770 lines
25 KiB
Fortran
770 lines
25 KiB
Fortran
*> \brief <b> SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
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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 SGEGV + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgegv.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/sgegv.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/sgegv.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 SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
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* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER JOBVL, JOBVR
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* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
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* ..
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* .. Array Arguments ..
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* REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
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* $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
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* $ VR( LDVR, * ), WORK( * )
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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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*> This routine is deprecated and has been replaced by routine SGGEV.
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*>
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*> SGEGV computes the eigenvalues and, optionally, the left and/or right
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*> eigenvectors of a real matrix pair (A,B).
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*> Given two square matrices A and B,
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*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
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*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
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*> that
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*>
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*> A*x = lambda*B*x.
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*>
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*> An alternate form is to find the eigenvalues mu and corresponding
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*> eigenvectors y such that
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*>
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*> mu*A*y = B*y.
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*>
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*> These two forms are equivalent with mu = 1/lambda and x = y if
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*> neither lambda nor mu is zero. In order to deal with the case that
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*> lambda or mu is zero or small, two values alpha and beta are returned
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*> for each eigenvalue, such that lambda = alpha/beta and
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*> mu = beta/alpha.
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*>
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*> The vectors x and y in the above equations are right eigenvectors of
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*> the matrix pair (A,B). Vectors u and v satisfying
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*>
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*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
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*>
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*> are left eigenvectors of (A,B).
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*>
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*> Note: this routine performs "full balancing" on A and B
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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] JOBVL
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*> \verbatim
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*> JOBVL is CHARACTER*1
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*> = 'N': do not compute the left generalized eigenvectors;
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*> = 'V': compute the left generalized eigenvectors (returned
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*> in VL).
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*> \endverbatim
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*>
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*> \param[in] JOBVR
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*> \verbatim
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*> JOBVR is CHARACTER*1
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*> = 'N': do not compute the right generalized eigenvectors;
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*> = 'V': compute the right generalized eigenvectors (returned
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*> in VR).
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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 A, B, VL, and VR. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is REAL array, dimension (LDA, N)
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*> On entry, the matrix A.
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*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
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*> contains the real Schur form of A from the generalized Schur
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*> factorization of the pair (A,B) after balancing.
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*> If no eigenvectors were computed, then only the diagonal
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*> blocks from the Schur form will be correct. See SGGHRD and
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*> SHGEQZ for details.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of A. LDA >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in,out] B
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*> \verbatim
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*> B is REAL array, dimension (LDB, N)
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*> On entry, the matrix B.
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*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
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*> upper triangular matrix obtained from B in the generalized
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*> Schur factorization of the pair (A,B) after balancing.
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*> If no eigenvectors were computed, then only those elements of
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*> B corresponding to the diagonal blocks from the Schur form of
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*> A will be correct. See SGGHRD and SHGEQZ for details.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of B. LDB >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] ALPHAR
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*> \verbatim
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*> ALPHAR is REAL array, dimension (N)
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*> The real parts of each scalar alpha defining an eigenvalue of
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*> GNEP.
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*> \endverbatim
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*>
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*> \param[out] ALPHAI
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*> \verbatim
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*> ALPHAI is REAL array, dimension (N)
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*> The imaginary parts of each scalar alpha defining an
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*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
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*> eigenvalue is real; if positive, then the j-th and
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*> (j+1)-st eigenvalues are a complex conjugate pair, with
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*> ALPHAI(j+1) = -ALPHAI(j).
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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 REAL array, dimension (N)
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*> The scalars beta that define the eigenvalues of GNEP.
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*>
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*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
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*> beta = BETA(j) represent the j-th eigenvalue of the matrix
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*> pair (A,B), in one of the forms lambda = alpha/beta or
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*> mu = beta/alpha. Since either lambda or mu may overflow,
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*> they should not, in general, be computed.
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*> \endverbatim
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*>
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*> \param[out] VL
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*> \verbatim
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*> VL is REAL array, dimension (LDVL,N)
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*> If JOBVL = 'V', the left eigenvectors u(j) are stored
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*> in the columns of VL, in the same order as their eigenvalues.
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*> If the j-th eigenvalue is real, then u(j) = VL(:,j).
