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#### Orthogonal Projection Methods.

Let be an complex matrix and be an -dimensional subspace of and consider the eigenvalue problem of finding belonging to and belonging to such that (7)

An orthogonal projection technique onto the subspace seeks an approximate eigenpair to the above problem, with in and in . This approximate eigenpair is obtained by imposing the following Galerkin condition: (8)

or, equivalently, (9)

In order to translate this into a matrix problem, assume that an orthonormal basis of is available. Denote by the matrix with column vectors , i.e., . Because we seek a , it can be written as (10)

Then, equation eq:4.16 becomes Therefore, and must satisfy (11)

with B_m = V^ A V . The approximate eigenvalues resulting from the projection process are all the eigenvalues of the matrix . The associated eigenvectors are the vectors in which is an eigenvector of associated with .

This procedure for numerically computing the Galerkin approximations to the eigenvalues/eigenvectors of is known as the Rayleigh-Ritz procedure.

RAYLEIGH-RITZ PROCEDURE
1. Compute an orthonormal basis of the subspace . Let .
2. Compute .
3. Compute the eigenvalues of and select the desired ones , where (for instance the largest ones).
4. Compute the eigenvectors , of associated with , and the corresponding approximate eigenvectors of , .

In implementations of this approach, one often does not compute the eigenpairs of for each set of generated basis vectors. The values computed from this procedure are referred to as Ritz values and the vectors are the associated Ritz vectors. The numerical solution of the eigenvalue problem in steps 3 and 4 can be treated by standard library subroutines such as those in LAPACK . Another important note is that in step 4 one can replace eigenvectors by Schur vectors to get approximate Schur vectors instead of approximate eigenvectors. Schur vectors can be obtained in a numerically stable way and, in general, eigenvectors are more sensitive to rounding errors than are Schur vectors.     Next: Oblique Projection Methods. Up: Basic Ideas Y. Saad Previous: Basic Ideas Y. Saad   Contents   Index
Susan Blackford 2000-11-20