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SUBROUTINE <a name="DGGESX.1"></a><a href="dggesx.f.html#DGGESX.1">DGGESX</a>( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
$ B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
$ VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
$ LIWORK, BWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK driver routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> CHARACTER JOBVSL, JOBVSR, SENSE, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
$ SDIM
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), RCONDE( 2 ),
$ RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ),
$ WORK( * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Function Arguments ..
</span> LOGICAL SELCTG
EXTERNAL SELCTG
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DGGESX.31"></a><a href="dggesx.f.html#DGGESX.1">DGGESX</a> computes for a pair of N-by-N real nonsymmetric matrices
</span><span class="comment">*</span><span class="comment"> (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
</span><span class="comment">*</span><span class="comment"> optionally, the left and/or right matrices of Schur vectors (VSL and
</span><span class="comment">*</span><span class="comment"> VSR). This gives the generalized Schur factorization
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Optionally, it also orders the eigenvalues so that a selected cluster
</span><span class="comment">*</span><span class="comment"> of eigenvalues appears in the leading diagonal blocks of the upper
</span><span class="comment">*</span><span class="comment"> quasi-triangular matrix S and the upper triangular matrix T; computes
</span><span class="comment">*</span><span class="comment"> a reciprocal condition number for the average of the selected
</span><span class="comment">*</span><span class="comment"> eigenvalues (RCONDE); and computes a reciprocal condition number for
</span><span class="comment">*</span><span class="comment"> the right and left deflating subspaces corresponding to the selected
</span><span class="comment">*</span><span class="comment"> eigenvalues (RCONDV). The leading columns of VSL and VSR then form
</span><span class="comment">*</span><span class="comment"> an orthonormal basis for the corresponding left and right eigenspaces
</span><span class="comment">*</span><span class="comment"> (deflating subspaces).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
</span><span class="comment">*</span><span class="comment"> or a ratio alpha/beta = w, such that A - w*B is singular. It is
</span><span class="comment">*</span><span class="comment"> usually represented as the pair (alpha,beta), as there is a
</span><span class="comment">*</span><span class="comment"> reasonable interpretation for beta=0 or for both being zero.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A pair of matrices (S,T) is in generalized real Schur form if T is
</span><span class="comment">*</span><span class="comment"> upper triangular with non-negative diagonal and S is block upper
</span><span class="comment">*</span><span class="comment"> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
</span><span class="comment">*</span><span class="comment"> to real generalized eigenvalues, while 2-by-2 blocks of S will be
</span><span class="comment">*</span><span class="comment"> "standardized" by making the corresponding elements of T have the
</span><span class="comment">*</span><span class="comment"> form:
</span><span class="comment">*</span><span class="comment"> [ a 0 ]
</span><span class="comment">*</span><span class="comment"> [ 0 b ]
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> and the pair of corresponding 2-by-2 blocks in S and T will have a
</span><span class="comment">*</span><span class="comment"> complex conjugate pair of generalized eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> JOBVSL (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'N': do not compute the left Schur vectors;
</span><span class="comment">*</span><span class="comment"> = 'V': compute the left Schur vectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> JOBVSR (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'N': do not compute the right Schur vectors;
</span><span class="comment">*</span><span class="comment"> = 'V': compute the right Schur vectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> SORT (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> Specifies whether or not to order the eigenvalues on the
</span><span class="comment">*</span><span class="comment"> diagonal of the generalized Schur form.
</span><span class="comment">*</span><span class="comment"> = 'N': Eigenvalues are not ordered;
</span><span class="comment">*</span><span class="comment"> = 'S': Eigenvalues are ordered (see SELCTG).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> SELCTG (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISION arguments
</span><span class="comment">*</span><span class="comment"> SELCTG must be declared EXTERNAL in the calling subroutine.
</span><span class="comment">*</span><span class="comment"> If SORT = 'N', SELCTG is not referenced.
</span><span class="comment">*</span><span class="comment"> If SORT = 'S', SELCTG is used to select eigenvalues to sort
</span><span class="comment">*</span><span class="comment"> to the top left of the Schur form.
