stgevc(3)

NAME

STGEVC - compute some or all of the right and/or left gen

eralized eigenvectors of a pair of real upper triangular matrices

(A,B)

SYNOPSIS

SUBROUTINE STGEVC( 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( * )
    REAL            A(  LDA, * ), B( LDB, * ), VL( LDVL, *
), VR( LDVR, * ), WORK( * )

PURPOSE

STGEVC computes some or all of the right and/or left gen

eralized eigenvectors of a pair of real upper triangular matrices

(A,B). The right generalized eigenvector x and the left general

ized eigenvector y of (A,B) corresponding to a generalized eigen

value 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 eigen

vectors 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 conjugate pair of eigenvalues and eigenvectors; only

one
eigenvector of the pair is computed, namely the one corre

sponding 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 eigenvec

tors, 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 eigen

value w(j), SELECT(j) must be set to .TRUE. To select the com

plex 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) REAL 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) REAL 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

diagonal with positive elements.

LDB (input) INTEGER

The leading dimension of array B. LDB >=

max(1,N).

VL (input/output) REAL array, dimension (LDVL,MM)

On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B',

VL must contain an N-by-N matrix Q (usually the orthogonal matrix

Q of left Schur vectors returned by SHGEQZ). 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 consecutive columns, the first hold

ing 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) REAL array, dimension (LDVR,MM)

On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B',

VR must contain an N-by-N matrix Q (usually the orthogonal matrix

Z of right Schur vectors returned by SHGEQZ). 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 consecutive columns, the first hold

ing 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 col

umn and each selected complex eigenvector occupies two columns.

WORK (workspace) REAL array, dimension (6*N)

INFO (output) INTEGER

= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille

gal 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 diag

onal
WORK( N+j ) = 1-norm of j-th column of B, above the di

agonal
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

STGEVC(3)