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C DLAQR0    SOURCE    BP208322  20/09/18    21:15:58     10718          *> \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*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr0.f">*> [TGZ]&lt;/a>*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr0.f">*> [ZIP]&lt;/a>*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr0.f">*> [TXT]&lt;/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 ..*       REAL*8   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 REAL*8 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 REAL*8 array, dimension (IHI)*> \endverbatim*>*> \param[out] WI*> \verbatim*> WI is REAL*8 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 REAL*8 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 REAL*8 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       IMPLICIT INTEGER(I-N)      IMPLICIT REAL*8(A-H,O-Z)**     .. Scalar Arguments ..      INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N      LOGICAL            WANTT, WANTZ*     ..*     .. Array Arguments ..      REAL*8   H( LDH, * ), WI( * ), WORK( * ), WR( * ),     $Z( LDZ, * )* ..** ================================================================** .. Parameters ..** ==== Matrices of order NTINY or smaller must be processed by* . DLAHQR because of insufficient subdiagonal scratch space.* . (This is a hard limit.) ==== INTEGER NTINY PARAMETER ( NTINY = 11 )** ==== Exceptional deflation windows: try to cure rare* . slow convergence by varying the size of the* . deflation window after KEXNW iterations. ==== INTEGER KEXNW PARAMETER ( KEXNW = 5 )** ==== Exceptional shifts: try to cure rare slow convergence* . with ad-hoc exceptional shifts every KEXSH iterations.* . ==== INTEGER KEXSH PARAMETER ( KEXSH = 6 )** ==== The constants WILK1 and WILK2 are used to form the* . exceptional shifts. ==== REAL*8 WILK1, WILK2 PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 ) REAL*8 ZERO, ONE PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )* ..* .. Local Scalars .. REAL*8 AA, BB, CC, CS, DD, SN, SS, SWAP INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,$                   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 ..      REAL*8   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

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