numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
| Name | Size | Mode | |
| .. | |||
| lapack/SRC/dlaqr0.f | 26809B | -rw-r--r-- |
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*> \brief \b DLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DLAQR0 + dependencies *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr0.f"> *> [TGZ]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr0.f"> *> [ZIP]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr0.f"> *> [TXT]</a> *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, * ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N * LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. * DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), * $ Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DLAQR0 computes the eigenvalues of a Hessenberg matrix H *> and, optionally, the matrices T and Z from the Schur decomposition *> H = Z T Z**T, where T is an upper quasi-triangular matrix (the *> Schur form), and Z is the orthogonal matrix of Schur vectors. *> *> Optionally Z may be postmultiplied into an input orthogonal *> matrix Q so that this routine can give the Schur factorization *> of a matrix A which has been reduced to the Hessenberg form H *> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. *> \endverbatim * * Arguments: * ========== * *> \param[in] WANTT *> \verbatim *> WANTT is LOGICAL *> = .TRUE. : the full Schur form T is required; *> = .FALSE.: only eigenvalues are required. *> \endverbatim *> *> \param[in] WANTZ *> \verbatim *> WANTZ is LOGICAL *> = .TRUE. : the matrix of Schur vectors Z is required; *> = .FALSE.: Schur vectors are not required. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix H. N >= 0. *> \endverbatim *> *> \param[in] ILO *> \verbatim *> ILO is INTEGER *> \endverbatim *> *> \param[in] IHI *> \verbatim *> IHI is INTEGER *> It is assumed that H is already upper triangular in rows *> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1, *> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a *> previous call to DGEBAL, and then passed to DGEHRD when the *> matrix output by DGEBAL is reduced to Hessenberg form. *> Otherwise, ILO and IHI should be set to 1 and N, *> respectively. If N > 0, then 1 <= ILO <= IHI <= N. *> If N = 0, then ILO = 1 and IHI = 0. *> \endverbatim *> *> \param[in,out] H *> \verbatim *> H is DOUBLE PRECISION array, dimension (LDH,N) *> On entry, the upper Hessenberg matrix H. *> On exit, if INFO = 0 and WANTT is .TRUE., then H contains *> the upper quasi-triangular matrix T from the Schur *> decomposition (the Schur form); 2-by-2 diagonal blocks *> (corresponding to complex conjugate pairs of eigenvalues) *> are returned in standard form, with H(i,i) = H(i+1,i+1) *> and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is *> .FALSE., then the contents of H are unspecified on exit. *> (The output value of H when INFO > 0 is given under the *> description of INFO below.) *> *> This subroutine may explicitly set H(i,j) = 0 for i > j and *> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. *> \endverbatim *> *> \param[in] LDH *> \verbatim *> LDH is INTEGER *> The leading dimension of the array H. LDH >= max(1,N). *> \endverbatim *> *> \param[out] WR *> \verbatim *> WR is DOUBLE PRECISION array, dimension (IHI) *> \endverbatim *> *> \param[out] WI *> \verbatim *> WI is DOUBLE PRECISION array, dimension (IHI) *> The real and imaginary parts, respectively, of the computed *> eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) *> and WI(ILO:IHI). If two eigenvalues are computed as a *> complex conjugate pair, they are stored in consecutive *> elements of WR and WI, say the i-th and (i+1)th, with *> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then *> the eigenvalues are stored in the same order