INDRNG).AND.(IBLOCK(WBEGIN).EQ.JBLK)) $ ) THEN M = M + 1 W( M ) = D( IBEGIN ) WERR(M) = ZERO * The gap for a single block doesn't matter for the later * algorithm and is assigned an arbitrary large value WGAP(M) = ZERO IBLOCK( M ) = JBLK INDEXW( M ) = 1 WBEGIN = WBEGIN + 1 ENDIF * E( IEND ) holds the shift for the initial RRR E( IEND ) = ZERO IBEGIN = IEND + 1 GO TO 170 END IF * * Blocks of size larger than 1x1 * * E( IEND ) will hold the shift for the initial RRR, for now set it =0 E( IEND ) = ZERO * * Find local outer bounds GL,GU for the block GL = D(IBEGIN) GU = D(IBEGIN) DO 15 I = IBEGIN , IEND GL = MIN( GERS( 2*I-1 ), GL ) GU = MAX( GERS( 2*I ), GU ) 15 CONTINUE SPDIAM = GU - GL IF(.NOT. ((IRANGE.EQ.ALLRNG).AND.(.NOT.FORCEB)) ) THEN * Count the number of eigenvalues in the current block. MB = 0 DO 20 I = WBEGIN,MM IF( IBLOCK(I).EQ.JBLK ) THEN MB = MB+1 ELSE GOTO 21 ENDIF 20 CONTINUE 21 CONTINUE IF( MB.EQ.0) THEN * No eigenvalue in the current block lies in the desired range * E( IEND ) holds the shift for the initial RRR E( IEND ) = ZERO IBEGIN = IEND + 1 GO TO 170 ELSE * Decide whether dqds or bisection is more efficient USEDQD = ( (MB .GT. FAC*IN) .AND. (.NOT.FORCEB) ) WEND = WBEGIN + MB - 1 * Calculate gaps for the current block * In later stages, when representations for individual * eigenvalues are different, we use SIGMA = E( IEND ). SIGMA = ZERO DO 30 I = WBEGIN, WEND - 1 WGAP( I ) = MAX( ZERO, $ W(I+1)-WERR(I+1) - (W(I)+WERR(I)) ) 30 CONTINUE WGAP( WEND ) = MAX( ZERO, $ VU - SIGMA - (W( WEND )+WERR( WEND ))) * Find local index of the first and last desired evalue. INDL = INDEXW(WBEGIN) INDU = INDEXW( WEND ) ENDIF ENDIF IF(( (IRANGE.EQ.ALLRNG) .AND. (.NOT. FORCEB) ).OR.USEDQD) THEN * Case of DQDS * Find approximations to the extremal eigenvalues of the block CALL SLARRK( IN, 1, GL, GU, D(IBEGIN), $ E2(IBEGIN), PIVMIN, RTL, TMP, TMP1, IINFO ) IF( IINFO.NE.0 ) THEN INFO = -1 RETURN ENDIF ISLEFT = MAX(GL, TMP - TMP1 $ - HNDRD * EPS* ABS(TMP - TMP1)) CALL SLARRK( IN, IN, GL, GU, D(IBEGIN), $ E2(IBEGIN), PIVMIN, RTL, TMP, TMP1, IINFO ) IF( IINFO.NE.0 ) THEN INFO = -1 RETURN ENDIF ISRGHT = MIN(GU, TMP + TMP1 $ + HNDRD * EPS * ABS(TMP + TMP1)) * Improve the estimate of the spectral diameter SPDIAM = ISRGHT - ISLEFT ELSE * Case of bisection * Find approximations to the wanted extremal eigenvalues ISLEFT = MAX(GL, W(WBEGIN) - WERR(WBEGIN) $ - HNDRD * EPS*ABS(W(WBEGIN)- WERR(WBEGIN) )) ISRGHT = MIN(GU,W(WEND) + WERR(WEND) $ + HNDRD * EPS * ABS(W(WEND)+ WERR(WEND))) ENDIF * Decide whether the base representation for the current block * L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I * should be on the left or the right end of the current block. * The strategy is to shift to the end which is "more populated" * Furthermore, decide whether to use DQDS for the computation of * the eigenvalue approximations at the end of SLARRE or bisection. * dqds is chosen if all eigenvalues are desired or the number of * eigenvalues to be computed is large compared to the blocksize. IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN * If all the eigenvalues have to be computed, we use dqd USEDQD = .TRUE. * INDL is the local index of the first eigenvalue to compute INDL = 1 INDU = IN * MB = number of eigenvalues to compute MB = IN WEND = WBEGIN + MB - 1 * Define 1/4 and 3/4 points of the spectrum S1 = ISLEFT + FOURTH * SPDIAM S2 = ISRGHT - FOURTH * SPDIAM ELSE * SLARRD has computed IBLOCK and INDEXW for each eigenvalue * approximation. * choose sigma IF( USEDQD ) THEN S1 = ISLEFT + FOURTH * SPDIAM S2 = ISRGHT - FOURTH * SPDIAM ELSE TMP = MIN(ISRGHT,VU) - MAX(ISLEFT,VL) S1 = MAX(ISLEFT,VL) + FOURTH * TMP S2 = MIN(ISRGHT,VU) - FOURTH * TMP ENDIF ENDIF * Compute the negcount at the 1/4 and 3/4 points IF(MB.GT.1) THEN CALL SLARRC( 'T', IN, S1, S2, D(IBEGIN), $ E(IBEGIN), PIVMIN, CNT, CNT1, CNT2, IINFO) ENDIF IF(MB.EQ.1) THEN SIGMA = GL SGNDEF = ONE ELSEIF( CNT1 - INDL .GE. INDU - CNT2 ) THEN IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN SIGMA = MAX(ISLEFT,GL) ELSEIF( USEDQD ) THEN * use Gerschgorin bound as shift to get pos def matrix * for dqds SIGMA = ISLEFT ELSE * use approximation of the first desired eigenvalue of the * block as shift SIGMA = MAX(ISLEFT,VL) ENDIF SGNDEF = ONE ELSE IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN SIGMA = MIN(ISRGHT,GU) ELSEIF( USEDQD ) THEN * use Gerschgorin bound as shift to get neg def matrix * for dqds SIGMA = ISRGHT ELSE * use approximation of the first desired eigenvalue of the * block as shift SIGMA = MIN(ISRGHT,VU) ENDIF SGNDEF = -ONE ENDIF * An initial SIGMA has been chosen that will be used for computing * T - SIGMA I = L D L^T * Define the increment TAU of the shift in case the initial shift * needs to be refined to obtain a factorization with not too much * element growth. IF( USEDQD ) THEN * The initial SIGMA was to the outer end of the spectrum * the matrix is definite and we need not retreat. TAU = SPDIAM*EPS*N + TWO*PIVMIN TAU = MAX( TAU,TWO*EPS*ABS(SIGMA) ) ELSE IF(MB.GT.1) THEN CLWDTH = W(WEND) + WERR(WEND) - W(WBEGIN) - WERR(WBEGIN) AVGAP = ABS(CLWDTH / REAL(WEND-WBEGIN)) IF( SGNDEF.EQ.ONE ) THEN TAU = HALF*MAX(WGAP(WBEGIN),AVGAP) TAU = MAX(TAU,WERR(WBEGIN)) ELSE TAU = HALF*MAX(WGAP(WEND-1),AVGAP) TAU = MAX(TAU,WERR(WEND)) ENDIF ELSE TAU = WERR(WBEGIN) ENDIF ENDIF * DO 80 IDUM = 1, MAXTRY * Compute L D L^T factorization of tridiagonal matrix T - sigma I. * Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of * pivots in WORK(2*IN+1:3*IN) DPIVOT = D( IBEGIN ) - SIGMA WORK( 1 ) = DPIVOT DMAX = ABS( WORK(1) ) J = IBEGIN DO 70 I = 1, IN - 1 WORK( 2*IN+I ) = ONE / WORK( I ) TMP = E( J )*WORK( 2*IN+I ) WORK( IN+I ) = TMP DPIVOT = ( D( J+1 )-SIGMA ) - TMP*E( J ) WORK( I+1 ) = DPIVOT DMAX = MAX( DMAX, ABS(DPIVOT) ) J = J + 1 70 CONTINUE * check for element growth IF( DMAX .GT. MAXGROWTH*SPDIAM ) THEN NOREP = .TRUE. ELSE NOREP = .FALSE. ENDIF IF( USEDQD .AND. .NOT.NOREP ) THEN * Ensure the definiteness of the representation * All entries of D (of L D L^T) must have the same sign DO 71 I = 1, IN TMP = SGNDEF*WORK( I ) IF( TMP.LT.ZERO ) NOREP = .TRUE. 71 CONTINUE ENDIF IF(NOREP) THEN * Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin * shift which makes the matrix definite. So we should end up * here really only in the case of IRANGE = VALRNG or INDRNG. IF( IDUM.EQ.MAXTRY-1 ) THEN IF( SGNDEF.EQ.ONE ) THEN * The fudged Gerschgorin shift should succeed SIGMA = $ GL - FUDGE*SPDIAM*EPS*N - FUDGE*TWO*PIVMIN ELSE SIGMA = $ GU + FUDGE*SPDIAM*EPS*N + FUDGE*TWO*PIVMIN END IF ELSE SIGMA = SIGMA - SGNDEF * TAU TAU = TWO * TAU END IF ELSE * an initial RRR is found GO TO 83 END IF 80 CONTINUE * if the program reaches this point, no base representation could be * found in MAXTRY iterations. INFO = 2 RETURN 83 CONTINUE * At this point, we have found an initial base representation * T - SIGMA I = L D L^T with not too much element growth. * Store the shift. E( IEND ) = SIGMA * Store D and L. CALL SCOPY( IN, WORK, 1, D( IBEGIN ), 1 ) CALL SCOPY( IN-1, WORK( IN+1 ), 1, E( IBEGIN ), 1 ) IF(MB.GT.1 ) THEN * * Perturb each entry of the base representation