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892 lines
28 KiB
892 lines
28 KiB
*> \brief \b CHGEQZ
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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 CHGEQZ + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chgeqz.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/chgeqz.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/chgeqz.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 CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
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* ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
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* RWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER COMPQ, COMPZ, JOB
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* INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
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* ..
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* .. Array Arguments ..
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* REAL RWORK( * )
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* COMPLEX ALPHA( * ), BETA( * ), H( LDH, * ),
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* $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
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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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*> CHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
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*> where H is an upper Hessenberg matrix and T is upper triangular,
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*> using the single-shift QZ method.
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*> Matrix pairs of this type are produced by the reduction to
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*> generalized upper Hessenberg form of a complex matrix pair (A,B):
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*>
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*> A = Q1*H*Z1**H, B = Q1*T*Z1**H,
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*>
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*> as computed by CGGHRD.
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*>
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*> If JOB='S', then the Hessenberg-triangular pair (H,T) is
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*> also reduced to generalized Schur form,
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*>
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*> H = Q*S*Z**H, T = Q*P*Z**H,
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*>
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*> where Q and Z are unitary matrices and S and P are upper triangular.
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*>
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*> Optionally, the unitary matrix Q from the generalized Schur
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*> factorization may be postmultiplied into an input matrix Q1, and the
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*> unitary matrix Z may be postmultiplied into an input matrix Z1.
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*> If Q1 and Z1 are the unitary matrices from CGGHRD that reduced
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*> the matrix pair (A,B) to generalized Hessenberg form, then the output
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*> matrices Q1*Q and Z1*Z are the unitary factors from the generalized
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*> Schur factorization of (A,B):
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*>
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*> A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
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*>
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*> To avoid overflow, eigenvalues of the matrix pair (H,T)
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*> (equivalently, of (A,B)) are computed as a pair of complex values
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*> (alpha,beta). If beta is nonzero, lambda = alpha / beta is an
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*> eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
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*> A*x = lambda*B*x
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*> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
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*> alternate form of the GNEP
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*> mu*A*y = B*y.
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*> The values of alpha and beta for the i-th eigenvalue can be read
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*> directly from the generalized Schur form: alpha = S(i,i),
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*> beta = P(i,i).
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*>
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*> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
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*> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
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*> pp. 241--256.
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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': Computer eigenvalues and the Schur form.
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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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*> = 'N': Left Schur vectors (Q) are not computed;
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*> = 'I': Q is initialized to the unit matrix and the matrix Q
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*> of left Schur vectors of (H,T) is returned;
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*> = 'V': Q must contain a unitary matrix Q1 on entry and
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*> the product Q1*Q is returned.
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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': Right Schur vectors (Z) are not computed;
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*> = 'I': Q is initialized to the unit matrix and the matrix Z
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*> of right Schur vectors of (H,T) is returned;
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*> = 'V': Z must contain a unitary matrix Z1 on entry and
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*> the product Z1*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 matrices H, T, Q, and Z. 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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*> ILO and IHI mark the rows and columns of H which are in
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*> Hessenberg form. It is assumed that A is already upper
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*> triangular in rows and columns 1:ILO-1 and IHI+1:N.
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*> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, 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 COMPLEX array, dimension (LDH, N)
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*> On entry, the N-by-N upper Hessenberg matrix H.
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*> On exit, if JOB = 'S', H contains the upper triangular
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*> matrix S from the generalized Schur factorization.
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*> If JOB = 'E', the diagonal of H matches that of S, but
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*> the rest of H is unspecified.
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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[in,out] T
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*> \verbatim
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*> T is COMPLEX array, dimension (LDT, N)
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*> On entry, the N-by-N upper triangular matrix T.
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*> On exit, if JOB = 'S', T contains the upper triangular
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*> matrix P from the generalized Schur factorization.
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*> If JOB = 'E', the diagonal of T matches that of P, but
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*> the rest of T is unspecified.
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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[out] ALPHA
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*> \verbatim
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*> ALPHA is COMPLEX array, dimension (N)
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*> The complex scalars alpha that define the eigenvalues of
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*> GNEP. ALPHA(i) = S(i,i) in the generalized Schur
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*> factorization.
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*> \endverbatim
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*>
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*> \param[out] BETA
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*> \verbatim
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*> BETA is COMPLEX array, dimension (N)
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*> The real non-negative scalars beta that define the
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*> eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized
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*> Schur factorization.
