version 1.1, 2010/01/26 15:22:46
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version 1.9, 2011/11/21 20:42:53
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*> \brief \b DHSEQR |
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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 DHSEQR + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhseqr.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/dhseqr.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/dhseqr.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 DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, |
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* LDZ, WORK, LWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N |
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* CHARACTER COMPZ, JOB |
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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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*> DHSEQR 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] JOB |
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*> \verbatim |
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*> JOB is CHARACTER*1 |
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*> = 'E': compute eigenvalues only; |
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*> = 'S': compute eigenvalues and the Schur form T. |
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*> \endverbatim |
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*> |
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*> \param[in] COMPZ |
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*> \verbatim |
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*> COMPZ is CHARACTER*1 |
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*> = 'N': no Schur vectors are computed; |
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*> = 'I': Z is initialized to the unit matrix and the matrix Z |
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*> of Schur vectors of H is returned; |
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*> = 'V': Z must contain an orthogonal matrix Q on entry, and |
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*> the product Q*Z is returned. |
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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 .GE. 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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*> |
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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. ILO and IHI are normally |
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*> set by a previous call to DGEBAL, and then passed to ZGEHRD |
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*> when the matrix output by DGEBAL is reduced to Hessenberg |
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*> form. Otherwise ILO and IHI should be set to 1 and N |
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*> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.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 JOB = 'S', then H contains the |
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*> upper quasi-triangular matrix T from the Schur decomposition |
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*> (the Schur form); 2-by-2 diagonal blocks (corresponding to |
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*> complex conjugate pairs of eigenvalues) are returned in |
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*> standard form, with H(i,i) = H(i+1,i+1) and |
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*> H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the |
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*> contents of H are unspecified on exit. (The output value of |
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*> H when INFO.GT.0 is given under the description of INFO |
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*> below.) |
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*> |
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*> Unlike earlier versions of DHSEQR, this subroutine may |
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*> explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 |
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*> 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 .GE. 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 (N) |
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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 (N) |
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*> |
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*> The real and imaginary parts, respectively, of the computed |
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*> eigenvalues. If two eigenvalues are computed as a complex |
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*> conjugate pair, they are stored in consecutive elements of |
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*> WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and |
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*> WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in |
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*> the same order as on the diagonal of the Schur form returned |
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*> in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 |
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*> diagonal 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,out] Z |
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*> \verbatim |
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*> Z is DOUBLE PRECISION array, dimension (LDZ,N) |
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*> If COMPZ = 'N', Z is not referenced. |
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*> If COMPZ = 'I', on entry Z need not be set and on exit, |
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*> if INFO = 0, Z contains the orthogonal matrix Z of the Schur |
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*> vectors of H. If COMPZ = 'V', on entry Z must contain an |
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*> N-by-N matrix Q, which is assumed to be equal to the unit |
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*> matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, |
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*> if INFO = 0, Z contains Q*Z. |
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*> Normally Q is the orthogonal matrix generated by DORGHR |
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*> after the call to DGEHRD which formed the Hessenberg matrix |
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*> H. (The output value of Z when INFO.GT.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 COMPZ = 'I' or |
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*> COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.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 INFO = 0, 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 .GE. max(1,N) |
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*> is sufficient and delivers very good and sometimes |
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*> optimal performance. However, LWORK as large as 11*N |
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*> may be required for optimal performance. A workspace |
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*> query is recommended to determine the optimal workspace |
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*> size. |
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*> |
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*> If LWORK = -1, then DHSEQR does a workspace query. |
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*> In this case, DHSEQR 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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*> .LT. 0: if INFO = -i, the i-th argument had an illegal |
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*> value |
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*> .GT. 0: if INFO = i, DHSEQR 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 .GT. 0 and JOB = 'E', then on exit, the |
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*> 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 .GT. 0 and JOB = 'S', 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 an orthogonal matrix. The final |
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*> value of H is upper Hessenberg and quasi-triangular |
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*> in rows and columns INFO+1 through IHI. |
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*> |
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*> If INFO .GT. 0 and COMPZ = 'V', then on exit |
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*> |
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*> (final value of Z) = (initial value of Z)*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 JOB.) |
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*> |