by a small * (but random) relative amount to overcome difficulties with * glued matrices. * DO 122 I = 1, 4 ISEED( I ) = 1 122 CONTINUE CALL SLARNV(2, ISEED, 2*IN-1, WORK(1)) DO 125 I = 1,IN-1 D(IBEGIN+I-1) = D(IBEGIN+I-1)*(ONE+EPS*PERT*WORK(I)) E(IBEGIN+I-1) = E(IBEGIN+I-1)*(ONE+EPS*PERT*WORK(IN+I)) 125 CONTINUE D(IEND) = D(IEND)*(ONE+EPS*FOUR*WORK(IN)) * ENDIF * * Don't update the Gerschgorin intervals because keeping track * of the updates would be too much work in SLARRV. * We update W instead and use it to locate the proper Gerschgorin * intervals. * Compute the required eigenvalues of L D L' by bisection or dqds IF ( .NOT.USEDQD ) THEN * If SLARRD has been used, shift the eigenvalue approximations * according to their representation. This is necessary for * a uniform SLARRV since dqds computes eigenvalues of the * shifted representation. In SLARRV, W will always hold the * UNshifted eigenvalue approximation. DO 134 J=WBEGIN,WEND W(J) = W(J) - SIGMA WERR(J) = WERR(J) + ABS(W(J)) * EPS 134 CONTINUE * call SLARRB to reduce eigenvalue error of the approximations * from SLARRD DO 135 I = IBEGIN, IEND-1 WORK( I ) = D( I ) * E( I )**2 135 CONTINUE * use bisection to find EV from INDL to INDU CALL SLARRB(IN, D(IBEGIN), WORK(IBEGIN), $ INDL, INDU, RTOL1, RTOL2, INDL-1, $ W(WBEGIN), WGAP(WBEGIN), WERR(WBEGIN), $ WORK( 2*N+1 ), IWORK, PIVMIN, SPDIAM, $ IN, IINFO ) IF( IINFO .NE. 0 ) THEN INFO = -4 RETURN END IF * SLARRB computes all gaps correctly except for the last one * Record distance to VU/GU WGAP( WEND ) = MAX( ZERO, $ ( VU-SIGMA ) - ( W( WEND ) + WERR( WEND ) ) ) DO 138 I = INDL, INDU M = M + 1 IBLOCK(M) = JBLK INDEXW(M) = I 138 CONTINUE ELSE * Call dqds to get all eigs (and then possibly delete unwanted * eigenvalues). * Note that dqds finds the eigenvalues of the L D L^T representation * of T to high relative accuracy. High relative accuracy * might be lost when the shift of the RRR is subtracted to obtain * the eigenvalues of T. However, T is not guaranteed to define its * eigenvalues to high relative accuracy anyway. * Set RTOL to the order of the tolerance used in SLASQ2 * This is an ESTIMATED error, the worst case bound is 4*N*EPS * which is usually too large and requires unnecessary work to be * done by bisection when computing the eigenvectors RTOL = LOG(REAL(IN)) * FOUR * EPS J = IBEGIN DO 140 I = 1, IN - 1 WORK( 2*I-1 ) = ABS( D( J ) ) WORK( 2*I ) = E( J )*E( J )*WORK( 2*I-1 ) J = J + 1 140 CONTINUE WORK( 2*IN-1 ) = ABS( D( IEND ) ) WORK( 2*IN ) = ZERO CALL SLASQ2( IN, WORK, IINFO ) IF( IINFO .NE. 0 ) THEN * If IINFO = -5 then an index is part of a tight cluster * and should be changed. The index is in IWORK(1) and the * gap is in WORK(N+1) INFO = -5 RETURN ELSE * Test that all eigenvalues are positive as expected DO 149 I = 1, IN IF( WORK( I ).LT.ZERO ) THEN INFO = -6 RETURN ENDIF 149 CONTINUE END IF IF( SGNDEF.GT.ZERO ) THEN DO 150 I = INDL, INDU M = M + 1 W( M ) = WORK( IN-I+1 ) IBLOCK( M ) = JBLK INDEXW( M ) = I 150 CONTINUE ELSE DO 160 I = INDL, INDU M = M + 1 W( M ) = -WORK( I ) IBLOCK( M ) = JBLK INDEXW( M ) = I 160 CONTINUE END IF DO 165 I = M - MB + 1, M * the value of RTOL below should be the tolerance in SLASQ2 WERR( I ) = RTOL * ABS( W(I) ) 165 CONTINUE DO 166 I = M - MB + 1, M - 1 * compute the right gap between the intervals WGAP( I ) = MAX( ZERO, $ W(I+1)-WERR(I+1) - (W(I)+WERR(I)) ) 166 CONTINUE WGAP( M ) = MAX( ZERO, $ ( VU-SIGMA ) - ( W( M ) + WERR( M ) ) ) END IF * proceed with next block IBEGIN = IEND + 1 WBEGIN = WEND + 1 170 CONTINUE * RETURN * * end of SLARRE * END