Purpose
=======
LA_GEGS computes for a pair of n-by-n real nonsymmetric matrices
A, B: the generalized eigenvalues (alphar alpha, beta),
the 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 SGEGV
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) )
=========
SUBROUTINE LA_GEGS( A, B, , BETA, VSL, VSR, INFO )
(), INTENT(INOUT) :: A(:,:), B(:,:)
(), INTENT(OUT), OPTIONAL :: , BETA(:)
(), INTENT(OUT), OPTIONAL :: VSL(:,:), VSR(:,:)
INTEGER, INTENT(OUT), OPTIONAL :: INFO
where
::= REAL | COMPLEX
::= KIND(1.0) | KIND(1.0D0)
::= ALPHAR, ALPHAI | ALPHA
::= ALPHAR(:), ALPHAI(:) | ALPHA(:)
Arguments
=========
A (input/output) REAL / COMPLEX array, shape (:,:),
SIZE(A,1) == SIZE(A,2) == 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.
B (input/output) REAL / COMPLEX array, shape (:,:),
SIZE(rBA,1) == SIZE(rBA,2) == 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.
ALPHAR Only for the real case. Optional (output) REAL array,
ALPHAI shape (:), SIZE(ALPHA') == n. ALPHA ::= ALPHAR | ALPHAI
ALPHA Only for the complex case. Optional (output) COMPLEX
array, shape (:), SIZE(ALPHAI) == n.
BETA Optional (output) REAL / COMPLEX array, shape (:),
SIZE(BETA) == 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).
VSL Optional (output) REAL / COMPLEX array, shape (:,:),
SIZE(VSL,1) == SIZE(VSL,2) == n.
VSL will contain the left Schur vectors. (See "Purpose", above.)
VSR Optional (output) REAL / COMPLEX array, shape (:,:),
SIZE(VSL,1) == SIZE(VSL,2) == n.
VSR will contain the right Schur vectors. (See "Purpose", above.)
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 LA_GGBAL
=n+2: error return from LA_GEQRF
=n+3: error return from LA_ORMQR
=n+4: error return from LA_ORGQR
=n+5: error return from LA_GGHRD
=n+6: error return from LA_HGEQZ (other than failed
iteration)
=n+7: error return from LA_GGBAK (computing VSL)
=n+8: error return from LA_GGBAK (computing VSR)
=n+9: error return from LA_LASCL (various places)
If INFO is not present and an error occurs, then the program is
terminated with an error message.