dggesx()

subroutine dggesx	(	character	jobvsl,
		character	jobvsr,
		character	sort,
		external	selctg,
		character	sense,
		integer	n,
		double precision, dimension( lda, * )	a,
		integer	lda,
		double precision, dimension( ldb, * )	b,
		integer	ldb,
		integer	sdim,
		double precision, dimension( * )	alphar,
		double precision, dimension( * )	alphai,
		double precision, dimension( * )	beta,
		double precision, dimension( ldvsl, * )	vsl,
		integer	ldvsl,
		double precision, dimension( ldvsr, * )	vsr,
		integer	ldvsr,
		double precision, dimension( 2 )	rconde,
		double precision, dimension( 2 )	rcondv,
		double precision, dimension( * )	work,
		integer	lwork,
		integer, dimension( * )	iwork,
		integer	liwork,
		logical, dimension( * )	bwork,
		integer	info )

DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

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Purpose:

!>
!> DGGESX computes for a pair of N-by-N real nonsymmetric matrices
!> (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
!> optionally, the left and/or right matrices of Schur vectors (VSL and
!> VSR).  This gives the generalized Schur factorization
!>
!>      (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )
!>
!> Optionally, it also orders the eigenvalues so that a selected cluster
!> of eigenvalues appears in the leading diagonal blocks of the upper
!> quasi-triangular matrix S and the upper triangular matrix T; computes
!> a reciprocal condition number for the average of the selected
!> eigenvalues (RCONDE); and computes a reciprocal condition number for
!> the right and left deflating subspaces corresponding to the selected
!> eigenvalues (RCONDV). The leading columns of VSL and VSR then form
!> an orthonormal basis for the corresponding left and right eigenspaces
!> (deflating subspaces).
!>
!> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
!> or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
!> usually represented as the pair (alpha,beta), as there is a
!> reasonable interpretation for beta=0 or for both being zero.
!>
!> A pair of matrices (S,T) is in generalized real Schur form if T is
!> upper triangular with non-negative diagonal and S is block upper
!> triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
!> to real generalized eigenvalues, while 2-by-2 blocks of S will be
!>  by making the corresponding elements of T have the
!> form:
!>         [  a  0  ]
!>         [  0  b  ]
!>
!> and the pair of corresponding 2-by-2 blocks in S and T will have a
!> complex conjugate pair of generalized eigenvalues.
!>
!> 

