
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 eigenvectors 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 realSchur factor
ization 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 1by1 and 2by2 diagonal blocks. Corresponding
to each 2by2 diagonal block is a complex conjugate 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 backtransform 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 computed. If HOWMNY='A' or
'B', SELECT is not referenced. To select the real eigenvector corresponding to
the real eigenvalue 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 quasitriangular 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 2by2 diagonal block, then the corre
sponding 2by2 block of B must be diagonal 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 contain an NbyN 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 referenced.
A complex eigenvector corresponding to a complex eigenvalue is stored in two con
secutive 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 contain an NbyN 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 referenced.
A complex eigenvector corresponding to a complex eigenvalue is stored in two con
secutive 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 eigen
vectors. 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 ith argument had an illegal value.
> 0: the 2by2 block (INFO:INFO+1) does not have a complex eigenvalue.
FURTHER DETAILS
Allocation of workspace:
  
WORK( j ) = 1norm of jth column of A, above the diagonal
WORK( N+j ) = 1norm of jth 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 backtransformed eigenvector
WORK( 5*N+1:6*N ) = imaginary part of backtransformed 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 ith right eigenvector (assume all eigenvalues are real). The equa
tion 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 = i1,. . .,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 jth 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 jth 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) 
