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514 lines
18 KiB
FortranFixed
514 lines
18 KiB
FortranFixed
*> \brief \b SHSEQR
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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 SHSEQR + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/shseqr.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/shseqr.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/shseqr.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 SHSEQR( 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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* REAL 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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*> SHSEQR 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 >= 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 SGEBAL, and then passed to ZGEHRD
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*> when the matrix output by SGEBAL 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 > 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 REAL 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) < 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 > 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 SHSEQR, this subroutine may
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*> explicitly H(i,j) = 0 for i > 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 >= 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 REAL 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 REAL 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) > 0 and
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*> WI(i+1) < 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 REAL 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 SORGHR
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*> after the call to SGEHRD which formed the Hessenberg matrix
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*> H. (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 COMPZ = 'I' or
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*> COMPZ = 'V', then LDZ >= MAX(1,N). 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 REAL 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 >= 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 SHSEQR does a workspace query.
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*> In this case, SHSEQR 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, the i-th argument had an illegal
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*> value
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*> > 0: if INFO = i, SHSEQR 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 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 > 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 > 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 > 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 > 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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*> \ingroup realOTHERcomputational
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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,'SHSEQR',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 SLAHQR vs SLAQR0 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) <= 500 is NS.
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*> The default for (IHI-ILO+1) > 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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*> SLAHQR 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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* =====================================================================
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SUBROUTINE SHSEQR( 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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* -- LAPACK computational routine --
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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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*
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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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REAL 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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*
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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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* . SLAHQR 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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* ==== NL allocates some local workspace to help small matrices
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* . through a rare SLAHQR failure. NL > NTINY = 15 is
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* . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
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* . mended. (The default value of NMIN is 75.) Using NL = 49
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* . allows up to six simultaneous shifts and a 16-by-16
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* . deflation window. ====
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INTEGER NL
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PARAMETER ( NL = 49 )
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0 )
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* ..
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* .. Local Arrays ..
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REAL HL( NL, NL ), WORKL( NL )
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* ..
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* .. Local Scalars ..
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INTEGER I, KBOT, NMIN
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LOGICAL INITZ, LQUERY, WANTT, WANTZ
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* ..
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* .. External Functions ..
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INTEGER ILAENV
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LOGICAL LSAME
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EXTERNAL ILAENV, LSAME
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* ..
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* .. External Subroutines ..
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EXTERNAL SLACPY, SLAHQR, SLAQR0, SLASET, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN, REAL
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* ..
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* .. Executable Statements ..
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*
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* ==== Decode and check the input parameters. ====
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*
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WANTT = LSAME( JOB, 'S' )
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INITZ = LSAME( COMPZ, 'I' )
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WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
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WORK( 1 ) = REAL( MAX( 1, N ) )
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LQUERY = LWORK.EQ.-1
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*
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INFO = 0
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IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
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INFO = -1
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ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
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INFO = -4
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ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
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INFO = -5
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ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
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INFO = -7
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ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
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INFO = -11
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ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
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INFO = -13
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END IF
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*
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IF( INFO.NE.0 ) THEN
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*
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* ==== Quick return in case of invalid argument. ====
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*
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CALL XERBLA( 'SHSEQR', -INFO )
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RETURN
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*
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ELSE IF( N.EQ.0 ) THEN
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*
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* ==== Quick return in case N = 0; nothing to do. ====
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*
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RETURN
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*
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ELSE IF( LQUERY ) THEN
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*
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* ==== Quick return in case of a workspace query ====
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*
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CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
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$ IHI, Z, LDZ, WORK, LWORK, INFO )
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* ==== Ensure reported workspace size is backward-compatible with
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* . previous LAPACK versions. ====
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WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) )
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RETURN
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*
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ELSE
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*
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* ==== copy eigenvalues isolated by SGEBAL ====
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*
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DO 10 I = 1, ILO - 1
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WR( I ) = H( I, I )
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WI( I ) = ZERO
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10 CONTINUE
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DO 20 I = IHI + 1, N
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WR( I ) = H( I, I )
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WI( I ) = ZERO
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20 CONTINUE
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*
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* ==== Initialize Z, if requested ====
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*
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IF( INITZ )
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$ CALL SLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
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*
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* ==== Quick return if possible ====
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*
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IF( ILO.EQ.IHI ) THEN
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WR( ILO ) = H( ILO, ILO )
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WI( ILO ) = ZERO
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RETURN
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END IF
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*
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* ==== SLAHQR/SLAQR0 crossover point ====
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*
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NMIN = ILAENV( 12, 'SHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
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$ ILO, IHI, LWORK )
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NMIN = MAX( NTINY, NMIN )
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*
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* ==== SLAQR0 for big matrices; SLAHQR for small ones ====
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*
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IF( N.GT.NMIN ) THEN
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CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
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$ IHI, Z, LDZ, WORK, LWORK, INFO )
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ELSE
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*
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* ==== Small matrix ====
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*
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CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
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$ IHI, Z, LDZ, INFO )
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*
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IF( INFO.GT.0 ) THEN
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*
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* ==== A rare SLAHQR failure! SLAQR0 sometimes succeeds
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* . when SLAHQR fails. ====
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*
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KBOT = INFO
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*
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IF( N.GE.NL ) THEN
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*
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* ==== Larger matrices have enough subdiagonal scratch
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|
* . space to call SLAQR0 directly. ====
|
|
*
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|
CALL SLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, WR,
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|
$ WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
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|
*
|
|
ELSE
|
|
*
|
|
* ==== Tiny matrices don't have enough subdiagonal
|
|
* . scratch space to benefit from SLAQR0. Hence,
|
|
* . tiny matrices must be copied into a larger
|
|
* . array before calling SLAQR0. ====
|
|
*
|
|
CALL SLACPY( 'A', N, N, H, LDH, HL, NL )
|
|
HL( N+1, N ) = ZERO
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|
CALL SLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
|
|
$ NL )
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|
CALL SLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, WR,
|
|
$ WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO )
|
|
IF( WANTT .OR. INFO.NE.0 )
|
|
$ CALL SLACPY( 'A', N, N, HL, NL, H, LDH )
|
|
END IF
|
|
END IF
|
|
END IF
|
|
*
|
|
* ==== Clear out the trash, if necessary. ====
|
|
*
|
|
IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
|
|
$ CALL SLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
|
|
*
|
|
* ==== Ensure reported workspace size is backward-compatible with
|
|
* . previous LAPACK versions. ====
|
|
*
|
|
WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) )
|
|
END IF
|
|
*
|
|
* ==== End of SHSEQR ====
|
|
*
|
|
END
|