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*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
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*> pair, then
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*> u(j) = VL(:,j) + i*VL(:,j+1)
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*> and
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*> u(j+1) = VL(:,j) - i*VL(:,j+1).
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*>
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*> Each eigenvector is scaled so that its largest component has
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*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
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*> corresponding to an eigenvalue with alpha = beta = 0, which
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*> are set to zero.
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*> Not referenced if JOBVL = 'N'.
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*> \endverbatim
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*>
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*> \param[in] LDVL
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*> \verbatim
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*> LDVL is INTEGER
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*> The leading dimension of the matrix VL. LDVL >= 1, and
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*> if JOBVL = 'V', LDVL >= N.
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*> \endverbatim
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*>
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*> \param[out] VR
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*> \verbatim
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*> VR is REAL array, dimension (LDVR,N)
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*> If JOBVR = 'V', the right eigenvectors x(j) are stored
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*> in the columns of VR, in the same order as their eigenvalues.
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*> If the j-th eigenvalue is real, then x(j) = VR(:,j).
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*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
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*> pair, then
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*> x(j) = VR(:,j) + i*VR(:,j+1)
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*> and
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*> x(j+1) = VR(:,j) - i*VR(:,j+1).
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*>
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*> Each eigenvector is scaled so that its largest component has
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*> abs(real part) + abs(imag. part) = 1, except for eigenvalues
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*> corresponding to an eigenvalue with alpha = beta = 0, which
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*> are set to zero.
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*> Not referenced if JOBVR = 'N'.
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*> \endverbatim
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*>
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*> \param[in] LDVR
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*> \verbatim
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*> LDVR is INTEGER
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*> The leading dimension of the matrix VR. LDVR >= 1, and
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*> if JOBVR = 'V', LDVR >= 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 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. LWORK >= max(1,8*N).
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*> For good performance, LWORK must generally be larger.
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*> To compute the optimal value of LWORK, call ILAENV to get
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*> blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute:
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*> NB -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR;
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*> The optimal LWORK is:
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*> 2*N + MAX( 6*N, N*(NB+1) ).
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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] 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:
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*> The QZ iteration failed. No eigenvectors have been
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*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
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*> should be correct for j=INFO+1,...,N.
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*> > N: errors that usually indicate LAPACK problems:
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*> =N+1: error return from SGGBAL
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*> =N+2: error return from SGEQRF
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*> =N+3: error return from SORMQR
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*> =N+4: error return from SORGQR
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*> =N+5: error return from SGGHRD
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*> =N+6: error return from SHGEQZ (other than failed
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*> iteration)
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*> =N+7: error return from STGEVC
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*> =N+8: error return from SGGBAK (computing VL)
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*> =N+9: error return from SGGBAK (computing VR)
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*> =N+10: error return from SLASCL (various calls)
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date November 2011
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*
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*> \ingroup realGEeigen
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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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*> Balancing
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*> ---------
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*>
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*> This driver calls SGGBAL to both permute and scale rows and columns
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*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
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*> and PL*B*R will be upper triangular except for the diagonal blocks
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*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
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*> possible. The diagonal scaling matrices DL and DR are chosen so
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*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
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*> one (except for the elements that start out zero.)
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*>
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*> After the eigenvalues and eigenvectors of the balanced matrices
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*> have been computed, SGGBAK transforms the eigenvectors back to what
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*> they would have been (in perfect arithmetic) if they had not been
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*> balanced.
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*>
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*> Contents of A and B on Exit
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*> -------- -- - --- - -- ----
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*>
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*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
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*> both), then on exit the arrays A and B will contain the real Schur
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*> form[*] of the "balanced" versions of A and B. If no eigenvectors
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*> are computed, then only the diagonal blocks will be correct.
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*>
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*> [*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
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*> by Golub & van Loan, pub. by Johns Hopkins U. Press.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
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$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
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*
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* -- LAPACK driver routine (version 3.4.0) --
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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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* November 2011
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*
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* .. Scalar Arguments ..
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CHARACTER JOBVL, JOBVR
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INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
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* ..
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* .. Array Arguments ..
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REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
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$ B( LDB, * ), BETA( * ), VL( LDVL, * ),
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$ VR( LDVR, * ), WORK( * )
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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 = 0.0E0, ONE = 1.0E0 )
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* ..
