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SUBROUTINE <a name="CGGES.1"></a><a href="cgges.f.html#CGGES.1">CGGES</a>( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
$ SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
$ LWORK, RWORK, 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, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> LOGICAL BWORK( * )
REAL RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), 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="CGGES.28"></a><a href="cgges.f.html#CGGES.1">CGGES</a> computes for a pair of N-by-N complex nonsymmetric matrices
</span><span class="comment">*</span><span class="comment"> (A,B), the generalized eigenvalues, the generalized complex Schur
</span><span class="comment">*</span><span class="comment"> form (S, T), and optionally left and/or right Schur vectors (VSL
</span><span class="comment">*</span><span class="comment"> and 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)**H, (VSL)*T*(VSR)**H )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where (VSR)**H is the conjugate-transpose of VSR.
</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"> triangular matrix S and the upper triangular matrix T. The leading
</span><span class="comment">*</span><span class="comment"> columns of VSL and VSR then form an unitary basis for the
</span><span class="comment">*</span><span class="comment"> corresponding left and right eigenspaces (deflating subspaces).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> (If only the generalized eigenvalues are needed, use the driver
</span><span class="comment">*</span><span class="comment"> <a name="CGGEV.44"></a><a href="cggev.f.html#CGGEV.1">CGGEV</a> instead, which is faster.)
</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, and even 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 complex Schur form if S
</span><span class="comment">*</span><span class="comment"> and T are upper triangular and, in addition, the diagonal elements
</span><span class="comment">*</span><span class="comment"> of T are non-negative real numbers.
</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 two COMPLEX 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 ALPHA(j)/BETA(j) is selected if
</span><span class="comment">*</span><span class="comment"> SELCTG(ALPHA(j),BETA(j)) is true.
</span><span class="comment">*</span><span class="comment">
</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(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
</span><span class="comment">*</span><span class="comment"> 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+2 (See INFO below).
</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) COMPLEX 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) COMPLEX 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.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ALPHA (output) COMPLEX array, dimension (N)
</span><span class="comment">*</span><span class="comment"> BETA (output) COMPLEX array, dimension (N)
</span><span class="comment">*</span><span class="comment"> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
</span><span class="comment">*</span><span class="comment"> generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j),
</span><span class="comment">*</span><span class="comment"> j=1,...,N are the diagonals of the complex Schur form (A,B)
</span><span class="comment">*</span><span class="comment"> output by <a name="CGGES.115"></a><a href="cgges.f.html#CGGES.1">CGGES</a>. The BETA(j) will be non-negative real.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Note: the quotients ALPHA(j)/BETA(j) may easily over- or
</span><span class="comment">*</span><span class="comment"> underflow, and BETA(j) may even be zero. Thus, the user
</span><span class="comment">*</span><span class="comment"> should avoid naively computing the ratio alpha/beta.
</span><span class="comment">*</span><span class="comment"> However, ALPHA will be always less than and usually
</span><span class="comment">*</span><span class="comment"> comparable with norm(A) in magnitude, and BETA always less
</span><span class="comment">*</span><span class="comment"> 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) COMPLEX 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.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VSR (output) COMPLEX array, dimension (LDVSR,N)
</span><span class="comment">*</span><span class="comment"> If JOBVSR = 'V', VSR will contain the right Schur vectors.
</span><span class="comment">*</span><span class="comment"> Not referenced if JOBVSR = 'N'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDVSR (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the matrix VSR. LDVSR >= 1, and
</span><span class="comment">*</span><span class="comment"> if JOBVSR = 'V', LDVSR >= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
</span><span class="comment">*</span><span class="comment"> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LWORK (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The dimension of the array WORK. LWORK >= max(1,2*N).
</span><span class="comment">*</span><span class="comment"> For good performance, LWORK must generally be larger.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If LWORK = -1, then a workspace query is assumed; the routine
</span><span class="comment">*</span><span class="comment"> only calculates the optimal size of the WORK array, returns
</span><span class="comment">*</span><span class="comment"> this value as the first entry of the WORK array, and no error
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