740 lines
26 KiB
Fortran
740 lines
26 KiB
Fortran
*> \brief \b DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download DLAQR4 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr4.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr4.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr4.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
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* ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
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* LOGICAL WANTT, WANTZ
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
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* $ Z( LDZ, * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> DLAQR4 implements one level of recursion for DLAQR0.
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*> It is a complete implementation of the small bulge multi-shift
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*> QR algorithm. It may be called by DLAQR0 and, for large enough
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*> deflation window size, it may be called by DLAQR3. This
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*> subroutine is identical to DLAQR0 except that it calls DLAQR2
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*> instead of DLAQR3.
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*>
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*> DLAQR4 computes the eigenvalues of a Hessenberg matrix H
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*> and, optionally, the matrices T and Z from the Schur decomposition
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*> H = Z T Z**T, where T is an upper quasi-triangular matrix (the
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*> Schur form), and Z is the orthogonal matrix of Schur vectors.
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*>
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*> Optionally Z may be postmultiplied into an input orthogonal
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*> matrix Q so that this routine can give the Schur factorization
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*> of a matrix A which has been reduced to the Hessenberg form H
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*> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] WANTT
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*> \verbatim
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*> WANTT is LOGICAL
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*> = .TRUE. : the full Schur form T is required;
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*> = .FALSE.: only eigenvalues are required.
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*> \endverbatim
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*>
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*> \param[in] WANTZ
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*> \verbatim
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*> WANTZ is LOGICAL
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*> = .TRUE. : the matrix of Schur vectors Z is required;
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*> = .FALSE.: Schur vectors are not required.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix H. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] ILO
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*> \verbatim
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*> ILO is INTEGER
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*> \endverbatim
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*>
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*> \param[in] IHI
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*> \verbatim
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*> IHI is INTEGER
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*> It is assumed that H is already upper triangular in rows
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*> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
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*> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
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*> previous call to DGEBAL, and then passed to DGEHRD when the
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*> matrix output by DGEBAL is reduced to Hessenberg form.
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*> Otherwise, ILO and IHI should be set to 1 and N,
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*> respectively. If N > 0, then 1 <= ILO <= IHI <= N.
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*> If N = 0, then ILO = 1 and IHI = 0.
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*> \endverbatim
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*>
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*> \param[in,out] H
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*> \verbatim
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*> H is DOUBLE PRECISION array, dimension (LDH,N)
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*> On entry, the upper Hessenberg matrix H.
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*> On exit, if INFO = 0 and WANTT is .TRUE., then H contains
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*> the upper quasi-triangular matrix T from the Schur
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*> decomposition (the Schur form); 2-by-2 diagonal blocks
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*> (corresponding to complex conjugate pairs of eigenvalues)
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*> are returned in standard form, with H(i,i) = H(i+1,i+1)
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*> and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is
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*> .FALSE., then the contents of H are unspecified on exit.
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*> (The output value of H when INFO > 0 is given under the
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*> description of INFO below.)
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*>
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*> This subroutine may explicitly set H(i,j) = 0 for i > j and
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*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
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*> \endverbatim
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*>
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*> \param[in] LDH
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*> \verbatim
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*> LDH is INTEGER
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*> The leading dimension of the array H. LDH >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] WR
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*> \verbatim
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*> WR is DOUBLE PRECISION array, dimension (IHI)
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*> \endverbatim
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*>
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*> \param[out] WI
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*> \verbatim
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*> WI is DOUBLE PRECISION array, dimension (IHI)
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*> The real and imaginary parts, respectively, of the computed
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*> eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
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*> and WI(ILO:IHI). If two eigenvalues are computed as a
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*> complex conjugate pair, they are stored in consecutive
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*> elements of WR and WI, say the i-th and (i+1)th, with
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*> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then
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*> the eigenvalues are stored in the same order as on the
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*> diagonal of the Schur form returned in H, with
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*> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
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*> block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
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*> WI(i+1) = -WI(i).
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*> \endverbatim
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*>
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*> \param[in] ILOZ
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*> \verbatim
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*> ILOZ is INTEGER
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*> \endverbatim
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*>
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*> \param[in] IHIZ
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*> \verbatim
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*> IHIZ is INTEGER
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*> Specify the rows of Z to which transformations must be
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*> applied if WANTZ is .TRUE..
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*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
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*> \endverbatim
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*>
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*> \param[in,out] Z
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*> \verbatim
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*> Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
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*> If WANTZ is .FALSE., then Z is not referenced.