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* .. Local Scalars ..
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LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
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CHARACTER CHTEMP
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INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
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$ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
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$ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
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REAL ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
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$ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
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$ SALFAI, SALFAR, SBETA, SCALE, TEMP
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* ..
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* .. Local Arrays ..
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LOGICAL LDUMMA( 1 )
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* ..
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* .. External Subroutines ..
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EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLACPY,
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$ SLASCL, SLASET, SORGQR, SORMQR, STGEVC, XERBLA
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILAENV
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REAL SLAMCH, SLANGE
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EXTERNAL ILAENV, LSAME, SLAMCH, SLANGE
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, INT, MAX
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* ..
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* .. Executable Statements ..
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*
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* Decode the input arguments
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*
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IF( LSAME( JOBVL, 'N' ) ) THEN
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IJOBVL = 1
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ILVL = .FALSE.
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ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
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IJOBVL = 2
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ILVL = .TRUE.
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ELSE
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IJOBVL = -1
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ILVL = .FALSE.
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END IF
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*
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IF( LSAME( JOBVR, 'N' ) ) THEN
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IJOBVR = 1
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ILVR = .FALSE.
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ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
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IJOBVR = 2
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ILVR = .TRUE.
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ELSE
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IJOBVR = -1
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ILVR = .FALSE.
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END IF
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ILV = ILVL .OR. ILVR
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*
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* Test the input arguments
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*
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LWKMIN = MAX( 8*N, 1 )
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LWKOPT = LWKMIN
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WORK( 1 ) = LWKOPT
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LQUERY = ( LWORK.EQ.-1 )
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INFO = 0
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IF( IJOBVL.LE.0 ) THEN
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INFO = -1
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ELSE IF( IJOBVR.LE.0 ) 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( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -5
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -7
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ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
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INFO = -12
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ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
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INFO = -14
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ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
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INFO = -16
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END IF
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*
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IF( INFO.EQ.0 ) THEN
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NB1 = ILAENV( 1, 'SGEQRF', ' ', N, N, -1, -1 )
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NB2 = ILAENV( 1, 'SORMQR', ' ', N, N, N, -1 )
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NB3 = ILAENV( 1, 'SORGQR', ' ', N, N, N, -1 )
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NB = MAX( NB1, NB2, NB3 )
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LOPT = 2*N + MAX( 6*N, N*(NB+1) )
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WORK( 1 ) = LOPT
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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( 'SGEGV ', -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( N.EQ.0 )
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$ RETURN
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*
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* Get machine constants
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*
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EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
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SAFMIN = SLAMCH( 'S' )
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SAFMIN = SAFMIN + SAFMIN
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SAFMAX = ONE / SAFMIN
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ONEPLS = ONE + ( 4*EPS )
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*
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* Scale A
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*
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ANRM = SLANGE( 'M', N, N, A, LDA, WORK )
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ANRM1 = ANRM
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ANRM2 = ONE
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IF( ANRM.LT.ONE ) THEN
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IF( SAFMAX*ANRM.LT.ONE ) THEN
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ANRM1 = SAFMIN
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ANRM2 = SAFMAX*ANRM
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END IF
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END IF
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*
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IF( ANRM.GT.ZERO ) THEN
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CALL SLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
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IF( IINFO.NE.0 ) THEN
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INFO = N + 10
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RETURN
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END IF
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END IF
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*
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* Scale B
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*
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BNRM = SLANGE( 'M', N, N, B, LDB, WORK )
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BNRM1 = BNRM
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BNRM2 = ONE
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IF( BNRM.LT.ONE ) THEN
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IF( SAFMAX*BNRM.LT.ONE ) THEN
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BNRM1 = SAFMIN
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BNRM2 = SAFMAX*BNRM
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END IF
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END IF
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*
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IF( BNRM.GT.ZERO ) THEN
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CALL SLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 10
|
|
RETURN
|
|
END IF
|
|
END IF
|
|
*
|
|
* Permute the matrix to make it more nearly triangular
|
|
* Workspace layout: (8*N words -- "work" requires 6*N words)
|
|
* left_permutation, right_permutation, work...