as on the *> diagonal of the Schur form returned in H, with *> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal *> block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and *> WI(i+1) = -WI(i). *> \endverbatim *> *> \param[in] ILOZ *> \verbatim *> ILOZ is INTEGER *> \endverbatim *> *> \param[in] IHIZ *> \verbatim *> IHIZ is INTEGER *> Specify the rows of Z to which transformations must be *> applied if WANTZ is .TRUE.. *> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. *> \endverbatim *> *> \param[in,out] Z *> \verbatim *> Z is DOUBLE PRECISION array, dimension (LDZ,IHI) *> If WANTZ is .FALSE., then Z is not referenced. *> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is *> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the *> orthogonal Schur factor of H(ILO:IHI,ILO:IHI). *> (The output value of Z when INFO > 0 is given under *> the description of INFO below.) *> \endverbatim *> *> \param[in] LDZ *> \verbatim *> LDZ is INTEGER *> The leading dimension of the array Z. if WANTZ is .TRUE. *> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension LWORK *> On exit, if LWORK = -1, WORK(1) returns an estimate of *> the optimal value for LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. LWORK >= max(1,N) *> is sufficient, but LWORK typically as large as 6*N may *> be required for optimal performance. A workspace query *> to determine the optimal workspace size is recommended. *> *> If LWORK = -1, then DLAQR0 does a workspace query. *> In this case, DLAQR0 checks the input parameters and *> estimates the optimal workspace size for the given *> values of N, ILO and IHI. The estimate is returned *> in WORK(1). No error message related to LWORK is *> issued by XERBLA. Neither H nor Z are accessed. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> > 0: if INFO = i, DLAQR0 failed to compute all of *> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR *> and WI contain those eigenvalues which have been *> successfully computed. (Failures are rare.) *> *> If INFO > 0 and WANT is .FALSE., then on exit, *> the remaining unconverged eigenvalues are the eigen- *> values of the upper Hessenberg matrix rows and *> columns ILO through INFO of the final, output *> value of H. *> *> If INFO > 0 and WANTT is .TRUE., then on exit *> *> (*) (initial value of H)*U = U*(final value of H) *> *> where U is an orthogonal matrix. The final *> value of H is upper Hessenberg and quasi-triangular *> in rows and columns INFO+1 through IHI. *> *> If INFO > 0 and WANTZ is .TRUE., then on exit *> *> (final value of Z(ILO:IHI,ILOZ:IHIZ) *> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U *> *> where U is the orthogonal matrix in (*) (regard- *> less of the value of WANTT.) *> *> If INFO > 0 and WANTZ is .FALSE., then Z is not *> accessed. *> \endverbatim * *> \par Contributors: * ================== *> *> Karen Braman and Ralph Byers, Department of Mathematics, *> University of Kansas, USA * *> \par References: * ================ *> *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages *> 929--947, 2002. *> \n *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal *> of Matrix Analysis, volume 23, pages 948--973, 2002. * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup laqr0 * * ===================================================================== SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, $ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO ) * * -- LAPACK auxiliary routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), $ Z( LDZ, * ) * .. * * ================================================================ * * .. Parameters .. * * ==== Matrices of order NTINY or smaller must be processed by * . DLAHQR because of insufficient subdiagonal scratch space. * . (This is a hard limit.) ==== INTEGER