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*>
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*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
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*> represent the j-th eigenvalue of the matrix pair (A,B), in
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*> one of the forms lambda = alpha/beta or mu = beta/alpha.
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*> Since either lambda or mu may overflow, they should not,
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*> in general, be computed.
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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 COMPLEX array, dimension (LDQ, N)
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*> On entry, if COMPQ = 'V', the unitary matrix Q1 used in the
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*> reduction of (A,B) to generalized Hessenberg form.
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*> On exit, if COMPQ = 'I', the unitary matrix of left Schur
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*> vectors of (H,T), and if COMPQ = 'V', the unitary matrix of
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*> left Schur vectors of (A,B).
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*> Not referenced if COMPQ = 'N'.
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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. LDQ >= 1.
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*> If COMPQ='V' or 'I', then LDQ >= N.
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*> \endverbatim
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*>
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*> \param[in,out] Z
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*> \verbatim
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*> Z is COMPLEX array, dimension (LDZ, N)
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*> On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
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*> reduction of (A,B) to generalized Hessenberg form.
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*> On exit, if COMPZ = 'I', the unitary matrix of right Schur
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*> vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
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*> right Schur vectors of (A,B).
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*> Not referenced if COMPZ = 'N'.
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*> \endverbatim
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*>
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*> \param[in] LDZ
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*> \verbatim
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*> LDZ is INTEGER
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*> The leading dimension of the array Z. LDZ >= 1.
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*> If COMPZ='V' or 'I', then LDZ >= N.
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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 COMPLEX 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. LWORK >= max(1,N).
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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] RWORK
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*> \verbatim
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*> RWORK is REAL array, dimension (N)
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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*> = 1,...,N: the QZ iteration did not converge. (H,T) is not
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*> in Schur form, but ALPHA(i) and BETA(i),
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*> i=INFO+1,...,N should be correct.
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*> = N+1,...,2*N: the shift calculation failed. (H,T) is not
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*> in Schur form, but ALPHA(i) and BETA(i),
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*> i=INFO-N+1,...,N should be correct.
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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 complexGEcomputational
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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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*> We assume that complex ABS works as long as its value is less than
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*> overflow.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
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$ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
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$ RWORK, 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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CHARACTER COMPQ, COMPZ, JOB
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INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
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* ..
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* .. Array Arguments ..
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REAL RWORK( * )
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COMPLEX ALPHA( * ), BETA( * ), H( LDH, * ),
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$ Q( LDQ, * ), T( LDT, * ), WORK( * ),
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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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COMPLEX CZERO, CONE
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PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
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$ CONE = ( 1.0E+0, 0.0E+0 ) )
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
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REAL HALF
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PARAMETER ( HALF = 0.5E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL ILAZR2, ILAZRO, ILQ, ILSCHR, ILZ, LQUERY
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INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
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$ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
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$ JR, MAXIT
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REAL ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL,
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$ C, SAFMIN, TEMP, TEMP2, TEMPR, ULP
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COMPLEX ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2,
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$ CTEMP3, ESHIFT, S, SHIFT, SIGNBC,
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$ U12, X, ABI12, Y
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* ..
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* .. External Functions ..
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COMPLEX CLADIV
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LOGICAL LSAME
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REAL CLANHS, SLAMCH
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EXTERNAL CLADIV, LSAME, CLANHS, SLAMCH
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* ..
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* .. External Subroutines ..
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EXTERNAL CLARTG, CLASET, CROT, CSCAL, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, MIN, REAL, SQRT
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* ..
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* .. Statement Functions ..
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REAL ABS1
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* ..
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* .. Statement Function definitions ..
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ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) )
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* ..
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* .. Executable Statements ..
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*
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* Decode JOB, COMPQ, COMPZ
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*
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IF( LSAME( JOB, 'E' ) ) THEN
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ILSCHR = .FALSE.
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ISCHUR = 1
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ELSE IF( LSAME( JOB, 'S' ) ) THEN
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ILSCHR = .TRUE.
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ISCHUR = 2
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ELSE
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ILSCHR = .TRUE.
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ISCHUR = 0
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END IF
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*
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IF( LSAME( COMPQ, 'N' ) ) THEN
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ILQ = .FALSE.
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ICOMPQ = 1
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ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
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ILQ = .TRUE.
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ICOMPQ = 2
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ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
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ILQ = .TRUE.