|
|
*
|
|
ILEFT = 1
|
|
IRIGHT = N + 1
|
|
IWORK = IRIGHT + N
|
|
CALL SGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
|
|
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 1
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
* Reduce B to triangular form, and initialize VL and/or VR
|
|
* Workspace layout: ("work..." must have at least N words)
|
|
* left_permutation, right_permutation, tau, work...
|
|
*
|
|
IROWS = IHI + 1 - ILO
|
|
IF( ILV ) THEN
|
|
ICOLS = N + 1 - ILO
|
|
ELSE
|
|
ICOLS = IROWS
|
|
END IF
|
|
ITAU = IWORK
|
|
IWORK = ITAU + IROWS
|
|
CALL SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
|
IF( IINFO.GE.0 )
|
|
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 2
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
CALL SORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
|
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
|
|
$ LWORK+1-IWORK, IINFO )
|
|
IF( IINFO.GE.0 )
|
|
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 3
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
IF( ILVL ) THEN
|
|
CALL SLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
|
|
CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
|
$ VL( ILO+1, ILO ), LDVL )
|
|
CALL SORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
|
|
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
|
|
$ IINFO )
|
|
IF( IINFO.GE.0 )
|
|
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 4
|
|
GO TO 120
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( ILVR )
|
|
$ CALL SLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
|
|
*
|
|
* Reduce to generalized Hessenberg form
|
|
*
|
|
IF( ILV ) THEN
|
|
*
|
|
* Eigenvectors requested -- work on whole matrix.
|
|
*
|
|
CALL SGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
|
|
$ LDVL, VR, LDVR, IINFO )
|
|
ELSE
|
|
CALL SGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
|
|
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
|
|
END IF
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 5
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
* Perform QZ algorithm
|
|
* Workspace layout: ("work..." must have at least 1 word)
|
|
* left_permutation, right_permutation, work...
|
|
*
|
|
IWORK = ITAU
|
|
IF( ILV ) THEN
|
|
CHTEMP = 'S'
|
|
ELSE
|
|
CHTEMP = 'E'
|
|
END IF
|
|
CALL SHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
|
|
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
|
|
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
|
|
IF( IINFO.GE.0 )
|
|
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
|
|
IF( IINFO.NE.0 ) THEN
|
|
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
|
|
INFO = IINFO
|
|
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
|
|
INFO = IINFO - N
|
|
ELSE
|
|
INFO = N + 6
|
|
END IF
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
IF( ILV ) THEN
|
|
*
|
|
* Compute Eigenvectors (STGEVC requires 6*N words of workspace)
|
|
*
|
|
IF( ILVL ) THEN
|
|
IF( ILVR ) THEN
|
|
CHTEMP = 'B'
|
|
ELSE
|
|
CHTEMP = 'L'
|
|
END IF
|
|
ELSE
|
|
CHTEMP = 'R'
|
|
END IF
|
|
*
|
|
CALL STGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
|
|
$ VR, LDVR, N, IN, WORK( IWORK ), IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 7
|
|
GO TO 120
|
|
END IF
|
|
*
|
|
* Undo balancing on VL and VR, rescale
|
|
*
|
|
IF( ILVL ) THEN
|
|
CALL SGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
|
|
$ WORK( IRIGHT ), N, VL, LDVL, IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 8
|
|
GO TO 120
|
|
END IF
|
|
DO 50 JC = 1, N
|
|
IF( ALPHAI( JC ).LT.ZERO )
|
|
$ GO TO 50
|
|
TEMP = ZERO
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 10 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
|
|
10 CONTINUE
|
|
ELSE
|
|
DO 20 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
|
|
$ ABS( VL( JR, JC+1 ) ) )
|
|
20 CONTINUE
