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NAME
DGEGS - a pair of N-by-N real nonsymmetric matrices A, B
SYNOPSIS
SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
WORK, LWORK, INFO )
CHARACTER JOBVSL, JOBVSR
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ),
ALPHAR( * ), B( LDB, * ), BETA( * ), VSL(
LDVSL, * ), VSR( LDVSR, * ), WORK( * )
PURPOSE
For a pair of N-by-N real nonsymmetric matrices A, B:
compute the generalized eigenvalues (alphar +/- alphai*i,
beta)
compute the real Schur form (A,B)
compute the left and/or right Schur vectors (VSL and VSR)
The last action is optional -- see the description of JOBVSL
and JOBVSR below. (If only the generalized eigenvalues are
needed, use the driver DGEGV instead.)
A generalized eigenvalue for a pair of matrices (A,B) is,
roughly speaking, 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 interpre-
tation for beta=0, and even for both being zero. A good
beginning reference is the book, "Matrix Computations", by
G. Golub & C. van Loan (Johns Hopkins U. Press)
The (generalized) Schur form of a pair of matrices is the
result of multiplying both matrices on the left by one
orthogonal matrix and both on the right by another orthogo-
nal matrix, these two orthogonal matrices being chosen so as
to bring the pair of matrices into (real) Schur form.
A pair of matrices A, B is in generalized real Schur form if
B is upper triangular with non-negative diagonal and A 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 A will be "standardized" by making
the corresponding entries of B have the form:
[ a 0 ]
[ 0 b ]
and the pair of corresponding 2-by-2 blocks in A and B will
have a complex conjugate pair of generalized eigenvalues.
The left and right Schur vectors are the columns of VSL and
VSR, respectively, where VSL and VSR are the orthogonal
matrices which reduce A and B to Schur form:
Schur form of (A,B) = ( (VSL)**T A (VSR), (VSL)**T B (VSR) )
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.
N (input) INTEGER
The number of rows and columns in 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 whose
generalized eigenvalues and (optionally) Schur vec-
tors are to be computed. On exit, the generalized
Schur form of A. Note: to avoid overflow, the Fro-
benius norm of the matrix A should be less than the
overflow threshold.
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 whose
generalized eigenvalues and (optionally) Schur vec-
tors are to be computed. On exit, the generalized
Schur form of B. Note: to avoid overflow, the Fro-
benius norm of the matrix B should be less than the
overflow threshold.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension
(N) BETA (output) DOUBLE PRECISION array, dimen-
sion (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
j=1,...,N, will be the generalized eigenvalues.
ALPHAR(j) + ALPHAI(j)*i, j=1,...,N and
BETA(j),j=1,...,N are the diagonals of the complex
Schur form (A,B) 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 com-
plex 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 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 alpha/beta. How-
ever, 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. (See "Purpose", above.) 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. (See "Purpose", above.) Not referenced if
JOBVSR = 'N'.
LDVSR (input) INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1,
and if JOBVSR = 'V', LDVSR >= N.
(LWORK)
WORK (workspace/output) DOUBLE PRECISION array, dimension
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >=
max(1,4*N). For good performance, LWORK must gen-
erally be larger. To compute the optimal value of
LWORK, call ILAENV to get blocksizes (for DGEQRF,
DORMQR, and DORGQR.) Then compute: NB -- MAX of
the blocksizes for DGEQRF, DORMQR, and DORGQR The
optimal LWORK is 2*N + N*(NB+1).
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: errors
that usually indicate LAPACK problems:
=N+1: error return from DGGBAL
=N+2: error return from DGEQRF
=N+3: error return from DORMQR
=N+4: error return from DORGQR
=N+5: error return from DGGHRD
=N+6: error return from DHGEQZ (other than failed
iteration) =N+7: error return from DGGBAK (computing
VSL)
=N+8: error return from DGGBAK (computing VSR)
=N+9: error return from DLASCL (various places)