NTINY PARAMETER ( NTINY = 15 ) * * ==== Exceptional deflation windows: try to cure rare * . slow convergence by varying the size of the * . deflation window after KEXNW iterations. ==== INTEGER KEXNW PARAMETER ( KEXNW = 5 ) * * ==== Exceptional shifts: try to cure rare slow convergence * . with ad-hoc exceptional shifts every KEXSH iterations. * . ==== INTEGER KEXSH PARAMETER ( KEXSH = 6 ) * * ==== The constants WILK1 and WILK2 are used to form the * . exceptional shifts. ==== DOUBLE PRECISION WILK1, WILK2 PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 ) * .. * .. Local Scalars .. DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS, $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS, $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS, $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD LOGICAL SORTED CHARACTER JBCMPZ*2 * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Local Arrays .. DOUBLE PRECISION ZDUM( 1, 1 ) * .. * .. External Subroutines .. EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR3, DLAQR4, $ DLAQR5 * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD * .. * .. Executable Statements .. INFO = 0 * * ==== Quick return for N = 0: nothing to do. ==== * IF( N.EQ.0 ) THEN WORK( 1 ) = ONE RETURN END IF * IF( N.LE.NTINY ) THEN * * ==== Tiny matrices must use DLAHQR. ==== * LWKOPT = 1 IF( LWORK.NE.-1 ) $ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, $ ILOZ, IHIZ, Z, LDZ, INFO ) ELSE * * ==== Use small bulge multi-shift QR with aggressive early * . deflation on larger-than-tiny matrices. ==== * * ==== Hope for the best. ==== * INFO = 0 * * ==== Set up job flags for ILAENV. ==== * IF( WANTT ) THEN JBCMPZ( 1: 1 ) = 'S' ELSE JBCMPZ( 1: 1 ) = 'E' END IF IF( WANTZ ) THEN JBCMPZ( 2: 2 ) = 'V' ELSE JBCMPZ( 2: 2 ) = 'N' END IF * * ==== NWR = recommended deflation window size. At this * . point, N .GT. NTINY = 15, so there is enough * . subdiagonal workspace for NWR.GE.2 as required. * . (In fact, there is enough subdiagonal space for * . NWR.GE.4.) ==== * NWR = ILAENV( 13, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) NWR = MAX( 2, NWR ) NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR ) * * ==== NSR = recommended number of simultaneous shifts. * . At this point N .GT. NTINY = 15, so there is at * . enough subdiagonal workspace for NSR to be even * . and greater than or equal to two as required. ==== * NSR = ILAENV( 15, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO ) NSR = MAX( 2, NSR-MOD( NSR, 2 ) ) * * ==== Estimate optimal workspace ==== * * ==== Workspace query call to DLAQR3 ==== * CALL DLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ, $ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH, $ N, H, LDH, WORK, -1 ) * * ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ==== * LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) ) * * ==== Quick return in case of workspace query. ==== * IF( LWORK.EQ.-1 ) THEN WORK( 1 ) = DBLE( LWKOPT ) RETURN END IF * * ==== DLAHQR/DLAQR0 crossover point ==== * NMIN = ILAENV( 12, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) NMIN = MAX( NTINY, NMIN ) * * ==== Nibble crossover point ==== * NIBBLE = ILAENV( 14, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) NIBBLE = MAX( 0, NIBBLE ) * * ==== Accumulate reflections during ttswp? Use block * . 