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ICOMPQ = 3
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ELSE
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ILQ = .TRUE.
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ICOMPQ = 0
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END IF
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*
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IF( LSAME( COMPZ, 'N' ) ) THEN
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ILZ = .FALSE.
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ICOMPZ = 1
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ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
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ILZ = .TRUE.
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ICOMPZ = 2
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ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
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ILZ = .TRUE.
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ICOMPZ = 3
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ELSE
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ILZ = .TRUE.
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ICOMPZ = 0
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END IF
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*
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* Check Argument Values
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*
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INFO = 0
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WORK( 1 ) = MAX( 1, N )
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LQUERY = ( LWORK.EQ.-1 )
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IF( ISCHUR.EQ.0 ) THEN
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INFO = -1
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ELSE IF( ICOMPQ.EQ.0 ) THEN
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INFO = -2
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ELSE IF( ICOMPZ.EQ.0 ) THEN
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INFO = -3
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ELSE IF( N.LT.0 ) THEN
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INFO = -4
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ELSE IF( ILO.LT.1 ) THEN
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INFO = -5
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ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
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INFO = -6
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ELSE IF( LDH.LT.N ) THEN
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INFO = -8
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ELSE IF( LDT.LT.N ) THEN
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INFO = -10
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ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
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INFO = -14
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ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
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INFO = -16
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ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
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INFO = -18
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CHGEQZ', -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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* WORK( 1 ) = CMPLX( 1 )
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IF( N.LE.0 ) THEN
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WORK( 1 ) = CMPLX( 1 )
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RETURN
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END IF
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*
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* Initialize Q and Z
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*
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IF( ICOMPQ.EQ.3 )
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$ CALL CLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
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IF( ICOMPZ.EQ.3 )
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$ CALL CLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
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*
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* Machine Constants
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*
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IN = IHI + 1 - ILO
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SAFMIN = SLAMCH( 'S' )
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ULP = SLAMCH( 'E' )*SLAMCH( 'B' )
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ANORM = CLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK )
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BNORM = CLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK )
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ATOL = MAX( SAFMIN, ULP*ANORM )
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BTOL = MAX( SAFMIN, ULP*BNORM )
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ASCALE = ONE / MAX( SAFMIN, ANORM )
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BSCALE = ONE / MAX( SAFMIN, BNORM )
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*
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*
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* Set Eigenvalues IHI+1:N
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*
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DO 10 J = IHI + 1, N
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ABSB = ABS( T( J, J ) )
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IF( ABSB.GT.SAFMIN ) THEN
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SIGNBC = CONJG( T( J, J ) / ABSB )
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T( J, J ) = ABSB
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IF( ILSCHR ) THEN
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CALL CSCAL( J-1, SIGNBC, T( 1, J ), 1 )
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CALL CSCAL( J, SIGNBC, H( 1, J ), 1 )
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ELSE
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CALL CSCAL( 1, SIGNBC, H( J, J ), 1 )
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END IF
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IF( ILZ )
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$ CALL CSCAL( N, SIGNBC, Z( 1, J ), 1 )
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ELSE
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T( J, J ) = CZERO
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END IF
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ALPHA( J ) = H( J, J )
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BETA( J ) = T( J, J )
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10 CONTINUE
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*
|
|
* If IHI < ILO, skip QZ steps
|
|
*
|
|
IF( IHI.LT.ILO )
|
|
$ GO TO 190
|
|
*
|
|
* MAIN QZ ITERATION LOOP
|
|
*
|
|
* Initialize dynamic indices
|
|
*
|
|
* Eigenvalues ILAST+1:N have been found.
|
|
* Column operations modify rows IFRSTM:whatever
|
|
* Row operations modify columns whatever:ILASTM
|
|
*
|
|
* If only eigenvalues are being computed, then
|
|
* IFRSTM is the row of the last splitting row above row ILAST;
|
|
* this is always at least ILO.
|
|
* IITER counts iterations since the last eigenvalue was found,
|
|
* to tell when to use an extraordinary shift.
|
|
* MAXIT is the maximum number of QZ sweeps allowed.
|
|
*
|
|
ILAST = IHI
|
|
IF( ILSCHR ) THEN
|
|
IFRSTM = 1
|
|
ILASTM = N
|
|
ELSE
|
|
IFRSTM = ILO
|
|
ILASTM = IHI
|
|
END IF
|
|
IITER = 0
|
|
ESHIFT = CZERO
|
|
MAXIT = 30*( IHI-ILO+1 )
|
|
*
|
|
DO 170 JITER = 1, MAXIT
|
|
*
|
|
* Check for too many iterations.