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*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
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*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
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*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
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*> (The output value of Z when INFO > 0 is given under
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*> the description of INFO below.)
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*> \endverbatim
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*>
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*> \param[in] LDZ
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*> \verbatim
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*> LDZ is INTEGER
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*> The leading dimension of the array Z. if WANTZ is .TRUE.
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*> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is DOUBLE PRECISION array, dimension LWORK
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*> On exit, if LWORK = -1, WORK(1) returns an estimate of
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*> the optimal value for LWORK.
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= max(1,N)
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*> is sufficient, but LWORK typically as large as 6*N may
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*> be required for optimal performance. A workspace query
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*> to determine the optimal workspace size is recommended.
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*>
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*> If LWORK = -1, then DLAQR4 does a workspace query.
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*> In this case, DLAQR4 checks the input parameters and
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*> estimates the optimal workspace size for the given
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*> values of N, ILO and IHI. The estimate is returned
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*> in WORK(1). No error message related to LWORK is
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*> issued by XERBLA. Neither H nor Z are accessed.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> > 0: if INFO = i, DLAQR4 failed to compute all of
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*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
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*> and WI contain those eigenvalues which have been
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*> successfully computed. (Failures are rare.)
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*>
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*> If INFO > 0 and WANT is .FALSE., then on exit,
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*> the remaining unconverged eigenvalues are the eigen-
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*> values of the upper Hessenberg matrix rows and
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*> columns ILO through INFO of the final, output
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*> value of H.
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*>
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*> If INFO > 0 and WANTT is .TRUE., then on exit
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*>
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*> (*) (initial value of H)*U = U*(final value of H)
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*>
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*> where U is a orthogonal matrix. The final
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*> value of H is upper Hessenberg and triangular in
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*> rows and columns INFO+1 through IHI.
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*>
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*> If INFO > 0 and WANTZ is .TRUE., then on exit
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*>
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*> (final value of Z(ILO:IHI,ILOZ:IHIZ)
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*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
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*>
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*> where U is the orthogonal matrix in (*) (regard-
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*> less of the value of WANTT.)
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*>
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*> If INFO > 0 and WANTZ is .FALSE., then Z is not
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*> accessed.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date December 2016
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*
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*> \ingroup doubleOTHERauxiliary
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*
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*> \par Contributors:
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* ==================
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*>
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*> Karen Braman and Ralph Byers, Department of Mathematics,
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*> University of Kansas, USA
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*
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*> \par References:
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* ================
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*>
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*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
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*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
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*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
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*> 929--947, 2002.
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*> \n
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*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
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*> Algorithm Part II: Aggressive Early Deflation, SIAM Journal
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*> of Matrix Analysis, volume 23, pages 948--973, 2002.
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*>
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* =====================================================================
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SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
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$ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
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*
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* -- LAPACK auxiliary routine (version 3.7.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* December 2016
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*
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* .. Scalar Arguments ..
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INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
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LOGICAL WANTT, WANTZ
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
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$ Z( LDZ, * )
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* ..
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*
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* ================================================================
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* .. Parameters ..
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*
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* ==== Matrices of order NTINY or smaller must be processed by
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* . DLAHQR because of insufficient subdiagonal scratch space.
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* . (This is a hard limit.) ====
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INTEGER NTINY
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PARAMETER ( NTINY = 15 )
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*
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* ==== Exceptional deflation windows: try to cure rare
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* . slow convergence by varying the size of the
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* . deflation window after KEXNW iterations. ====
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INTEGER KEXNW
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PARAMETER ( KEXNW = 5 )
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*
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* ==== Exceptional shifts: try to cure rare slow convergence
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* . with ad-hoc exceptional shifts every KEXSH iterations.
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* . ====
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INTEGER KEXSH
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PARAMETER ( KEXSH = 6 )
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*
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* ==== The constants WILK1 and WILK2 are used to form the
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* . exceptional shifts. ====
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DOUBLE PRECISION WILK1, WILK2
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PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 )
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
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* ..
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* .. Local Scalars ..
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DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
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INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
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$ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
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$ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
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$ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
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LOGICAL SORTED
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CHARACTER JBCMPZ*2
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* ..
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* .. External Functions ..
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INTEGER ILAENV
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EXTERNAL ILAENV
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* ..
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* .. Local Arrays ..
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DOUBLE PRECISION ZDUM( 1, 1 )
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* ..
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* .. External Subroutines ..