2-by-2 structure during matrix-matrix multiply? ==== * KACC22 = ILAENV( 16, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) KACC22 = MAX( 0, KACC22 ) KACC22 = MIN( 2, KACC22 ) * * ==== NWMAX = the largest possible deflation window for * . which there is sufficient workspace. ==== * NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 ) NW = NWMAX * * ==== NSMAX = the Largest number of simultaneous shifts * . for which there is sufficient workspace. ==== * NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 ) NSMAX = NSMAX - MOD( NSMAX, 2 ) * * ==== NDFL: an iteration count restarted at deflation. ==== * NDFL = 1 * * ==== ITMAX = iteration limit ==== * ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) ) * * ==== Last row and column in the active block ==== * KBOT = IHI * * ==== Main Loop ==== * DO 80 IT = 1, ITMAX * * ==== Done when KBOT falls below ILO ==== * IF( KBOT.LT.ILO ) $ GO TO 90 * * ==== Locate active block ==== * DO 10 K = KBOT, ILO + 1, -1 IF( H( K, K-1 ).EQ.ZERO ) $ GO TO 20 10 CONTINUE K = ILO 20 CONTINUE KTOP = K * * ==== Select deflation window size: * . Typical Case: * . If possible and advisable, nibble the entire * . active block. If not, use size MIN(NWR,NWMAX) * . or MIN(NWR+1,NWMAX) depending upon which has * . the smaller corresponding subdiagonal entry * . (a heuristic). * . * . Exceptional Case: * . If there have been no deflations in KEXNW or * . more iterations, then vary the deflation window * . size. At first, because, larger windows are, * . in general, more powerful than smaller ones, * . rapidly increase the window to the maximum possible. * . Then, gradually reduce the window size. ==== * NH = KBOT - KTOP + 1 NWUPBD = MIN( NH, NWMAX ) IF( NDFL.LT.KEXNW ) THEN NW = MIN( NWUPBD, NWR ) ELSE NW = MIN( NWUPBD, 2*NW ) END IF IF( NW.LT.NWMAX ) THEN IF( NW.GE.NH-1 ) THEN NW = NH ELSE KWTOP = KBOT - NW + 1 IF( ABS( H( KWTOP, KWTOP-1 ) ).GT. $ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1 END IF END IF IF( NDFL.LT.KEXNW ) THEN NDEC = -1 ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN NDEC = NDEC + 1 IF( NW-NDEC.LT.2 ) $ NDEC = 0 NW = NW - NDEC END IF * * ==== Aggressive early deflation: * . split workspace under the subdiagonal into * . - an nw-by-nw work array V in the lower * . left-hand-corner, * . - an NW-by-at-least-NW-but-more-is-better * . (NW-by-NHO) horizontal work array along * . the bottom edge, * . - an at-least-NW-but-more-is-better (NHV-by-NW) * . vertical work array along the left-hand-edge. * . ==== * KV = N - NW + 1 KT = NW + 1 NHO = ( N-NW-1 ) - KT + 1 KWV = NW + 2 NVE = ( N-NW ) - KWV + 1 * * ==== Aggressive early deflation ==== * CALL DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, $ ILOZ, $ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH, $ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, $ WORK, LWORK ) * * ==== Adjust KBOT accounting for new deflations. ==== * KBOT = KBOT - LD * * ==== KS points to the shifts. ==== * KS = KBOT - LS + 1 * * ==== Skip an expensive QR sweep if there is a (partly * . heuristic) reason to expect that many eigenvalues * . will deflate without it. Here, the QR sweep is * . skipped if many eigenvalues have just been deflated * . or if the remaining active block is small. * IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT- $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN * * ==== NS = nominal number of simultaneous shifts. * . This may be lowered (slightly) if DLAQR3 * . did not provide that many shifts. ==== * NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) ) NS = NS - MOD( NS, 2 ) * * ==== If there have been no deflations * . in a multiple of KEXSH iterations, * . then try exceptional shifts. * . Otherwise use shifts provided by * . DLAQR3 above or from the eigenvalues * . of a trailing principal submatrix. ==== * IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN KS = KBOT - NS + 1 DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2 SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) ) AA = WILK1*SS + H( I, I ) BB = SS CC = WILK2*SS DD = AA CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), $ WI( I-1 ), $ WR( I ), WI( I ), CS, SN ) 30 CONTINUE IF( KS.EQ.KTOP ) THEN WR( KS+1 ) = H( KS+1, KS+1 ) WI( KS+1 ) = ZERO WR( KS ) = WR( KS+1 ) WI( KS ) = WI( KS+1 ) END IF ELSE * * ==== Got NS/2 or fewer shifts? Use DLAQR4 or * . DLAHQR on a trailing principal submatrix to * . get more. (Since NS.LE.NSMAX.LE.