DTGEVC(1) LAPACK routine (version 3.2) DTGEVC(1)NAMEDTGEVC - computes some or all of the right and/or left eigenvectors of
a pair of real matrices (S,P), where S is a quasi-triangular matrix and
P is upper triangular
SYNOPSIS
SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL,
VR, LDVR, MM, M, WORK, INFO )
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDP, LDS, LDVL, LDVR, M, MM, N
LOGICAL SELECT( * )
DOUBLE PRECISION P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
VR( LDVR, * ), WORK( * )
PURPOSEDTGEVC computes some or all of the right and/or left eigenvectors of a
pair of real matrices (S,P), where S is a quasi-triangular matrix and P
is upper triangular. Matrix pairs of this type are produced by the
generalized Schur factorization of a matrix pair (A,B):
A = Q*S*Z**T, B = Q*P*Z**T
as computed by DGGHRD + DHGEQZ.
The right eigenvector x and the left eigenvector y of (S,P) correspond‐
ing to an eigenvalue w are defined by:
S*x = w*P*x, (y**H)*S = w*(y**H)*P,
where y**H denotes the conjugate tranpose of y.
The eigenvalues are not input to this routine, but are computed
directly from the diagonal blocks of S and P.
This routine returns the matrices X and/or Y of right and left eigen‐
vectors of (S,P), or the products Z*X and/or Q*Y,
where Z and Q are input matrices.
If Q and Z are the orthogonal factors from the generalized Schur fac‐
torization of a matrix pair (A,B), then Z*X and Q*Y
are the matrices of right and left eigenvectors of (A,B).
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, backtrans‐
formed by the matrices 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 w(j) is a real eigenvalue, the corresponding real
eigenvector is computed if SELECT(j) is .TRUE.. If w(j) and
w(j+1) are the real and imaginary parts of a complex eigenval‐
ue, the corresponding complex eigenvector is computed if either
SELECT(j) or SELECT(j+1) is .TRUE., and on exit SELECT(j) is
set to .TRUE. and SELECT(j+1) is set to .FALSE.. Not refer‐
enced if HOWMNY = 'A' or 'B'.
N (input) INTEGER
The order of the matrices S and P. N >= 0.
S (input) DOUBLE PRECISION array, dimension (LDS,N)
The upper quasi-triangular matrix S from a generalized Schur
factorization, as computed by DHGEQZ.
LDS (input) INTEGER
The leading dimension of array S. LDS >= max(1,N).
P (input) DOUBLE PRECISION array, dimension (LDP,N)
The upper triangular matrix P from a generalized Schur factor‐
ization, as computed by DHGEQZ. 2-by-2 diagonal blocks of P
corresponding to 2-by-2 blocks of S must be in positive diago‐
nal form.
LDP (input) INTEGER
The leading dimension of array P. LDP >= 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 (S,P); if HOWMNY = 'B', the matrix Q*Y; if
HOWMNY = 'S', the left eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of VL, in the same
order as their eigenvalues. A complex eigenvector correspond‐
ing to a complex eigenvalue is stored in two consecutive col‐
umns, the first holding the real part, and the second the imag‐
inary part. Not referenced if SIDE = 'R'.
LDVL (input) INTEGER
The leading dimension of array VL. LDVL >= 1, and if SIDE =
'L' or 'B', LDVL >= N.
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 Z (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 (S,P); if HOWMNY = 'B' or 'b', the matrix Z*X;
if HOWMNY = 'S' or 's', the right eigenvectors of (S,P) speci‐
fied by SELECT, stored consecutively in the columns of VR, in
the same order as their eigenvalues. A complex eigenvector
corresponding to a complex eigenvalue is stored in two consecu‐
tive columns, the first holding the real part and the second
the imaginary part. Not referenced if SIDE = 'L'.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1, and if SIDE
= 'R' or 'B', LDVR >= N.
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 LDS (and LDP) words apart. When finding left eigenvec‐
tors, 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 routine (version 3.2) November 2008 DTGEVC(1)