SGGESX(l) ) SGGESX(l)
NAME
SGGESX - 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 SGGESX( JOBVSL, JOBVSR, SORT, SELCTG, 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( * )
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), RCONDE( 2 ), RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, *
), WORK( * )
LOGICAL SELCTG
EXTERNAL SELCTG
PURPOSE
SGGESX 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 eigen-
values (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigen-
values (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 usu-
ally 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 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 complex 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 diagonal of the generalized 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 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.
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) 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 sorting) 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 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) REAL 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) REAL 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) REAL array, dimension ( 2 )
If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenval-
ues. Not referenced if SENSE = 'N' or 'V'.
RCONDV (output) REAL 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) 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+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 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 in STGSEN.
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 SGGESX(l)