572 lines
18 KiB
Fortran
572 lines
18 KiB
Fortran
*> \brief \b STRSEN
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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 STRSEN + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/strsen.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/strsen.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/strsen.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 STRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
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* M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER COMPQ, JOB
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* INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
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* REAL S, SEP
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* ..
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* .. Array Arguments ..
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* LOGICAL SELECT( * )
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* INTEGER IWORK( * )
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* REAL Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
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* $ WR( * )
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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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*> STRSEN reorders the real Schur factorization of a real matrix
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*> A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
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*> the leading diagonal blocks of the upper quasi-triangular matrix T,
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*> and the leading columns of Q form an orthonormal basis of the
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*> corresponding right invariant subspace.
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*>
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*> Optionally the routine computes the reciprocal condition numbers of
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*> the cluster of eigenvalues and/or the invariant subspace.
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*>
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*> T must be in Schur canonical form (as returned by SHSEQR), that is,
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*> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
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*> 2-by-2 diagonal block has its diagonal elements equal and its
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*> off-diagonal elements of opposite sign.
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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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*> Specifies whether condition numbers are required for the
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*> cluster of eigenvalues (S) or the invariant subspace (SEP):
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*> = 'N': none;
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*> = 'E': for eigenvalues only (S);
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*> = 'V': for invariant subspace only (SEP);
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*> = 'B': for both eigenvalues and invariant subspace (S and
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*> SEP).
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*> \endverbatim
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*>
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*> \param[in] COMPQ
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*> \verbatim
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*> COMPQ is CHARACTER*1
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*> = 'V': update the matrix Q of Schur vectors;
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*> = 'N': do not update Q.
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*> \endverbatim
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*>
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*> \param[in] SELECT
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*> \verbatim
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*> SELECT is LOGICAL array, dimension (N)
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*> SELECT specifies the eigenvalues in the selected cluster. To
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*> select a real eigenvalue w(j), SELECT(j) must be set to
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*> .TRUE.. To select a complex conjugate pair of eigenvalues
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*> w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
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*> either SELECT(j) or SELECT(j+1) or both must be set to
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*> .TRUE.; a complex conjugate pair of eigenvalues must be
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*> either both included in the cluster or both excluded.
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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 T. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] T
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*> \verbatim
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*> T is REAL array, dimension (LDT,N)
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*> On entry, the upper quasi-triangular matrix T, in Schur
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*> canonical form.
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*> On exit, T is overwritten by the reordered matrix T, again in
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*> Schur canonical form, with the selected eigenvalues in the
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*> leading diagonal blocks.
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*> \endverbatim
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*>
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*> \param[in] LDT
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*> \verbatim
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*> LDT is INTEGER
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*> The leading dimension of the array T. LDT >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in,out] Q
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*> \verbatim
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*> Q is REAL array, dimension (LDQ,N)
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*> On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
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*> On exit, if COMPQ = 'V', Q has been postmultiplied by the
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*> orthogonal transformation matrix which reorders T; the
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*> leading M columns of Q form an orthonormal basis for the
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*> specified invariant subspace.
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*> If COMPQ = 'N', Q is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDQ
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*> \verbatim
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*> LDQ is INTEGER
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*> The leading dimension of the array Q.
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*> LDQ >= 1; and if COMPQ = 'V', LDQ >= 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 reordered
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*> eigenvalues of T. The eigenvalues are stored in the same
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*> order as on the diagonal of T, with WR(i) = T(i,i) and, if
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*> T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
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*> WI(i+1) = -WI(i). Note that if a complex eigenvalue is
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*> sufficiently ill-conditioned, then its value may differ
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*> significantly from its value before reordering.
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*> \endverbatim
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*>
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*> \param[out] M
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*> \verbatim
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*> M is INTEGER
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*> The dimension of the specified invariant subspace.
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*> 0 < = M <= N.
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*> \endverbatim
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*>
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*> \param[out] S
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*> \verbatim
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*> S is REAL
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*> If JOB = 'E' or 'B', S is a lower bound on the reciprocal
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*> condition number for the selected cluster of eigenvalues.
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*> S cannot underestimate the true reciprocal condition number
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*> by more than a factor of sqrt(N). If M = 0 or N, S = 1.
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*> If JOB = 'N' or 'V', S is not referenced.
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*> \endverbatim
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*>
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*> \param[out] SEP
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*> \verbatim
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*> SEP is REAL
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*> If JOB = 'V' or 'B', SEP is the estimated reciprocal
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*> condition number of the specified invariant subspace. If
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*> M = 0 or N, SEP = norm(T).
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*> If JOB = 'N' or 'E', SEP is not referenced.
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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 (MAX(1,LWORK))
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*> On exit, if INFO = 0, WORK(1) returns the optimal 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.
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*> If JOB = 'N', LWORK >= max(1,N);
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*> if JOB = 'E', LWORK >= max(1,M*(N-M));
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*> if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
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*>
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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*> this value as the first entry of the WORK array, and no error
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*> message related to LWORK is issued by XERBLA.