|
|
*
|
|
IF( JITER.GT.MAXIT )
|
|
$ GO TO 180
|
|
*
|
|
* Split the matrix if possible.
|
|
*
|
|
* Two tests:
|
|
* 1: H(j,j-1)=0 or j=ILO
|
|
* 2: T(j,j)=0
|
|
*
|
|
* Special case: j=ILAST
|
|
*
|
|
IF( ILAST.EQ.ILO ) THEN
|
|
GO TO 60
|
|
ELSE
|
|
IF( ABS1( H( ILAST, ILAST-1 ) ).LE.MAX( SAFMIN, ULP*(
|
|
$ ABS1( H( ILAST, ILAST ) ) + ABS1( H( ILAST-1, ILAST-1 )
|
|
$ ) ) ) ) THEN
|
|
H( ILAST, ILAST-1 ) = CZERO
|
|
GO TO 60
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
|
|
T( ILAST, ILAST ) = CZERO
|
|
GO TO 50
|
|
END IF
|
|
*
|
|
* General case: j<ILAST
|
|
*
|
|
DO 40 J = ILAST - 1, ILO, -1
|
|
*
|
|
* Test 1: for H(j,j-1)=0 or j=ILO
|
|
*
|
|
IF( J.EQ.ILO ) THEN
|
|
ILAZRO = .TRUE.
|
|
ELSE
|
|
IF( ABS1( H( J, J-1 ) ).LE.MAX( SAFMIN, ULP*(
|
|
$ ABS1( H( J, J ) ) + ABS1( H( J-1, J-1 ) )
|
|
$ ) ) ) THEN
|
|
H( J, J-1 ) = CZERO
|
|
ILAZRO = .TRUE.
|
|
ELSE
|
|
ILAZRO = .FALSE.
|
|
END IF
|
|
END IF
|
|
*
|
|
* Test 2: for T(j,j)=0
|
|
*
|
|
IF( ABS( T( J, J ) ).LT.BTOL ) THEN
|
|
T( J, J ) = CZERO
|
|
*
|
|
* Test 1a: Check for 2 consecutive small subdiagonals in A
|
|
*
|
|
ILAZR2 = .FALSE.
|
|
IF( .NOT.ILAZRO ) THEN
|
|
IF( ABS1( H( J, J-1 ) )*( ASCALE*ABS1( H( J+1,
|
|
$ J ) ) ).LE.ABS1( H( J, J ) )*( ASCALE*ATOL ) )
|
|
$ ILAZR2 = .TRUE.
|
|
END IF
|
|
*
|
|
* If both tests pass (1 & 2), i.e., the leading diagonal
|
|
* element of B in the block is zero, split a 1x1 block off
|
|
* at the top. (I.e., at the J-th row/column) The leading
|
|
* diagonal element of the remainder can also be zero, so
|
|
* this may have to be done repeatedly.
|
|
*
|
|
IF( ILAZRO .OR. ILAZR2 ) THEN
|
|
DO 20 JCH = J, ILAST - 1
|
|
CTEMP = H( JCH, JCH )
|
|
CALL CLARTG( CTEMP, H( JCH+1, JCH ), C, S,
|
|
$ H( JCH, JCH ) )
|
|
H( JCH+1, JCH ) = CZERO
|
|
CALL CROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
|
|
$ H( JCH+1, JCH+1 ), LDH, C, S )
|
|
CALL CROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
|
|
$ T( JCH+1, JCH+1 ), LDT, C, S )
|
|
IF( ILQ )
|
|
$ CALL CROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
|
|
$ C, CONJG( S ) )
|
|
IF( ILAZR2 )
|
|
$ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
|
|
ILAZR2 = .FALSE.
