DGEGS(3F)DGEGS(3F)NAMEDGEGS - compute for 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( * )
PURPOSEDGEGS computes for a pair of N-by-N real nonsymmetric matrices A, B: the
generalized eigenvalues (alphar +/- alphai*i, beta), the real Schur form
(A, B), and optionally left and/or right Schur vectors (VSL and VSR).
(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 interpretation 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 orthogonal 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 elements 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) )
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DGEGS(3F)DGEGS(3F)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 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 whose generalized
eigenvalues and (optionally) Schur vectors are to be computed.
On exit, the generalized Schur form of A. Note: to avoid
overflow, the Frobenius 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 vectors are to be computed.
On exit, the generalized Schur form of B. Note: to avoid
overflow, the Frobenius 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, dimension (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 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 alpha/beta.
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).
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DGEGS(3F)DGEGS(3F)
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.
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 >= max(1,4*N). For good
performance, LWORK must generally 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)
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