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*> \endverbatim
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
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*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
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*> \endverbatim
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*>
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*> \param[in] LIWORK
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*> \verbatim
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*> LIWORK is INTEGER
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*> The dimension of the array IWORK.
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*> If JOB = 'N' or 'E', LIWORK >= 1;
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*> if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
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*>
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*> If LIWORK = -1, then a workspace query is assumed; the
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*> routine only calculates the optimal size of the IWORK array,
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*> returns this value as the first entry of the IWORK array, and
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*> no error message related to LIWORK is issued by XERBLA.
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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 value
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*> = 1: reordering of T failed because some eigenvalues are too
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*> close to separate (the problem is very ill-conditioned);
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*> T may have been partially reordered, and WR and WI
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*> contain the eigenvalues in the same order as in T; S and
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*> SEP (if requested) are set to zero.
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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 April 2012
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*
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*> \ingroup realOTHERcomputational
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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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*> STRSEN first collects the selected eigenvalues by computing an
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*> orthogonal transformation Z to move them to the top left corner of T.
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*> In other words, the selected eigenvalues are the eigenvalues of T11
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*> in:
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*>
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*> Z**T * T * Z = ( T11 T12 ) n1
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*> ( 0 T22 ) n2
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*> n1 n2
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*>
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*> where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
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*> of Z span the specified invariant subspace of T.
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*>
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*> If T has been obtained from the real Schur factorization of a matrix
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*> A = Q*T*Q**T, then the reordered real Schur factorization of A is given
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*> by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
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*> the corresponding invariant subspace of A.
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*>
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*> The reciprocal condition number of the average of the eigenvalues of
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*> T11 may be returned in S. S lies between 0 (very badly conditioned)
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*> and 1 (very well conditioned). It is computed as follows. First we
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*> compute R so that
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*>
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*> P = ( I R ) n1
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*> ( 0 0 ) n2
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*> n1 n2
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*>
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*> is the projector on the invariant subspace associated with T11.
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*> R is the solution of the Sylvester equation:
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*>
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*> T11*R - R*T22 = T12.
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*>
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*> Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
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*> the two-norm of M. Then S is computed as the lower bound
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*>
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*> (1 + F-norm(R)**2)**(-1/2)
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*>
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*> on the reciprocal of 2-norm(P), the true reciprocal condition number.
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*> S cannot underestimate 1 / 2-norm(P) by more than a factor of
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*> sqrt(N).
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*>
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*> An approximate error bound for the computed average of the
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*> eigenvalues of T11 is
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*>
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*> EPS * norm(T) / S
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*>
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*> where EPS is the machine precision.
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*>
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*> The reciprocal condition number of the right invariant subspace
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*> spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
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*> SEP is defined as the separation of T11 and T22:
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*>
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*> sep( T11, T22 ) = sigma-min( C )
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*>
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*> where sigma-min(C) is the smallest singular value of the
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*> n1*n2-by-n1*n2 matrix
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*>
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*> C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
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*>
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*> I(m) is an m by m identity matrix, and kprod denotes the Kronecker
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*> product. We estimate sigma-min(C) by the reciprocal of an estimate of
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*> the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
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*> cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
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*>
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*> When SEP is small, small changes in T can cause large changes in
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*> the invariant subspace. An approximate bound on the maximum angular
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*> error in the computed right invariant subspace is
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*>
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*> EPS * norm(T) / SEP
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE STRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
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$ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
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*
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* -- LAPACK computational routine (version 3.7.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* April 2012
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*
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* .. Scalar Arguments ..
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CHARACTER COMPQ, JOB
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INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
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REAL S, SEP
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* ..
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* .. Array Arguments ..
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LOGICAL SELECT( * )
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INTEGER IWORK( * )
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REAL Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
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$ WR( * )
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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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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
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$ WANTSP
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INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
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$ NN
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REAL EST, RNORM, SCALE
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* ..
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* .. Local Arrays ..
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INTEGER ISAVE( 3 )
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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REAL SLANGE
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EXTERNAL LSAME, SLANGE
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* ..
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* .. External Subroutines ..
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EXTERNAL SLACN2, SLACPY, STREXC, STRSYL, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, SQRT
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* ..
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* .. Executable Statements ..
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*
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* Decode and test the input parameters
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*
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WANTBH = LSAME( JOB, 'B' )
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WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
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WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
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WANTQ = LSAME( COMPQ, 'V' )
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*
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INFO = 0
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LQUERY = ( LWORK.EQ.-1 )
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IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
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$ THEN
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INFO = -1
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ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
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INFO = -6
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ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
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INFO = -8
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ELSE
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*
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* Set M to the dimension of the specified invariant subspace,
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* and test LWORK and LIWORK.
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*
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M = 0
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PAIR = .FALSE.
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DO 10 K = 1, N
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IF( PAIR ) THEN
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PAIR = .FALSE.
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ELSE
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IF( K.LT.N ) THEN
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IF( T( K+1, K ).EQ.ZERO ) THEN
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IF( SELECT( K ) )
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$ M = M + 1
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ELSE
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PAIR = .TRUE.