|
|
IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
|
|
IF( JCH+1.GE.ILAST ) THEN
|
|
GO TO 60
|
|
ELSE
|
|
IFIRST = JCH + 1
|
|
GO TO 70
|
|
END IF
|
|
END IF
|
|
T( JCH+1, JCH+1 ) = CZERO
|
|
20 CONTINUE
|
|
GO TO 50
|
|
ELSE
|
|
*
|
|
* Only test 2 passed -- chase the zero to T(ILAST,ILAST)
|
|
* Then process as in the case T(ILAST,ILAST)=0
|
|
*
|
|
DO 30 JCH = J, ILAST - 1
|
|
CTEMP = T( JCH, JCH+1 )
|
|
CALL CLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S,
|
|
$ T( JCH, JCH+1 ) )
|
|
T( JCH+1, JCH+1 ) = CZERO
|
|
IF( JCH.LT.ILASTM-1 )
|
|
$ CALL CROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
|
|
$ T( JCH+1, JCH+2 ), LDT, C, S )
|
|
CALL CROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
|
|
$ H( JCH+1, JCH-1 ), LDH, C, S )
|
|
IF( ILQ )
|
|
$ CALL CROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
|
|
$ C, CONJG( S ) )
|
|
CTEMP = H( JCH+1, JCH )
|
|
CALL CLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S,
|
|
$ H( JCH+1, JCH ) )
|
|
H( JCH+1, JCH-1 ) = CZERO
|
|
CALL CROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
|
|
$ H( IFRSTM, JCH-1 ), 1, C, S )
|
|
CALL CROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
|
|
$ T( IFRSTM, JCH-1 ), 1, C, S )
|
|
IF( ILZ )
|
|
$ CALL CROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
|
|
$ C, S )
|
|
30 CONTINUE
|
|
GO TO 50
|
|
END IF
|
|
ELSE IF( ILAZRO ) THEN
|
|
*
|
|
* Only test 1 passed -- work on J:ILAST
|
|
*
|
|
IFIRST = J
|
|
GO TO 70
|
|
END IF
|
|
*
|
|
* Neither test passed -- try next J
|
|
*
|
|
40 CONTINUE
|
|
*
|
|
* (Drop-through is "impossible")
|
|
*
|
|
INFO = 2*N + 1
|
|
GO TO 210
|
|
*
|
|
* T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
|
|
* 1x1 block.
|
|
*
|
|
50 CONTINUE
|
|
CTEMP = H( ILAST, ILAST )
|
|
CALL CLARTG( CTEMP, H( ILAST, ILAST-1 ), C, S,
|
|
$ H( ILAST, ILAST ) )
|
|
H( ILAST, ILAST-1 ) = CZERO
|
|
CALL CROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
|
|
$ H( IFRSTM, ILAST-1 ), 1, C, S )
|
|
CALL CROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
|
|
$ T( IFRSTM, ILAST-1 ), 1, C, S )
|
|
IF( ILZ )
|
|
$ CALL CROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
|
|
*
|
|
* H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
|
|
*
|
|
60 CONTINUE
|
|
ABSB = ABS( T( ILAST, ILAST ) )
|
|
IF( ABSB.GT.SAFMIN ) THEN
|
|
SIGNBC = CONJG( T( ILAST, ILAST ) / ABSB )
|
|
T( ILAST, ILAST ) = ABSB
|
|
IF( ILSCHR ) THEN
|
|
CALL CSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 )
|
|
CALL CSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ),
|
|
$ 1 )
|
|
ELSE
|
|
CALL CSCAL( 1, SIGNBC, H( ILAST, ILAST ), 1 )
|
|
END IF
|
|
IF( ILZ )
|
|
$ CALL CSCAL( N, SIGNBC, Z( 1, ILAST ), 1 )
|
|
ELSE
|
|
T( ILAST, ILAST ) = CZERO
|
|
END IF
|
|
ALPHA( ILAST ) = H( ILAST, ILAST )
|
|
BETA( ILAST ) = T( ILAST, ILAST )
|
|
*
|
|
* Go to next block -- exit if finished.
|
|
*
|
|
ILAST = ILAST - 1
|
|
IF( ILAST.LT.ILO )
|
|
$ GO TO 190
|
|
*
|
|
* Reset counters
|
|
*
|
|
IITER = 0
|
|
ESHIFT = CZERO
|
|
IF( .NOT.ILSCHR ) THEN
|
|
ILASTM = ILAST
|
|
IF( IFRSTM.GT.ILAST )
|
|
$ IFRSTM = ILO
|
|
END IF
|
|
GO TO 160
|
|
*
|
|
* QZ step
|
|
*
|
|
* This iteration only involves rows/columns IFIRST:ILAST. We
|
|
* assume IFIRST < ILAST, and that the diagonal of B is non-zero.
