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version 1.14, 2015/11/26 11:44:24
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*> \brief \b ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. |
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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 ZLAHQR + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlahqr.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/zlahqr.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/zlahqr.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 ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, |
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* IHIZ, Z, LDZ, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N |
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* LOGICAL WANTT, WANTZ |
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* .. |
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* .. Array Arguments .. |
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* COMPLEX*16 H( LDH, * ), W( * ), 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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*> ZLAHQR is an auxiliary routine called by CHSEQR to update the |
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*> eigenvalues and Schur decomposition already computed by CHSEQR, by |
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*> dealing with the Hessenberg submatrix in rows and columns ILO to |
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*> IHI. |
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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 and |
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*> columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). |
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*> ZLAHQR works primarily with the Hessenberg submatrix in rows |
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*> and columns ILO to IHI, but applies transformations to all of |
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*> H if WANTT is .TRUE.. |
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*> 1 <= ILO <= max(1,IHI); IHI <= N. |
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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 COMPLEX*16 array, dimension (LDH,N) |
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*> On entry, the upper Hessenberg matrix H. |
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*> On exit, if INFO is zero and if WANTT is .TRUE., then H |
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*> is upper triangular in rows and columns ILO:IHI. If INFO |
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*> is zero and if WANTT is .FALSE., then the contents of H |
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*> are unspecified on exit. The output state of H in case |
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*> INF is positive is below under the description of INFO. |
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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] W |
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*> \verbatim |
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*> W is COMPLEX*16 array, dimension (N) |
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*> The computed eigenvalues ILO to IHI are stored in the |
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*> corresponding elements of W. If WANTT is .TRUE., the |
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*> eigenvalues are stored in the same order as on the diagonal |
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*> of the Schur form returned in H, with W(i) = H(i,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 COMPLEX*16 array, dimension (LDZ,N) |
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*> If WANTZ is .TRUE., on entry Z must contain the current |
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*> matrix Z of transformations accumulated by CHSEQR, and on |
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*> exit Z has been updated; transformations are applied only to |
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*> the submatrix Z(ILOZ:IHIZ,ILO:IHI). |
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*> If WANTZ is .FALSE., Z is not referenced. |
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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. LDZ >= max(1,N). |
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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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*> .GT. 0: if INFO = i, ZLAHQR failed to compute all the |
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*> eigenvalues ILO to IHI in a total of 30 iterations |
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*> per eigenvalue; elements i+1:ihi of W contain |
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*> those eigenvalues which have been successfully |
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*> computed. |
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*> |
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*> If INFO .GT. 0 and WANTT is .FALSE., then on exit, |
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*> the remaining unconverged eigenvalues are the |
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*> eigenvalues of the upper Hessenberg matrix |
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*> rows and columns ILO thorugh INFO of the final, |
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*> output value of H. |
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*> |
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*> If INFO .GT. 0 and WANTT is .TRUE., then on exit |
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*> (*) (initial value of H)*U = U*(final value of H) |
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*> where U is an orthognal 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 .GT. 0 and WANTZ is .TRUE., then on exit |
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*> (final value of Z) = (initial value of Z)*U |
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*> where U is the orthogonal matrix in (*) |
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*> (regardless of the value of WANTT.) |
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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 November 2015 |
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* |
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*> \ingroup complex16OTHERauxiliary |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> \verbatim |
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*> |
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*> 02-96 Based on modifications by |
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*> David Day, Sandia National Laboratory, USA |
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*> |
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*> 12-04 Further modifications by |
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*> Ralph Byers, University of Kansas, USA |
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*> This is a modified version of ZLAHQR from LAPACK version 3.0. |
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*> It is (1) more robust against overflow and underflow and |
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*> (2) adopts the more conservative Ahues & Tisseur stopping |
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*> criterion (LAWN 122, 1997). |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, |
SUBROUTINE ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, |
$ IHIZ, Z, LDZ, INFO ) |
$ IHIZ, Z, LDZ, INFO ) |
* |
* |
* -- LAPACK auxiliary routine (version 3.2) -- |
* -- LAPACK auxiliary routine (version 3.6.0) -- |
* Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.. |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* November 2006 |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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* November 2015 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N |
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N |
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COMPLEX*16 H( LDH, * ), W( * ), Z( LDZ, * ) |
COMPLEX*16 H( LDH, * ), W( * ), Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* Purpose |
* ========================================================= |
* ======= |
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* |
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* ZLAHQR is an auxiliary routine called by CHSEQR to update the |
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* eigenvalues and Schur decomposition already computed by CHSEQR, by |
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* dealing with the Hessenberg submatrix in rows and columns ILO to |
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* IHI. |
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* |
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* Arguments |
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* ========= |
