#include "ESMF_LapackBlas.inc" !> \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 .GE. 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.GT.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.GT.0, then 1.LE.ILO.LE.IHI.LE.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).LT.0. If INFO = 0 and WANTT is !> .FALSE., then the contents of H are unspecified on exit. !> (The output value of H when INFO.GT.0 is given under the !> description of INFO below.) !> !> This subroutine may explicitly set H(i,j) = 0 for i.GT.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 .GE. 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) .GT. 0 and WI(i+1) .LT. 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 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. 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.GT.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.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.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 .GE. 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 !> .GT. 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 .GT. 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 .GT. 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 .GT. 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 .GT. 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. ! !> \date December 2016 ! !> \ingroup doubleOTHERauxiliary ! ! ===================================================================== SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, & ILOZ, IHIZ, Z, LDZ, WORK, LWORK, 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..-- ! December 2016 ! ! .. 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 = 11 ) ! ! ==== 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 = 11, so there is enough ! . subdiagonal workspace for NWR.GE.2 as required. ! . (In fact, there is enough subdiagonal space for ! . NWR.GE.3.) ==== ! 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 = 11, 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+6 ) / 9, 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+6 ) / 9, 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+6)/9, ! . 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 magnatiude ! . 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 = 3*NS - 3 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