dgeesx()

subroutine dgeesx	(	character	jobvs,
		character	sort,
		external	select,
		character	sense,
		integer	n,
		double precision, dimension( lda, * )	a,
		integer	lda,
		integer	sdim,
		double precision, dimension( * )	wr,
		double precision, dimension( * )	wi,
		double precision, dimension( ldvs, * )	vs,
		integer	ldvs,
		double precision	rconde,
		double precision	rcondv,
		double precision, dimension( * )	work,
		integer	lwork,
		integer, dimension( * )	iwork,
		integer	liwork,
		logical, dimension( * )	bwork,
		integer	info )

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

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Purpose:

!>
!> DGEESX computes for an N-by-N real nonsymmetric matrix A, the
!> eigenvalues, the real Schur form T, and, optionally, the matrix of
!> Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).
!>
!> Optionally, it also orders the eigenvalues on the diagonal of the
!> real Schur form so that selected eigenvalues are at the top left;
!> computes a reciprocal condition number for the average of the
!> selected eigenvalues (RCONDE); and computes a reciprocal condition
!> number for the right invariant subspace corresponding to the
!> selected eigenvalues (RCONDV).  The leading columns of Z form an
!> orthonormal basis for this invariant subspace.
!>
!> For further explanation of the reciprocal condition numbers RCONDE
!> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
!> these quantities are called s and sep respectively).
!>
!> A real matrix is in real Schur form if it is upper quasi-triangular
!> with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
!> the form
!>           [  a  b  ]
!>           [  c  a  ]
!>
!> where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
!> 

Parameters

[in]	JOBVS	!> JOBVS is CHARACTER*1 !> = 'N': Schur vectors are not computed; !> = 'V': Schur vectors are computed. !>
[in]	SORT	!> SORT is CHARACTER*1 !> Specifies whether or not to order the eigenvalues on the !> diagonal of the Schur form. !> = 'N': Eigenvalues are not ordered; !> = 'S': Eigenvalues are ordered (see SELECT). !>
[in]	SELECT	!> SELECT is a LOGICAL FUNCTION of two DOUBLE PRECISION arguments !> SELECT must be declared EXTERNAL in the calling subroutine. !> If SORT = 'S', SELECT is used to select eigenvalues to sort !> to the top left of the Schur form. !> If SORT = 'N', SELECT is not referenced. !> An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if !> SELECT(WR(j),WI(j)) is true; i.e., if either one of a !> complex conjugate pair of eigenvalues is selected, then both !> are. Note that a selected complex eigenvalue may no longer !> satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since !> ordering may change the value of complex eigenvalues !> (especially if the eigenvalue is ill-conditioned); in this !> case INFO may be set to N+3 (see INFO below). !>
[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 right invariant subspace only; !> = 'B': Computed for both. !> If SENSE = 'E', 'V' or 'B', SORT must equal 'S'. !>
[in]	N	!> N is INTEGER !> The order of the matrix A. N >= 0. !>
[in,out]	A	!> A is DOUBLE PRECISION array, dimension (LDA, N) !> On entry, the N-by-N matrix A. !> On exit, A is overwritten by its real Schur form T. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !>
[out]	SDIM	!> SDIM is INTEGER !> If SORT = 'N', SDIM = 0. !> If SORT = 'S', SDIM = number of eigenvalues (after sorting) !> for which SELECT is true. (Complex conjugate !> pairs for which SELECT is true for either !> eigenvalue count as 2.) !>
[out]	WR	!> WR is DOUBLE PRECISION array, dimension (N) !>
[out]	WI	!> WI is DOUBLE PRECISION array, dimension (N) !> WR and WI contain the real and imaginary parts, respectively, !> of the computed eigenvalues, in the same order that they !> appear on the diagonal of the output Schur form T. Complex !> conjugate pairs of eigenvalues appear consecutively with the !> eigenvalue having the positive imaginary part first. !>
[out]	VS	!> VS is DOUBLE PRECISION array, dimension (LDVS,N) !> If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur !> vectors. !> If JOBVS = 'N', VS is not referenced. !>
[in]	LDVS	!> LDVS is INTEGER !> The leading dimension of the array VS. LDVS >= 1, and if !> JOBVS = 'V', LDVS >= N. !>
[out]	RCONDE	!> RCONDE is DOUBLE PRECISION !> If SENSE = 'E' or 'B', RCONDE contains the reciprocal !> condition number for the average of the selected eigenvalues. !> Not referenced if SENSE = 'N' or 'V'. !>
[out]	RCONDV	!> RCONDV is DOUBLE PRECISION !> If SENSE = 'V' or 'B', RCONDV contains the reciprocal !> condition number for the selected right invariant subspace. !> 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. LWORK >= max(1,3*N). !> Also, if SENSE = 'E' or 'V' or 'B', !> LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of !> selected eigenvalues computed by this routine. Note that !> N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only !> returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or !> 'B' this may not be large enough. !> For good performance, LWORK must generally be larger. !> !> If LWORK = -1, then a workspace query is assumed; the routine !> only calculates upper bounds on the optimal sizes of the !> arrays WORK and IWORK, returns these values as the first !> entries of the WORK and IWORK arrays, and no error messages !> related to LWORK or LIWORK are issued by XERBLA. !>
[out]	IWORK	!> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) !> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. !>
[in]	LIWORK	!> LIWORK is INTEGER !> The dimension of the array IWORK. !> LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM). !> Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is !> only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this !> may not be large enough. !> !> If LIWORK = -1, then a workspace query is assumed; the !> routine only calculates upper bounds on the optimal sizes of !> the arrays WORK and IWORK, returns these values as the first !> entries of the WORK and IWORK arrays, and no error messages !> related to LWORK or LIWORK are 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. !> > 0: if INFO = i, and i is !> <= N: the QR algorithm failed to compute all the !> eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI !> contain those eigenvalues which have converged; if !> JOBVS = 'V', VS contains the transformation which !> reduces A to its partially converged Schur form. !> = N+1: the eigenvalues could not be reordered because some !> eigenvalues were too close to separate (the problem !> is very ill-conditioned); !> = N+2: after reordering, roundoff changed values of some !> complex eigenvalues so that leading eigenvalues in !> the Schur form no longer satisfy SELECT=.TRUE. This !> could also be caused by underflow due to scaling. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.