DTGEVC - compute some or all of the right and/or left generalized eigenvectors of a pair of real upper triangular matrices (A,B)

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( * )

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 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 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.

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 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 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 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 referenced.

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 contain 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 referenced.

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.

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.