|
|
*
|
|
70 CONTINUE
|
|
IITER = IITER + 1
|
|
IF( .NOT.ILSCHR ) THEN
|
|
IFRSTM = IFIRST
|
|
END IF
|
|
*
|
|
* Compute the Shift.
|
|
*
|
|
* At this point, IFIRST < ILAST, and the diagonal elements of
|
|
* T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
|
|
* magnitude)
|
|
*
|
|
IF( ( IITER / 10 )*10.NE.IITER ) THEN
|
|
*
|
|
* The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
|
|
* the bottom-right 2x2 block of A inv(B) which is nearest to
|
|
* the bottom-right element.
|
|
*
|
|
* We factor B as U*D, where U has unit diagonals, and
|
|
* compute (A*inv(D))*inv(U).
|
|
*
|
|
U12 = ( BSCALE*T( ILAST-1, ILAST ) ) /
|
|
$ ( BSCALE*T( ILAST, ILAST ) )
|
|
AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
|
|
$ ( BSCALE*T( ILAST-1, ILAST-1 ) )
|
|
AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
|
|
$ ( BSCALE*T( ILAST-1, ILAST-1 ) )
|
|
AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
|
|
$ ( BSCALE*T( ILAST, ILAST ) )
|
|
AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
|
|
$ ( BSCALE*T( ILAST, ILAST ) )
|
|
ABI22 = AD22 - U12*AD21
|
|
ABI12 = AD12 - U12*AD11
|
|
*
|
|
SHIFT = ABI22
|
|
CTEMP = SQRT( ABI12 )*SQRT( AD21 )
|
|
TEMP = ABS1( CTEMP )
|
|
IF( CTEMP.NE.ZERO ) THEN
|
|
X = HALF*( AD11-SHIFT )
|
|
TEMP2 = ABS1( X )
|
|
TEMP = MAX( TEMP, ABS1( X ) )
|
|
Y = TEMP*SQRT( ( X / TEMP )**2+( CTEMP / TEMP )**2 )
|
|
IF( TEMP2.GT.ZERO ) THEN
|
|
IF( REAL( X / TEMP2 )*REAL( Y )+
|
|
$ AIMAG( X / TEMP2 )*AIMAG( Y ).LT.ZERO )Y = -Y
|
|
END IF
|
|
SHIFT = SHIFT - CTEMP*CLADIV( CTEMP, ( X+Y ) )
|
|
END IF
|
|
ELSE
|
|
*
|
|
* Exceptional shift. Chosen for no particularly good reason.
|
|
*
|
|
IF( ( IITER / 20 )*20.EQ.IITER .AND.
|
|
$ BSCALE*ABS1(T( ILAST, ILAST )).GT.SAFMIN ) THEN
|
|
ESHIFT = ESHIFT + ( ASCALE*H( ILAST,
|
|
$ ILAST ) )/( BSCALE*T( ILAST, ILAST ) )
|
|
ELSE
|
|
ESHIFT = ESHIFT + ( ASCALE*H( ILAST,
|
|
$ ILAST-1 ) )/( BSCALE*T( ILAST-1, ILAST-1 ) )
|
|
END IF
|
|
SHIFT = ESHIFT
|
|
END IF
|
|
*
|
|
* Now check for two consecutive small subdiagonals.
|
|
*
|
|
DO 80 J = ILAST - 1, IFIRST + 1, -1
|
|
ISTART = J
|
|
CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) )
|
|
TEMP = ABS1( CTEMP )
|
|
TEMP2 = ASCALE*ABS1( H( J+1, J ) )
|
|
TEMPR = MAX( TEMP, TEMP2 )
|
|
IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
|
|
TEMP = TEMP / TEMPR
|
|
TEMP2 = TEMP2 / TEMPR
|
|
END IF
|
|
IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL )
|
|
$ GO TO 90
|
|
80 CONTINUE
|
|
*
|
|
ISTART = IFIRST
|
|
CTEMP = ASCALE*H( IFIRST, IFIRST ) -
|
|
$ SHIFT*( BSCALE*T( IFIRST, IFIRST ) )
|
|
90 CONTINUE
|
|
*
|
|
* Do an implicit-shift QZ sweep.