(N-3)/6, * . there is enough space below the subdiagonal * . to fit an NS-by-NS scratch array.) ==== * IF( KBOT-KS+1.LE.NS / 2 ) THEN KS = KBOT - NS + 1 KT = N - NS + 1 CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH, $ H( KT, 1 ), LDH ) IF( NS.GT.NMIN ) THEN CALL DLAQR4( .false., .false., NS, 1, NS, $ H( KT, 1 ), LDH, WR( KS ), $ WI( KS ), 1, 1, ZDUM, 1, WORK, $ LWORK, INF ) ELSE CALL DLAHQR( .false., .false., NS, 1, NS, $ H( KT, 1 ), LDH, WR( KS ), $ WI( KS ), 1, 1, ZDUM, 1, INF ) END IF KS = KS + INF * * ==== In case of a rare QR failure use * . eigenvalues of the trailing 2-by-2 * . principal submatrix. ==== * IF( KS.GE.KBOT ) THEN AA = H( KBOT-1, KBOT-1 ) CC = H( KBOT, KBOT-1 ) BB = H( KBOT-1, KBOT ) DD = H( KBOT, KBOT ) CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ), $ WI( KBOT-1 ), WR( KBOT ), $ WI( KBOT ), CS, SN ) KS = KBOT - 1 END IF END IF * IF( KBOT-KS+1.GT.NS ) THEN * * ==== Sort the shifts (Helps a little) * . Bubble sort keeps complex conjugate * . pairs together. ==== * SORTED = .false. DO 50 K = KBOT, KS + 1, -1 IF( SORTED ) $ GO TO 60 SORTED = .true. DO 40 I = KS, K - 1 IF( ABS( WR( I ) )+ABS( WI( I ) ).LT. $ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN SORTED = .false. * SWAP = WR( I ) WR( I ) = WR( I+1 ) WR( I+1 ) = SWAP * SWAP = WI( I ) WI( I ) = WI( I+1 ) WI( I+1 ) = SWAP END IF 40 CONTINUE 50 CONTINUE 60 CONTINUE END IF * * ==== Shuffle shifts into pairs of real shifts * . and pairs of complex conjugate shifts * . assuming complex conjugate shifts are * . already adjacent to one another. (Yes, * . they are.) ==== * DO 70 I = KBOT, KS + 2, -2 IF( WI( I ).NE.-WI( I-1 ) ) THEN * SWAP = WR( I ) WR( I ) = WR( I-1 ) WR( I-1 ) = WR( I-2 ) WR( I-2 ) = SWAP * SWAP = WI( I ) WI( I ) = WI( I-1 ) WI( I-1 ) = WI( I-2 ) WI( I-2 ) = SWAP END IF 70 CONTINUE END IF * * ==== If there are only two shifts and both are * . real, then use only one. ==== * IF( KBOT-KS+1.EQ.2 ) THEN IF( WI( KBOT ).EQ.ZERO ) THEN IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT. $ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN WR( KBOT-1 ) = WR( KBOT ) ELSE WR( KBOT ) = WR( KBOT-1 ) END IF END IF END IF * * ==== Use up to NS of the the smallest magnitude * . shifts. If there aren't NS shifts available, * . then use them all, possibly dropping one to * . make the number of shifts even. ==== * NS = MIN( NS, KBOT-KS+1 ) NS = NS - MOD( NS, 2 ) KS = KBOT - NS + 1 * * ==== Small-bulge multi-shift QR sweep: * . split workspace under the subdiagonal into * . - a KDU-by-KDU work array U in the lower * . left-hand-corner, * . - a KDU-by-at-least-KDU-but-more-is-better * . (KDU-by-NHo) horizontal work array WH along * . the bottom edge, * . - and an at-least-KDU-but-more-is-better-by-KDU * . (NVE-by-KDU) vertical work WV arrow along * . the left-hand-edge. ==== * KDU = 2*NS KU = N - KDU + 1 KWH = KDU + 1 NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1 KWV = KDU + 4 NVE = N - KDU - KWV + 1 * * ==== Small-bulge multi-shift QR sweep ==== * CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS, $ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z, $ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE, $ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH ) END IF * * ==== Note progress (or the lack of it). ==== * IF( LD.GT.0 ) THEN NDFL = 1 ELSE NDFL = NDFL + 1 END IF * * ==== End of main loop ==== 80 CONTINUE * * ==== Iteration limit exceeded. Set INFO to show where * . the problem occurred and exit. ==== * INFO = KBOT 90 CONTINUE END IF * * ==== Return the optimal value of LWORK. ==== * WORK( 1 ) = DBLE( LWKOPT ) * * ==== End of DLAQR0 ==== * END