DTGEVC(l) ) DTGEVC(l)NAME
DTGEVC - compute some or all of the right and/or left generalized
eigenvectors of a pair of real upper triangular matrices (A,B)
SYNOPSIS
SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL,
VR, LDVR, MM, M, WORK, INFO )
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDA, LDB, LDVL, LDVR, M, MM, N
LOGICAL SELECT( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
VR( LDVR, * ), WORK( * )
PURPOSE
DTGEVC computes some or all of the right and/or left generalized eigen‐
vectors of a pair of real upper triangular matrices (A,B). The right
generalized eigenvector x and the left generalized eigenvector y of
(A,B) corresponding to a generalized eigenvalue w are defined by:
(A - wB) * x = 0 and y**H * (A - wB) = 0
where y**H denotes the conjugate tranpose of y.
If an eigenvalue w is determined by zero diagonal elements of both A
and B, a unit vector is returned as the corresponding eigenvector.
If all eigenvectors are requested, the routine may either return the
matrices X and/or Y of right or left eigenvectors of (A,B), or the
products Z*X and/or Q*Y, where Z and Q are input orthogonal matrices.
If (A,B) was obtained from the generalized real-Schur factorization of
an original pair of matrices
(A0,B0) = (Q*A*Z**H,Q*B*Z**H),
then Z*X and Q*Y are the matrices of right or left eigenvectors of A.
A must be block upper triangular, with 1-by-1 and 2-by-2 diagonal
blocks. Corresponding to each 2-by-2 diagonal block is a complex con‐
jugate pair of eigenvalues and eigenvectors; only one
eigenvector of the pair is computed, namely the one corresponding to
the eigenvalue with positive imaginary part.
ARGUMENTS
SIDE (input) CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
HOWMNY (input) CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors, and back‐
transform them using the input matrices supplied in VR and/or
VL; = 'S': compute selected right and/or left eigenvectors,
specified by the logical array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY='S', SELECT specifies the eigenvectors to be com‐
puted. If HOWMNY='A' or 'B', SELECT is not referenced. To
select the real eigenvector corresponding to the real eigenval‐
ue w(j), SELECT(j) must be set to .TRUE. To select the complex
eigenvector corresponding to a complex conjugate pair w(j) and
w(j+1), either SELECT(j) or SELECT(j+1) must be set to .TRUE..
N (input) INTEGER
The order of the matrices A and B. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The upper quasi-triangular matrix A.
LDA (input) INTEGER
The leading dimension of array A. LDA >= max(1, N).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The upper triangular matrix B. If A has a 2-by-2 diagonal
block, then the corresponding 2-by-2 block of B must be diago‐
nal with positive elements.
LDB (input) INTEGER
The leading dimension of array B. LDB >= max(1,N).
VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must con‐
tain an N-by-N matrix Q (usually the orthogonal matrix Q of
left Schur vectors returned by DHGEQZ). On exit, if SIDE = 'L'
or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left
eigenvectors of (A,B); if HOWMNY = 'B', the matrix Q*Y; if
HOWMNY = 'S', the left eigenvectors of (A,B) specified by
SELECT, stored consecutively in the columns of VL, in the same
order as their eigenvalues. If SIDE = 'R', VL is not refer‐
enced.
A complex eigenvector corresponding to a complex eigenvalue is
stored in two consecutive columns, the first holding the real
part, and the second the imaginary part.
LDVL (input) INTEGER
The leading dimension of array VL. LDVL >= max(1,N) if SIDE =
'L' or 'B'; LDVL >= 1 otherwise.
VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must con‐
tain an N-by-N matrix Q (usually the orthogonal matrix Z of
right Schur vectors returned by DHGEQZ). On exit, if SIDE =
'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right
eigenvectors of (A,B); if HOWMNY = 'B', the matrix Z*X; if
HOWMNY = 'S', the right eigenvectors of (A,B) specified by
SELECT, stored consecutively in the columns of VR, in the same
order as their eigenvalues. If SIDE = 'L', VR is not refer‐
enced.
A complex eigenvector corresponding to a complex eigenvalue is
stored in two consecutive columns, the first holding the real
part and the second the imaginary part.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= max(1,N) if
SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
MM (input) INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M (output) INTEGER
The number of columns in the arrays VL and/or VR actually used
to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to
N. Each selected real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.
WORK (workspace) DOUBLE PRECISION array, dimension (6*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: the 2-by-2 block (INFO:INFO+1) does not have a complex
eigenvalue.
FURTHER DETAILS
Allocation of workspace:
---------- -- ---------
WORK( j ) = 1-norm of j-th column of A, above the diagonal
WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
WORK( 2*N+1:3*N ) = real part of eigenvector
WORK( 3*N+1:4*N ) = imaginary part of eigenvector
WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
Rowwise vs. columnwise solution methods:
------- -- ---------- -------- -------
Finding a generalized eigenvector consists basically of solving the
singular triangular system
(A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)
Consider finding the i-th right eigenvector (assume all eigenvalues are
real). The equation to be solved is:
n i
0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1
k=j k=j
where C = (A - w B) (The components v(i+1:n) are 0.)
The "rowwise" method is:
(1) v(i) := 1
for j = i-1,. . .,1:
i
(2) compute s = - sum C(j,k) v(k) and
k=j+1
(3) v(j) := s / C(j,j)
Step 2 is sometimes called the "dot product" step, since it is an inner
product between the j-th row and the portion of the eigenvector that
has been computed so far.
The "columnwise" method consists basically in doing the sums for all
the rows in parallel. As each v(j) is computed, the contribution of
v(j) times the j-th column of C is added to the partial sums. Since
FORTRAN arrays are stored columnwise, this has the advantage that at
each step, the elements of C that are accessed are adjacent to one
another, whereas with the rowwise method, the elements accessed at a
step are spaced LDA (and LDB) words apart.
When finding left eigenvectors, the matrix in question is the transpose
of the one in storage, so the rowwise method then actually accesses
columns of A and B at each step, and so is the preferred method.
LAPACK version 3.0 15 June 2000 DTGEVC(l)