|
|
*
|
|
* Initial Q
|
|
*
|
|
CTEMP2 = ASCALE*H( ISTART+1, ISTART )
|
|
CALL CLARTG( CTEMP, CTEMP2, C, S, CTEMP3 )
|
|
*
|
|
* Sweep
|
|
*
|
|
DO 150 J = ISTART, ILAST - 1
|
|
IF( J.GT.ISTART ) THEN
|
|
CTEMP = H( J, J-1 )
|
|
CALL CLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
|
|
H( J+1, J-1 ) = CZERO
|
|
END IF
|
|
*
|
|
DO 100 JC = J, ILASTM
|
|
CTEMP = C*H( J, JC ) + S*H( J+1, JC )
|
|
H( J+1, JC ) = -CONJG( S )*H( J, JC ) + C*H( J+1, JC )
|
|
H( J, JC ) = CTEMP
|
|
CTEMP2 = C*T( J, JC ) + S*T( J+1, JC )
|
|
T( J+1, JC ) = -CONJG( S )*T( J, JC ) + C*T( J+1, JC )
|
|
T( J, JC ) = CTEMP2
|
|
100 CONTINUE
|
|
IF( ILQ ) THEN
|
|
DO 110 JR = 1, N
|
|
CTEMP = C*Q( JR, J ) + CONJG( S )*Q( JR, J+1 )
|
|
Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
|
|
Q( JR, J ) = CTEMP
|
|
110 CONTINUE
|
|
END IF
|
|
*
|
|
CTEMP = T( J+1, J+1 )
|
|
CALL CLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
|
|
T( J+1, J ) = CZERO
|
|
*
|
|
DO 120 JR = IFRSTM, MIN( J+2, ILAST )
|
|
CTEMP = C*H( JR, J+1 ) + S*H( JR, J )
|
|
H( JR, J ) = -CONJG( S )*H( JR, J+1 ) + C*H( JR, J )
|
|
H( JR, J+1 ) = CTEMP
|
|
120 CONTINUE
|
|
DO 130 JR = IFRSTM, J
|
|
CTEMP = C*T( JR, J+1 ) + S*T( JR, J )
|
|
T( JR, J ) = -CONJG( S )*T( JR, J+1 ) + C*T( JR, J )
|
|
T( JR, J+1 ) = CTEMP
|
|
130 CONTINUE
|
|
IF( ILZ ) THEN
|
|
DO 140 JR = 1, N
|
|
CTEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
|
|
Z( JR, J ) = -CONJG( S )*Z( JR, J+1 ) + C*Z( JR, J )
|
|
Z( JR, J+1 ) = CTEMP
|
|
140 CONTINUE
|
|
END IF
|
|
150 CONTINUE
|
|
*
|
|
160 CONTINUE
|
|
*
|
|
170 CONTINUE
|
|
*
|
|
* Drop-through = non-convergence
|
|
*
|
|
180 CONTINUE
|
|
INFO = ILAST
|
|
GO TO 210
|
|
*
|
|
* Successful completion of all QZ steps
|
|
*
|
|
190 CONTINUE
|
|
*
|
|
* Set Eigenvalues 1:ILO-1
|
|
*
|
|
DO 200 J = 1, ILO - 1
|
|
ABSB = ABS( T( J, J ) )
|
|
IF( ABSB.GT.SAFMIN ) THEN
|
|
SIGNBC = CONJG( T( J, J ) / ABSB )
|
|
T( J, J ) = ABSB
|
|
IF( ILSCHR ) THEN
|
|
CALL CSCAL( J-1, SIGNBC, T( 1, J ), 1 )
|
|
CALL CSCAL( J, SIGNBC, H( 1, J ), 1 )
|
|
ELSE
|
|
CALL CSCAL( 1, SIGNBC, H( J, J ), 1 )
|
|
END IF
|
|
IF( ILZ )
|
|
$ CALL CSCAL( N, SIGNBC, Z( 1, J ), 1 )
|
|
ELSE
|
|
T( J, J ) = CZERO
|
|
END IF
|
|
ALPHA( J ) = H( J, J )
|
|
BETA( J ) = T( J, J )
|
|
200 CONTINUE
|
|
*
|
|
* Normal Termination
|
|
*
|
|
INFO = 0
|
|
*
|
|
* Exit (other than argument error) -- return optimal workspace size
|
|
*
|
|
210 CONTINUE
|
|
WORK( 1 ) = CMPLX( N )
|
|
RETURN
|
|
*
|
|
* End of CHGEQZ
|
|
*
|
|
END
|
|
|