|
|
END IF
|
|
IF( TEMP.LT.SAFMIN )
|
|
$ GO TO 50
|
|
TEMP = ONE / TEMP
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 30 JR = 1, N
|
|
VL( JR, JC ) = VL( JR, JC )*TEMP
|
|
30 CONTINUE
|
|
ELSE
|
|
DO 40 JR = 1, N
|
|
VL( JR, JC ) = VL( JR, JC )*TEMP
|
|
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
|
|
40 CONTINUE
|
|
END IF
|
|
50 CONTINUE
|
|
END IF
|
|
IF( ILVR ) THEN
|
|
CALL SGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
|
|
$ WORK( IRIGHT ), N, VR, LDVR, IINFO )
|
|
IF( IINFO.NE.0 ) THEN
|
|
INFO = N + 9
|
|
GO TO 120
|
|
END IF
|
|
DO 100 JC = 1, N
|
|
IF( ALPHAI( JC ).LT.ZERO )
|
|
$ GO TO 100
|
|
TEMP = ZERO
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 60 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
|
|
60 CONTINUE
|
|
ELSE
|
|
DO 70 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
|
|
$ ABS( VR( JR, JC+1 ) ) )
|
|
70 CONTINUE
|
|
END IF
|
|
IF( TEMP.LT.SAFMIN )
|
|
$ GO TO 100
|
|
TEMP = ONE / TEMP
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 80 JR = 1, N
|
|
VR( JR, JC ) = VR( JR, JC )*TEMP
|
|
80 CONTINUE
|
|
ELSE
|
|
DO 90 JR = 1, N
|
|
VR( JR, JC ) = VR( JR, JC )*TEMP
|
|
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
|
|
90 CONTINUE
|
|
END IF
|
|
100 CONTINUE
|
|
END IF
|
|
*
|
|
* End of eigenvector calculation
|
|
*
|
|
END IF
|
|
*
|
|
* Undo scaling in alpha, beta
|
|
*
|
|
* Note: this does not give the alpha and beta for the unscaled
|
|
* problem.
|
|
*
|
|
* Un-scaling is limited to avoid underflow in alpha and beta
|
|
* if they are significant.
|
|
*
|
|
DO 110 JC = 1, N
|
|
ABSAR = ABS( ALPHAR( JC ) )
|
|
ABSAI = ABS( ALPHAI( JC ) )
|
|
ABSB = ABS( BETA( JC ) )
|
|
SALFAR = ANRM*ALPHAR( JC )
|
|
SALFAI = ANRM*ALPHAI( JC )
|
|
SBETA = BNRM*BETA( JC )
|
|
ILIMIT = .FALSE.
|
|
SCALE = ONE
|
|
*
|
|
* Check for significant underflow in ALPHAI
|
|
*
|
|
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
|
|
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
|
|
ILIMIT = .TRUE.
|
|
SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
|
|
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
|
|
*
|
|
ELSE IF( SALFAI.EQ.ZERO ) THEN
|
|
*
|
|
* If insignificant underflow in ALPHAI, then make the
|
|
* conjugate eigenvalue real.
|
|
*
|
|
IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
|
|
ALPHAI( JC-1 ) = ZERO
|
|
ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
|
|
ALPHAI( JC+1 ) = ZERO
|
|
END IF
|
|
END IF
|
|
*
|
|
* Check for significant underflow in ALPHAR
|
|
*
|
|
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
|
|
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
|
|
ILIMIT = .TRUE.
|
|
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
|
|
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
|
|
END IF
|
|
*
|
|
* Check for significant underflow in BETA
|
|
*
|
|
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
|
|
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
|
|
ILIMIT = .TRUE.
|
|
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
|
|
$ MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
|
|
END IF
|
|
*
|
|
* Check for possible overflow when limiting scaling
|
|
*
|
|
IF( ILIMIT ) THEN
|
|
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
|
|
$ ABS( SBETA ) )
|
|
IF( TEMP.GT.ONE )
|
|
$ SCALE = SCALE / TEMP
|
|
IF( SCALE.LT.ONE )
|
|
$ ILIMIT = .FALSE.
|
|
END IF
|
|
*
|
|
* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
|
|
*
|
|
IF( ILIMIT ) THEN
|
|
SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
|
|
SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
|
|
SBETA = ( SCALE*BETA( JC ) )*BNRM
|
|
END IF
|
|
ALPHAR( JC ) = SALFAR
|
|
ALPHAI( JC ) = SALFAI
|
|
BETA( JC ) = SBETA
|
|
110 CONTINUE
|
|
*
|
|
120 CONTINUE
|
|
WORK( 1 ) = LWKOPT
|
|
*
|
|
RETURN
|
|
*
|
|
* End of SGEGV
|
|
*
|
|
END
|