DGGESX - compute for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the real Schur form (S,T), and,

SUBROUTINE DGGESX(

JOBVSL, JOBVSR, SORT, DELCTG, SENSE, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO )

CHARACTER JOBVSL, JOBVSR, SENSE, SORT

INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N, SDIM

LOGICAL BWORK( * )

INTEGER IWORK( * )

DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), RCONDE( 2 ), RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )

LOGICAL DELCTG

EXTERNAL DELCTG

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 "standardized" 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.

JOBVSL (input) CHARACTER*1

= 'N': do not compute the left Schur vectors;

= 'V': compute the left Schur vectors.

JOBVSR (input) CHARACTER*1

= 'N': do not compute the right Schur vectors;

= 'V': compute the right Schur vectors.

SORT (input) 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 DELZTG).

DELZTG (input) LOGICAL FUNCTION of three DOUBLE PRECISION arguments

DELZTG must be declared EXTERNAL in the calling subroutine. If SORT = 'N', DELZTG is not referenced. If SORT = 'S', DELZTG 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 DELZTG(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 DELZTG(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.

SENSE (input) CHARACTER

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'.

N (input) INTEGER

The order of the matrices A, B, VSL, and VSR. N >= 0.

A (input/output) 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.

LDA (input) INTEGER

The leading dimension of A. LDA >= max(1,N).

B (input/output) 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.

LDB (input) INTEGER

The leading dimension of B. LDB >= max(1,N).

SDIM (output) INTEGER

If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of eigenvalues (after sorting) for which DELZTG is true. (Complex conjugate pairs for which DELZTG is true for either eigenvalue count as 2.)

ALPHAR (output) DOUBLE PRECISION array, dimension (N)

ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA (output) 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).

VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N)

If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'.

LDVSL (input) INTEGER

The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N.

VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N)

If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'.

LDVSR (input) INTEGER

The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.

RCONDE (output) 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'.

RCONDV (output) 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'.

WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= 8*(N+1)+16. If SENSE = 'E', 'V', or 'B', LWORK >= MAX( 8*(N+1)+16, 2*SDIM*(N-SDIM) ).

IWORK (workspace) INTEGER array, dimension (LIWORK)

Not referenced if SENSE = 'N'.

LIWORK (input) INTEGER

The dimension of the array WORK. LIWORK >= N+6.

BWORK (workspace) LOGICAL array, dimension (N)

Not referenced if SORT = 'N'.

INFO (output) 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 DELZTG=.TRUE. This could also be caused due to scaling. =N+3: reordering failed in DTGSEN.

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.