</span><span class="comment">*</span><span class="comment"> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
</span><span class="comment">*</span><span class="comment"> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
</span><span class="comment">*</span><span class="comment"> one of a complex conjugate pair of eigenvalues is selected,
</span><span class="comment">*</span><span class="comment"> then both complex eigenvalues are selected.
</span><span class="comment">*</span><span class="comment"> Note that a selected complex eigenvalue may no longer satisfy
</span><span class="comment">*</span><span class="comment"> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
</span><span class="comment">*</span><span class="comment"> since ordering may change the value of complex eigenvalues
</span><span class="comment">*</span><span class="comment"> (especially if the eigenvalue is ill-conditioned), in this
</span><span class="comment">*</span><span class="comment"> case INFO is set to N+3.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> SENSE (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> Determines which reciprocal condition numbers are computed.
</span><span class="comment">*</span><span class="comment"> = 'N' : None are computed;
</span><span class="comment">*</span><span class="comment"> = 'E' : Computed for average of selected eigenvalues only;
</span><span class="comment">*</span><span class="comment"> = 'V' : Computed for selected deflating subspaces only;
</span><span class="comment">*</span><span class="comment"> = 'B' : Computed for both.
</span><span class="comment">*</span><span class="comment"> If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrices A, B, VSL, and VSR. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
</span><span class="comment">*</span><span class="comment"> On entry, the first of the pair of matrices.
</span><span class="comment">*</span><span class="comment"> On exit, A has been overwritten by its generalized Schur
</span><span class="comment">*</span><span class="comment"> form S.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of A. LDA >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
</span><span class="comment">*</span><span class="comment"> On entry, the second of the pair of matrices.
</span><span class="comment">*</span><span class="comment"> On exit, B has been overwritten by its generalized Schur
</span><span class="comment">*</span><span class="comment"> form T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of B. LDB >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> SDIM (output) INTEGER
</span><span class="comment">*</span><span class="comment"> If SORT = 'N', SDIM = 0.
</span><span class="comment">*</span><span class="comment"> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
</span><span class="comment">*</span><span class="comment"> for which SELCTG is true. (Complex conjugate pairs for which
</span><span class="comment">*</span><span class="comment"> SELCTG is true for either eigenvalue count as 2.)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ALPHAR (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> ALPHAI (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> BETA (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
</span><span class="comment">*</span><span class="comment"> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i
</span><span class="comment">*</span><span class="comment"> and BETA(j),j=1,...,N are the diagonals of the complex Schur
</span><span class="comment">*</span><span class="comment"> form (S,T) that would result if the 2-by-2 diagonal blocks of
</span><span class="comment">*</span><span class="comment"> the real Schur form of (A,B) were further reduced to
</span><span class="comment">*</span><span class="comment"> triangular form using 2-by-2 complex unitary transformations.
</span><span class="comment">*</span><span class="comment"> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
</span><span class="comment">*</span><span class="comment"> positive, then the j-th and (j+1)-st eigenvalues are a
</span><span class="comment">*</span><span class="comment"> complex conjugate pair, with ALPHAI(j+1) negative.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
</span><span class="comment">*</span><span class="comment"> may easily over- or underflow, and BETA(j) may even be zero.
</span><span class="comment">*</span><span class="comment"> Thus, the user should avoid naively computing the ratio.
</span><span class="comment">*</span><span class="comment"> However, ALPHAR and ALPHAI will be always less than and
</span><span class="comment">*</span><span class="comment"> usually comparable with norm(A) in magnitude, and BETA always
</span><span class="comment">*</span><span class="comment"> less than and usually comparable with norm(B).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N)
</span><span class="comment">*</span><span class="comment"> If JOBVSL = 'V', VSL will contain the left Schur vectors.
</span><span class="comment">*</span><span class="comment"> Not referenced if JOBVSL = 'N'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDVSL (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the matrix VSL. LDVSL >=1, and
</span><span class="comment">*</span><span class="comment"> if JOBVSL = 'V', LDVSL >= N.
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