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* |
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* WANTT (input) 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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* |
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* WANTZ (input) 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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* |
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* N (input) INTEGER |
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* The order of the matrix H. N >= 0. |
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* |
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* ILO (input) INTEGER |
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* IHI (input) INTEGER |
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* It is assumed that H is already upper triangular in rows and |
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* columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). |
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* ZLAHQR works primarily with the Hessenberg submatrix in rows |
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* and columns ILO to IHI, but applies transformations to all of |
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* H if WANTT is .TRUE.. |
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* 1 <= ILO <= max(1,IHI); IHI <= N. |
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* |
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* H (input/output) COMPLEX*16 array, dimension (LDH,N) |
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* On entry, the upper Hessenberg matrix H. |
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* On exit, if INFO is zero and if WANTT is .TRUE., then H |
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* is upper triangular in rows and columns ILO:IHI. If INFO |
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* is zero and if WANTT is .FALSE., then the contents of H |
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* are unspecified on exit. The output state of H in case |
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* INF is positive is below under the description of INFO. |
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* |
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* LDH (input) INTEGER |
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* The leading dimension of the array H. LDH >= max(1,N). |
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* |
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* W (output) COMPLEX*16 array, dimension (N) |
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* The computed eigenvalues ILO to IHI are stored in the |
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* corresponding elements of W. If WANTT is .TRUE., the |
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* eigenvalues are stored in the same order as on the diagonal |
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* of the Schur form returned in H, with W(i) = H(i,i). |
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* |
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* ILOZ (input) INTEGER |
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* IHIZ (input) 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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* |
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* Z (input/output) COMPLEX*16 array, dimension (LDZ,N) |
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* If WANTZ is .TRUE., on entry Z must contain the current |
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* matrix Z of transformations accumulated by CHSEQR, and on |
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* exit Z has been updated; transformations are applied only to |
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* the submatrix Z(ILOZ:IHIZ,ILO:IHI). |
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* If WANTZ is .FALSE., Z is not referenced. |
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* |
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* LDZ (input) INTEGER |
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* The leading dimension of the array Z. LDZ >= max(1,N). |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* .GT. 0: if INFO = i, ZLAHQR failed to compute all the |
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* eigenvalues ILO to IHI in a total of 30 iterations |
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* per eigenvalue; elements i+1:ihi of W contain |
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* those eigenvalues which have been successfully |
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* computed. |
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* |
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* If INFO .GT. 0 and WANTT is .FALSE., then on exit, |
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* the remaining unconverged eigenvalues are the |
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* eigenvalues of the upper Hessenberg matrix |
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* rows and columns ILO thorugh INFO of the final, |
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* output value of H. |
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* |
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* If INFO .GT. 0 and WANTT is .TRUE., then on exit |
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* (*) (initial value of H)*U = U*(final value of H) |
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* where U is an orthognal 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 .GT. 0 and WANTZ is .TRUE., then on exit |
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* (final value of Z) = (initial value of Z)*U |
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* where U is the orthogonal matrix in (*) |
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* (regardless of the value of WANTT.) |
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* |
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* Further Details |
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* =============== |
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* |
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* 02-96 Based on modifications by |
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* David Day, Sandia National Laboratory, USA |
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* |
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* 12-04 Further modifications by |
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* Ralph Byers, University of Kansas, USA |
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* This is a modified version of ZLAHQR from LAPACK version 3.0. |
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* It is (1) more robust against overflow and underflow and |
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* (2) adopts the more conservative Ahues & Tisseur stopping |
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* criterion (LAWN 122, 1997). |
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* |
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* ========================================================= |
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* |
* |
* .. Parameters .. |
* .. Parameters .. |
INTEGER ITMAX |
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PARAMETER ( ITMAX = 30 ) |
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COMPLEX*16 ZERO, ONE |
COMPLEX*16 ZERO, ONE |
PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ), |
PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ), |
$ ONE = ( 1.0d0, 0.0d0 ) ) |
$ ONE = ( 1.0d0, 0.0d0 ) ) |
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$ V2, X, Y |
$ V2, X, Y |
DOUBLE PRECISION AA, AB, BA, BB, H10, H21, RTEMP, S, SAFMAX, |
DOUBLE PRECISION AA, AB, BA, BB, H10, H21, RTEMP, S, SAFMAX, |
$ SAFMIN, SMLNUM, SX, T2, TST, ULP |
$ SAFMIN, SMLNUM, SX, T2, TST, ULP |
INTEGER I, I1, I2, ITS, J, JHI, JLO, K, L, M, NH, NZ |
INTEGER I, I1, I2, ITS, ITMAX, J, JHI, JLO, K, L, M, |
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$ NH, NZ |
* .. |
* .. |
* .. Local Arrays .. |
* .. Local Arrays .. |
COMPLEX*16 V( 2 ) |
COMPLEX*16 V( 2 ) |
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I2 = N |
I2 = N |
END IF |
END IF |
* |
* |
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* ITMAX is the total number of QR iterations allowed. |
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* |
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ITMAX = 30 * MAX( 10, NH ) |
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* |
* The main loop begins here. I is the loop index and decreases from |
* The main loop begins here. I is the loop index and decreases from |
* IHI to ILO in steps of 1. Each iteration of the loop works |
* IHI to ILO in steps of 1. Each iteration of the loop works |
* with the active submatrix in rows and columns L to I. |
* with the active submatrix in rows and columns L to I. |