DGGESX(l) ) DGGESX(l)NAME
DGGESX - compute for a pair of N-by-N real nonsymmetric matrices (A,B),
the generalized eigenvalues, the real Schur form (S,T), and,
SYNOPSIS
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
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, option‐
ally, 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 sub‐
spaces).
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 inter‐
pretation 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 trian‐
gular 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 "standard‐
ized" 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 com‐
plex conjugate pair of generalized eigenvalues.
ARGUMENTS
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 diago‐
nal 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 ei‐
genvalues (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 gen‐
eralized 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 posi‐
tive, then the j-th and (j+1)-st eigenvalues are a complex con‐
jugate 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 compara‐
ble 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 JOB‐
VSL = '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 sub‐
spaces. 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.
LAPACK version 3.0 15 June 2000 DGGESX(l)