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EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR2, DLAQR5
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD
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* ..
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* .. Executable Statements ..
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INFO = 0
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*
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* ==== Quick return for N = 0: nothing to do. ====
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*
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IF( N.EQ.0 ) THEN
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WORK( 1 ) = ONE
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RETURN
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END IF
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*
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IF( N.LE.NTINY ) THEN
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*
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* ==== Tiny matrices must use DLAHQR. ====
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*
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LWKOPT = 1
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IF( LWORK.NE.-1 )
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$ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
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$ ILOZ, IHIZ, Z, LDZ, INFO )
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ELSE
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*
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* ==== Use small bulge multi-shift QR with aggressive early
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* . deflation on larger-than-tiny matrices. ====
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*
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* ==== Hope for the best. ====
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*
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INFO = 0
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*
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* ==== Set up job flags for ILAENV. ====
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*
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IF( WANTT ) THEN
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JBCMPZ( 1: 1 ) = 'S'
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ELSE
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JBCMPZ( 1: 1 ) = 'E'
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END IF
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IF( WANTZ ) THEN
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JBCMPZ( 2: 2 ) = 'V'
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ELSE
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JBCMPZ( 2: 2 ) = 'N'
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END IF
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*
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* ==== NWR = recommended deflation window size. At this
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* . point, N .GT. NTINY = 15, so there is enough
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* . subdiagonal workspace for NWR.GE.2 as required.
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* . (In fact, there is enough subdiagonal space for
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* . NWR.GE.4.) ====
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*
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NWR = ILAENV( 13, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
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NWR = MAX( 2, NWR )
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NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
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*
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* ==== NSR = recommended number of simultaneous shifts.
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* . At this point N .GT. NTINY = 15, so there is at
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* . enough subdiagonal workspace for NSR to be even
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* . and greater than or equal to two as required. ====
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*
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NSR = ILAENV( 15, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
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NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
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NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
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*
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* ==== Estimate optimal workspace ====
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*
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* ==== Workspace query call to DLAQR2 ====
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*
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CALL DLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
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$ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
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$ N, H, LDH, WORK, -1 )
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*
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* ==== Optimal workspace = MAX(DLAQR5, DLAQR2) ====
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*
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LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
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*
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* ==== Quick return in case of workspace query. ====
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*
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IF( LWORK.EQ.-1 ) THEN
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WORK( 1 ) = DBLE( LWKOPT )
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RETURN
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END IF
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*
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* ==== DLAHQR/DLAQR0 crossover point ====
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*
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NMIN = ILAENV( 12, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
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NMIN = MAX( NTINY, NMIN )
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*
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* ==== Nibble crossover point ====
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*
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NIBBLE = ILAENV( 14, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
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NIBBLE = MAX( 0, NIBBLE )
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*
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* ==== Accumulate reflections during ttswp? Use block
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* . 2-by-2 structure during matrix-matrix multiply? ====
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*
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KACC22 = ILAENV( 16, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
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KACC22 = MAX( 0, KACC22 )
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KACC22 = MIN( 2, KACC22 )
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*
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* ==== NWMAX = the largest possible deflation window for
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* . which there is sufficient workspace. ====
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*
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NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
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NW = NWMAX
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*
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* ==== NSMAX = the Largest number of simultaneous shifts
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* . for which there is sufficient workspace. ====
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*
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NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
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NSMAX = NSMAX - MOD( NSMAX, 2 )
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*
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* ==== NDFL: an iteration count restarted at deflation. ====
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*
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NDFL = 1
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*
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* ==== ITMAX = iteration limit ====
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*
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ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
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*
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* ==== Last row and column in the active block ====
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*
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KBOT = IHI
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*
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* ==== Main Loop ====
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*
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DO 80 IT = 1, ITMAX
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*
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* ==== Done when KBOT falls below ILO ====
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*
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IF( KBOT.LT.ILO )
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$ GO TO 90
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*
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* ==== Locate active block ====
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*
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DO 10 K = KBOT, ILO + 1, -1
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IF( H( K, K-1 ).EQ.ZERO )
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|
$ 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 DLAQR2( 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 DLAQR2
|
|
* . 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
|
|
* . DLAQR2 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 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 )
|
|
CALL DLAHQR( .false., .false., NS, 1, NS,
|
|
$ H( KT, 1 ), LDH, WR( KS ), WI( 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. ====
|
|
*
|
|
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 DLAQR4 ====
|
|
*
|
|
END
|