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*> If INFO .GT. 0 and COMPZ = 'I', then on exit |
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*> (final value of Z) = U |
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*> where U is the orthogonal matrix in (*) (regard- |
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*> less of the value of JOB.) |
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*> |
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*> If INFO .GT. 0 and COMPZ = 'N', 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 November 2011 |
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* |
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*> \ingroup doubleOTHERcomputational |
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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 Further Details: |
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* ===================== |
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*> |
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*> \verbatim |
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*> |
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*> Default values supplied by |
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*> ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK). |
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*> It is suggested that these defaults be adjusted in order |
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*> to attain best performance in each particular |
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*> computational environment. |
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*> |
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*> ISPEC=12: The DLAHQR vs DLAQR0 crossover point. |
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*> Default: 75. (Must be at least 11.) |
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*> |
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*> ISPEC=13: Recommended deflation window size. |
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*> This depends on ILO, IHI and NS. NS is the |
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*> number of simultaneous shifts returned |
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*> by ILAENV(ISPEC=15). (See ISPEC=15 below.) |
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*> The default for (IHI-ILO+1).LE.500 is NS. |
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*> The default for (IHI-ILO+1).GT.500 is 3*NS/2. |
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*> |
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*> ISPEC=14: Nibble crossover point. (See IPARMQ for |
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*> details.) Default: 14% of deflation window |
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*> size. |
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*> |
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*> ISPEC=15: Number of simultaneous shifts in a multishift |
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*> QR iteration. |
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*> |
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*> If IHI-ILO+1 is ... |
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*> |
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*> greater than ...but less ... the |
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*> or equal to ... than default is |
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*> |
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*> 1 30 NS = 2(+) |
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*> 30 60 NS = 4(+) |
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*> 60 150 NS = 10(+) |
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*> 150 590 NS = ** |
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*> 590 3000 NS = 64 |
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*> 3000 6000 NS = 128 |
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*> 6000 infinity NS = 256 |
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*> |
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*> (+) By default some or all matrices of this order |
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*> are passed to the implicit double shift routine |
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*> DLAHQR and this parameter is ignored. See |
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*> ISPEC=12 above and comments in IPARMQ for |
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*> details. |
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*> |
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*> (**) The asterisks (**) indicate an ad-hoc |
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*> function of N increasing from 10 to 64. |
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*> |
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*> ISPEC=16: Select structured matrix multiply. |
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*> If the number of simultaneous shifts (specified |
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*> by ISPEC=15) is less than 14, then the default |
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*> for ISPEC=16 is 0. Otherwise the default for |
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*> ISPEC=16 is 2. |
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*> \endverbatim |
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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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* ===================================================================== |
SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, |
SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, |
$ LDZ, WORK, LWORK, INFO ) |
$ LDZ, WORK, LWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.4.0) -- |
* 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 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N |
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N |
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Line 329
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DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), |
DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), |
$ Z( LDZ, * ) |
$ Z( LDZ, * ) |
* .. |
* .. |
* Purpose |
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* ======= |
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* |
* |
* DHSEQR computes the eigenvalues of a Hessenberg matrix H |
* ===================================================================== |
* 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. |
|
* |
|
* Optionally Z may be postmultiplied into an input orthogonal |
|
* matrix Q so that this routine can give the Schur factorization |
|
* of a matrix A which has been reduced to the Hessenberg form H |
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* by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. |
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* |
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* Arguments |
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* ========= |
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* |
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* JOB (input) CHARACTER*1 |
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* = 'E': compute eigenvalues only; |
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* = 'S': compute eigenvalues and the Schur form T. |
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* |
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* COMPZ (input) CHARACTER*1 |
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* = 'N': no Schur vectors are computed; |
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* = 'I': Z is initialized to the unit matrix and the matrix Z |
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* of Schur vectors of H is returned; |
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* = 'V': Z must contain an orthogonal matrix Q on entry, and |
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* the product Q*Z is returned. |
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* |
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* N (input) INTEGER |
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* The order of the matrix H. N .GE. 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 |
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* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally |
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* set by a previous call to DGEBAL, and then passed to DGEHRD |
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* when the matrix output by DGEBAL is reduced to Hessenberg |
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* form. Otherwise ILO and IHI should be set to 1 and N |
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* respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. |
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* If N = 0, then ILO = 1 and IHI = 0. |
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* |