Parameters

[in]	JOBVSL	!> JOBVSL is CHARACTER*1 !> = 'N': do not compute the left Schur vectors; !> = 'V': compute the left Schur vectors. !>
[in]	JOBVSR	!> JOBVSR is CHARACTER*1 !> = 'N': do not compute the right Schur vectors; !> = 'V': compute the right Schur vectors. !>
[in]	SORT	!> SORT is CHARACTER*1 !> Specifies whether or not to order the eigenvalues on the !> diagonal of the generalized Schur form. !> = 'N': Eigenvalues are not ordered; !> = 'S': Eigenvalues are ordered (see SELCTG). !>
[in]	SELCTG	!> SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments !> SELCTG must be declared EXTERNAL in the calling subroutine. !> If SORT = 'N', SELCTG is not referenced. !> If SORT = 'S', SELCTG is used to select eigenvalues to sort !> to the top left of the Schur form. !> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if !> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either !> one of a complex conjugate pair of eigenvalues is selected, !> then both complex eigenvalues are selected. !> Note that a selected complex eigenvalue may no longer satisfy !> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, !> since ordering may change the value of complex eigenvalues !> (especially if the eigenvalue is ill-conditioned), in this !> case INFO is set to N+3. !>
[in]	SENSE	!> SENSE is CHARACTER*1 !> Determines which reciprocal condition numbers are computed. !> = 'N': None are computed; !> = 'E': Computed for average of selected eigenvalues only; !> = 'V': Computed for selected deflating subspaces only; !> = 'B': Computed for both. !> If SENSE = 'E', 'V', or 'B', SORT must equal 'S'. !>
[in]	N	!> N is INTEGER !> The order of the matrices A, B, VSL, and VSR. N >= 0. !>
[in,out]	A	!> A is DOUBLE PRECISION array, dimension (LDA, N) !> On entry, the first of the pair of matrices. !> On exit, A has been overwritten by its generalized Schur !> form S. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of A. LDA >= max(1,N). !>
[in,out]	B	!> B is DOUBLE PRECISION array, dimension (LDB, N) !> On entry, the second of the pair of matrices. !> On exit, B has been overwritten by its generalized Schur !> form T. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of B. LDB >= max(1,N). !>
[out]	SDIM	!> SDIM is INTEGER !> If SORT = 'N', SDIM = 0. !> If SORT = 'S', SDIM = number of eigenvalues (after sorting) !> for which SELCTG is true. (Complex conjugate pairs for which !> SELCTG is true for either eigenvalue count as 2.) !>
[out]	ALPHAR	!> ALPHAR is DOUBLE PRECISION array, dimension (N) !>
[out]	ALPHAI	!> ALPHAI is DOUBLE PRECISION array, dimension (N) !>
[out]	BETA	!> BETA is DOUBLE PRECISION array, dimension (N) !> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will !> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i !> and BETA(j),j=1,...,N are the diagonals of the complex Schur !> form (S,T) that would result if the 2-by-2 diagonal blocks of !> the real Schur form of (A,B) were further reduced to !> triangular form using 2-by-2 complex unitary transformations. !> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if !> positive, then the j-th and (j+1)-st eigenvalues are a !> complex conjugate pair, with ALPHAI(j+1) negative. !> !> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) !> may easily over- or underflow, and BETA(j) may even be zero. !> Thus, the user should avoid naively computing the ratio. !> However, ALPHAR and ALPHAI will be always less than and !> usually comparable with norm(A) in magnitude, and BETA always !> less than and usually comparable with norm(B). !>
[out]	VSL	!> VSL is DOUBLE PRECISION array, dimension (LDVSL,N) !> If JOBVSL = 'V', VSL will contain the left Schur vectors. !> Not referenced if JOBVSL = 'N'. !>
[in]	LDVSL	!> LDVSL is INTEGER !> The leading dimension of the matrix VSL. LDVSL >=1, and !> if JOBVSL = 'V', LDVSL >= N. !>
[out]	VSR	!> VSR is DOUBLE PRECISION array, dimension (LDVSR,N) !> If JOBVSR = 'V', VSR will contain the right Schur vectors. !> Not referenced if JOBVSR = 'N'. !>
[in]	LDVSR	!> LDVSR is INTEGER !> The leading dimension of the matrix VSR. LDVSR >= 1, and !> if JOBVSR = 'V', LDVSR >= N. !>
[out]	RCONDE	!> RCONDE is DOUBLE PRECISION array, dimension ( 2 ) !> If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the !> reciprocal condition numbers for the average of the selected !> eigenvalues. !> Not referenced if SENSE = 'N' or 'V'. !>
[out]	RCONDV	!> RCONDV is DOUBLE PRECISION array, dimension ( 2 ) !> If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the !> reciprocal condition numbers for the selected deflating !> subspaces. !> Not referenced if SENSE = 'N' or 'E'. !>
[out]	WORK	!> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
[in]	LWORK	!> LWORK is INTEGER !> The dimension of the array WORK. !> If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', !> LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else !> LWORK >= max( 8*N, 6*N+16 ). !> Note that 2*SDIM*(N-SDIM) <= N*N/2. !> Note also that an error is only returned if !> LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B' !> this may not be large enough. !> !> If LWORK = -1, then a workspace query is assumed; the routine !> only calculates the bound on the optimal size of the WORK !> array and the minimum size of the IWORK array, returns these !> values as the first entries of the WORK and IWORK arrays, and !> no error message related to LWORK or LIWORK is issued by !> XERBLA. !>
[out]	IWORK	!> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) !> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. !>
[in]	LIWORK	!> LIWORK is INTEGER !> The dimension of the array IWORK. !> If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise !> LIWORK >= N+6. !> !> If LIWORK = -1, then a workspace query is assumed; the !> routine only calculates the bound on the optimal size of the !> WORK array and the minimum size of the IWORK array, returns !> these values as the first entries of the WORK and IWORK !> arrays, and no error message related to LWORK or LIWORK is !> issued by XERBLA. !>
[out]	BWORK	!> BWORK is LOGICAL array, dimension (N) !> Not referenced if SORT = 'N'. !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value. !> = 1,...,N: !> The QZ iteration failed. (A,B) are not in Schur !> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should !> be correct for j=INFO+1,...,N. !> > N: =N+1: other than QZ iteration failed in DHGEQZ !> =N+2: after reordering, roundoff changed values of !> some complex eigenvalues so that leading !> eigenvalues in the Generalized Schur form no !> longer satisfy SELCTG=.TRUE. This could also !> be caused due to scaling. !> =N+3: reordering failed in DTGSEN. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

!>
!>  An approximate (asymptotic) bound on the average absolute error of
!>  the selected eigenvalues is
!>
!>       EPS * norm((A, B)) / RCONDE( 1 ).
!>
!>  An approximate (asymptotic) bound on the maximum angular error in
!>  the computed deflating subspaces is
!>
!>       EPS * norm((A, B)) / RCONDV( 2 ).
!>
!>  See LAPACK User's Guide, section 4.11 for more information.
!>