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|
*> \brief \b CLAQR4 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 CLAQR4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/claqr4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/claqr4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/claqr4.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
* IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CLAQR4 implements one level of recursion for CLAQR0.
*> It is a complete implementation of the small bulge multi-shift
*> QR algorithm. It may be called by CLAQR0 and, for large enough
*> deflation window size, it may be called by CLAQR3. This
*> subroutine is identical to CLAQR0 except that it calls CLAQR2
*> instead of CLAQR3.
*>
*> CLAQR4 computes the eigenvalues of a Hessenberg matrix H
*> and, optionally, the matrices T and Z from the Schur decomposition
*> H = Z T Z**H, where T is an upper triangular matrix (the
*> Schur form), and Z is the unitary matrix of Schur vectors.
*>
*> Optionally Z may be postmultiplied into an input unitary
*> 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 unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.
*> \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 CGEBAL, and then passed to CGEHRD when the
*> matrix output by CGEBAL 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 COMPLEX 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 triangular matrix T from the Schur
*> decomposition (the Schur form). If INFO = 0 and WANT 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] W
*> \verbatim
*> W is COMPLEX array, dimension (N)
*> The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
*> in W(ILO:IHI). 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 W(i) = H(i,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 COMPLEX 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 COMPLEX 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 CLAQR4 does a workspace query.
*> In this case, CLAQR4 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, CLAQR4 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 a unitary matrix. The final
*> value of H is upper Hessenberg and 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 unitary matrix in (*) (regard-
*> less of the value of WANTT.)
*>
*> If INFO > 0 and WANTZ is .FALSE., then Z is not
*> accessed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complexOTHERauxiliary
*
*> \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.
*>
* =====================================================================
SUBROUTINE CLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, 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 ..
COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
* ================================================================
*
* .. Parameters ..
*
* ==== Matrices of order NTINY or smaller must be processed by
* . CLAHQR 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 constant WILK1 is used to form the exceptional
* . shifts. ====
REAL WILK1
PARAMETER ( WILK1 = 0.75e0 )
COMPLEX ZERO, ONE
PARAMETER ( ZERO = ( 0.0e0, 0.0e0 ),
$ ONE = ( 1.0e0, 0.0e0 ) )
REAL TWO
PARAMETER ( TWO = 2.0e0 )
* ..
* .. Local Scalars ..
COMPLEX AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2
REAL S
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 ..
COMPLEX ZDUM( 1, 1 )
* ..
* .. External Subroutines ..
EXTERNAL CLACPY, CLAHQR, CLAQR2, CLAQR5
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CMPLX, INT, MAX, MIN, MOD, REAL,
$ SQRT
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) )
* ..
* .. 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 CLAHQR. ====
*
LWKOPT = 1
IF( LWORK.NE.-1 )
$ CALL CLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, 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, 'CLAQR4', 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, 'CLAQR4', 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 CLAQR2 ====
*
CALL CLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H,
$ LDH, WORK, -1 )
*
* ==== Optimal workspace = MAX(CLAQR5, CLAQR2) ====
*
LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = CMPLX( LWKOPT, 0 )
RETURN
END IF
*
* ==== CLAHQR/CLAQR0 crossover point ====
*
NMIN = ILAENV( 12, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NMIN = MAX( NTINY, NMIN )
*
* ==== Nibble crossover point ====
*
NIBBLE = ILAENV( 14, 'CLAQR4', 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, 'CLAQR4', 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 70 IT = 1, ITMAX
*
* ==== Done when KBOT falls below ILO ====
*
IF( KBOT.LT.ILO )
$ GO TO 80
*
* ==== 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( CABS1( H( KWTOP, KWTOP-1 ) ).GT.
$ CABS1( 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 CLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, LS, LD, W, 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 CLAQR2
* . 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
* . CLAQR2 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, KS + 1, -2
W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) )
W( I-1 ) = W( I )
30 CONTINUE
ELSE
*
* ==== Got NS/2 or fewer shifts? Use CLAHQR
* . 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 CLACPY( 'A', NS, NS, H( KS, KS ), LDH,
$ H( KT, 1 ), LDH )
CALL CLAHQR( .false., .false., NS, 1, NS,
$ H( KT, 1 ), LDH, W( KS ), 1, 1, ZDUM,
$ 1, INF )
KS = KS + INF
*
* ==== In case of a rare QR failure use
* . eigenvalues of the trailing 2-by-2
* . principal submatrix. Scale to avoid
* . overflows, underflows and subnormals.
* . (The scale factor S can not be zero,
* . because H(KBOT,KBOT-1) is nonzero.) ====
*
IF( KS.GE.KBOT ) THEN
S = CABS1( H( KBOT-1, KBOT-1 ) ) +
$ CABS1( H( KBOT, KBOT-1 ) ) +
$ CABS1( H( KBOT-1, KBOT ) ) +
$ CABS1( H( KBOT, KBOT ) )
AA = H( KBOT-1, KBOT-1 ) / S
CC = H( KBOT, KBOT-1 ) / S
BB = H( KBOT-1, KBOT ) / S
DD = H( KBOT, KBOT ) / S
TR2 = ( AA+DD ) / TWO
DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC
RTDISC = SQRT( -DET )
W( KBOT-1 ) = ( TR2+RTDISC )*S
W( KBOT ) = ( TR2-RTDISC )*S
*
KS = KBOT - 1
END IF
END IF
*
IF( KBOT-KS+1.GT.NS ) THEN
*
* ==== Sort the shifts (Helps a little) ====
*
SORTED = .false.
DO 50 K = KBOT, KS + 1, -1
IF( SORTED )
$ GO TO 60
SORTED = .true.
DO 40 I = KS, K - 1
IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) )
$ THEN
SORTED = .false.
SWAP = W( I )
W( I ) = W( I+1 )
W( I+1 ) = SWAP
END IF
40 CONTINUE
50 CONTINUE
60 CONTINUE
END IF
END IF
*
* ==== If there are only two shifts, then use
* . only one. ====
*
IF( KBOT-KS+1.EQ.2 ) THEN
IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT.
$ CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
W( KBOT-1 ) = W( KBOT )
ELSE
W( KBOT ) = W( KBOT-1 )
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 CLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
$ W( 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 ====
70 CONTINUE
*
* ==== Iteration limit exceeded. Set INFO to show where
* . the problem occurred and exit. ====
*
INFO = KBOT
80 CONTINUE
END IF
*
* ==== Return the optimal value of LWORK. ====
*
WORK( 1 ) = CMPLX( LWKOPT, 0 )
*
* ==== End of CLAQR4 ====
*
END
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