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* H (input/output) 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 JOB = 'S', then H contains the |
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* upper quasi-triangular matrix T from the Schur decomposition |
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* (the Schur form); 2-by-2 diagonal blocks (corresponding to |
|
* complex conjugate pairs of eigenvalues) are returned in |
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* standard form, with H(i,i) = H(i+1,i+1) and |
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* H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the |
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* contents of H are unspecified on exit. (The output value of |
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* H when INFO.GT.0 is given under the description of INFO |
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* below.) |
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* |
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* Unlike earlier versions of DHSEQR, this subroutine may |
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* explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 |
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* or j = IHI+1, IHI+2, ... N. |
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* |
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* LDH (input) INTEGER |
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* The leading dimension of the array H. LDH .GE. max(1,N). |
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* |
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* WR (output) DOUBLE PRECISION array, dimension (N) |
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* WI (output) DOUBLE PRECISION array, dimension (N) |
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* The real and imaginary parts, respectively, of the computed |
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* eigenvalues. If two eigenvalues are computed as a complex |
|
* conjugate pair, they are stored in consecutive elements of |
|
* WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and |
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* WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in |
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* the same order as on the diagonal of the Schur form returned |
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* in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 |
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* diagonal 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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* |
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* Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) |
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* If COMPZ = 'N', Z is not referenced. |
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* If COMPZ = 'I', on entry Z need not be set and on exit, |
|
* if INFO = 0, Z contains the orthogonal matrix Z of the Schur |
|
* vectors of H. If COMPZ = 'V', on entry Z must contain an |
|
* N-by-N matrix Q, which is assumed to be equal to the unit |
|
* matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, |
|
* if INFO = 0, Z contains Q*Z. |
|
* Normally Q is the orthogonal matrix generated by DORGHR |
|
* after the call to DGEHRD which formed the Hessenberg matrix |
|
* H. (The output value of Z when INFO.GT.0 is given under |
|
* the description of INFO below.) |
|
* |
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* LDZ (input) INTEGER |
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* The leading dimension of the array Z. if COMPZ = 'I' or |
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* COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1. |
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* |
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* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) |
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* On exit, if INFO = 0, WORK(1) returns an estimate of |
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* the optimal value for LWORK. |
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* |
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* LWORK (input) INTEGER |
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* The dimension of the array WORK. LWORK .GE. max(1,N) |
|
* is sufficient and delivers very good and sometimes |
|
* optimal performance. However, LWORK as large as 11*N |
|
* may be required for optimal performance. A workspace |
|
* query is recommended to determine the optimal workspace |
|
* size. |
|
* |
|
* If LWORK = -1, then DHSEQR does a workspace query. |
|
* In this case, DHSEQR 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. |
|
* |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* .LT. 0: if INFO = -i, the i-th argument had an illegal |
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* value |
|
* .GT. 0: if INFO = i, DHSEQR failed to compute all of |
|
* the eigenvalues. Elements 1:ilo-1 and i+1:n of WR |
|
* and WI contain those eigenvalues which have been |
|
* successfully computed. (Failures are rare.) |
|
* |
|
* If INFO .GT. 0 and JOB = 'E', then on exit, the |
|
* remaining unconverged eigenvalues are the eigen- |
|
* values of the upper Hessenberg matrix rows and |
|
* columns ILO through INFO of the final, output |
|
* value of H. |
|
* |
|
* If INFO .GT. 0 and JOB = 'S', then on exit |
|
* |
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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 an orthogonal matrix. The final |
|
* value of H is upper Hessenberg and quasi-triangular |
|
* in rows and columns INFO+1 through IHI. |
|
* |
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* If INFO .GT. 0 and COMPZ = 'V', then on exit |
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* |
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* (final value of Z) = (initial value of Z)*U |
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* |
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* where U is the orthogonal matrix in (*) (regard- |
|
* less of the value of JOB.) |
|
* |
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* If INFO .GT. 0 and COMPZ = 'I', then on exit |
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* (final value of Z) = U |
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* where U is the orthogonal matrix in (*) (regard- |
|
* less of the value of JOB.) |
|
* |
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* If INFO .GT. 0 and COMPZ = 'N', then Z is not |
|
* accessed. |
|
* |
|
* ================================================================ |
|
* Default values supplied by |
|
* ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK). |
|
* It is suggested that these defaults be adjusted in order |
|
* to attain best performance in each particular |
|
* computational environment. |
|
* |
|
* ISPEC=12: The DLAHQR vs DLAQR0 crossover point. |
|
* Default: 75. (Must be at least 11.) |
|
* |
|
* ISPEC=13: Recommended deflation window size. |
|
* This depends on ILO, IHI and NS. NS is the |
|
* number of simultaneous shifts returned |
|
* by ILAENV(ISPEC=15). (See ISPEC=15 below.) |
|
* The default for (IHI-ILO+1).LE.500 is NS. |
|
* The default for (IHI-ILO+1).GT.500 is 3*NS/2. |
|
* |
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* ISPEC=14: Nibble crossover point. (See IPARMQ for |
|
* details.) Default: 14% of deflation window |
|
* size. |
|
* |
|
* ISPEC=15: Number of simultaneous shifts in a multishift |
|
* QR iteration. |
|
* |
|
* If IHI-ILO+1 is ... |
|
* |
|
* greater than ...but less ... the |
|
* or equal to ... than default is |
|
* |
|
* 1 30 NS = 2(+) |
|
* 30 60 NS = 4(+) |
|
* 60 150 NS = 10(+) |
|
* 150 590 NS = ** |
|
* 590 3000 NS = 64 |
|
* 3000 6000 NS = 128 |
|
* 6000 infinity NS = 256 |
|
* |
|
* (+) By default some or all matrices of this order |
|
* are passed to the implicit double shift routine |
|
* DLAHQR and this parameter is ignored. See |
|
* ISPEC=12 above and comments in IPARMQ for |
|
* details. |
|
* |
|
* (**) The asterisks (**) indicate an ad-hoc |
|
* function of N increasing from 10 to 64. |
|
* |
|
* ISPEC=16: Select structured matrix multiply. |
|
* If the number of simultaneous shifts (specified |
|
* by ISPEC=15) is less than 14, then the default |
|
* for ISPEC=16 is 0. Otherwise the default for |
|
* ISPEC=16 is 2. |
|
* |
|
* ================================================================ |
|
* Based on contributions by |
|
* Karen Braman and Ralph Byers, Department of Mathematics, |
|
* University of Kansas, USA |
|
* |
|
* ================================================================ |
|
* 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. |
|
* |
|
* 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. |
|
* |
* |
* ================================================================ |
|
* .. Parameters .. |
* .. Parameters .. |
* |
* |
* ==== Matrices of order NTINY or smaller must be processed by |
* ==== Matrices of order NTINY or smaller must be processed by |