SGGES(l) ) SGGES(l)
SGGES - compute for a pair of N-by-N real nonsymmetric matrices (A,B),
SUBROUTINE SGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI,
BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO )
CHARACTER JOBVSL, JOBVSR, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
LOGICAL BWORK( * )
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), VSL(
LDVSL, * ), VSR( LDVSR, * ), WORK( * )
SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized ei-
genvalues, the generalized real Schur form (S,T), optionally, the left and/or right matri-
ces 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.The leading columns of VSL and VSR then form an orthonormal
basis for the corresponding left and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver SGGEV instead, which is
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 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 general-
ized Schur form. = 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELCTG);
SELCTG (input) LOGICAL FUNCTION of three REAL 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
Note that in the ill-conditioned case, a selected complex eigenvalue may no longer
satisfy SELCTG(ALPHAR(j),ALPHAI(j), BETA(j)) = .TRUE. after ordering. INFO is to
be set to N+2 in this case.
N (input) INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
A (input/output) REAL 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) REAL 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 sort-
ing) for which SELCTG is true. (Complex conjugate pairs for which SELCTG is true
for either eigenvalue count as 2.)
ALPHAR (output) REAL array, dimension (N)
ALPHAI (output) REAL array, dimension (N) BETA (output) REAL 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 conju-
gate 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 com-
puting 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 compa-
rable with norm(B).
VSL (output) REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOB-
VSL = 'N'.
LDVSL (input) INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >=
VSR (output) REAL array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOB-
VSR = 'N'.
LDVSR (input) INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >=
WORK (workspace/output) REAL 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+16.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued by XERBLA.
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 SHGEQZ.
=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
LAPACK version 3.0 15 June 2000 SGGES(l)