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IF( SELECT( K ) .OR. SELECT( K+1 ) )
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$ M = M + 2
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END IF
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ELSE
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IF( SELECT( N ) )
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$ M = M + 1
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END IF
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END IF
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10 CONTINUE
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*
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N1 = M
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N2 = N - M
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NN = N1*N2
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*
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IF( WANTSP ) THEN
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LWMIN = MAX( 1, 2*NN )
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LIWMIN = MAX( 1, NN )
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ELSE IF( LSAME( JOB, 'N' ) ) THEN
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LWMIN = MAX( 1, N )
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LIWMIN = 1
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ELSE IF( LSAME( JOB, 'E' ) ) THEN
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LWMIN = MAX( 1, NN )
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LIWMIN = 1
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END IF
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*
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IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
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INFO = -15
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ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
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INFO = -17
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END IF
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END IF
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*
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IF( INFO.EQ.0 ) THEN
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WORK( 1 ) = LWMIN
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IWORK( 1 ) = LIWMIN
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'STRSEN', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* Quick return if possible.
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*
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IF( M.EQ.N .OR. M.EQ.0 ) THEN
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IF( WANTS )
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$ S = ONE
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IF( WANTSP )
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$ SEP = SLANGE( '1', N, N, T, LDT, WORK )
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GO TO 40
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END IF
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*
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* Collect the selected blocks at the top-left corner of T.
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*
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KS = 0
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PAIR = .FALSE.
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DO 20 K = 1, N
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IF( PAIR ) THEN
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PAIR = .FALSE.
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ELSE
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SWAP = SELECT( K )
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IF( K.LT.N ) THEN
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IF( T( K+1, K ).NE.ZERO ) THEN
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PAIR = .TRUE.
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SWAP = SWAP .OR. SELECT( K+1 )
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END IF
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END IF
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IF( SWAP ) THEN
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KS = KS + 1
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*
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* Swap the K-th block to position KS.
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*
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IERR = 0
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KK = K
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IF( K.NE.KS )
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$ CALL STREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK,
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$ IERR )
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IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
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*
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* Blocks too close to swap: exit.
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*
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INFO = 1
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IF( WANTS )
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$ S = ZERO
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IF( WANTSP )
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$ SEP = ZERO
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GO TO 40
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END IF
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IF( PAIR )
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$ KS = KS + 1
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END IF
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END IF
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20 CONTINUE
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*
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IF( WANTS ) THEN
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|
*
|
|
* Solve Sylvester equation for R:
|
|
*
|
|
* T11*R - R*T22 = scale*T12
|
|
*
|
|
CALL SLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
|
|
CALL STRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
|
|
$ LDT, WORK, N1, SCALE, IERR )
|
|
*
|
|
* Estimate the reciprocal of the condition number of the cluster
|
|
* of eigenvalues.
|
|
*
|
|
RNORM = SLANGE( 'F', N1, N2, WORK, N1, WORK )
|
|
IF( RNORM.EQ.ZERO ) THEN
|
|
S = ONE
|
|
ELSE
|
|
S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
|
|
$ SQRT( RNORM ) )
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( WANTSP ) THEN
|
|
*
|
|
* Estimate sep(T11,T22).
|
|
*
|
|
EST = ZERO
|
|
KASE = 0
|
|
30 CONTINUE
|
|
CALL SLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE, ISAVE )
|
|
IF( KASE.NE.0 ) THEN
|
|
IF( KASE.EQ.1 ) THEN
|
|
*
|
|
* Solve T11*R - R*T22 = scale*X.
|
|
*
|
|
CALL STRSYL( 'N', 'N', -1, N1, N2, T, LDT,
|
|
$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
|
|
$ IERR )
|
|
ELSE
|
|
*
|
|
* Solve T11**T*R - R*T22**T = scale*X.
|
|
*
|
|
CALL STRSYL( 'T', 'T', -1, N1, N2, T, LDT,
|
|
$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
|
|
$ IERR )
|
|
END IF
|
|
GO TO 30
|
|
END IF
|
|
*
|
|
SEP = SCALE / EST
|
|
END IF
|
|
*
|
|
40 CONTINUE
|
|
*
|
|
* Store the output eigenvalues in WR and WI.
|
|
*
|
|
DO 50 K = 1, N
|
|
WR( K ) = T( K, K )
|
|
WI( K ) = ZERO
|
|
50 CONTINUE
|
|
DO 60 K = 1, N - 1
|
|
IF( T( K+1, K ).NE.ZERO ) THEN
|
|
WI( K ) = SQRT( ABS( T( K, K+1 ) ) )*
|
|
$ SQRT( ABS( T( K+1, K ) ) )
|
|
WI( K+1 ) = -WI( K )
|
|
END IF
|
|
60 CONTINUE
|
|
*
|
|
WORK( 1 ) = LWMIN
|
|
IWORK( 1 ) = LIWMIN
|
|
*
|
|
RETURN
|
|
*
|
|
* End of STRSEN
|
|
*
|
|
END
|