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##

Subspace Iteration

Another important improvement over the power method permits
us to compute a -dimensional invariant subspace,
rather than one eigenvector at a time. It is called *orthogonal iteration* (and sometimes *subspace
iteration* or *simultaneous iteration*).

This algorithm is a straightforward generalization of the power
method. The QR factorization is a normalization
process that is similar to the normalization used in the
power method. The eigenvalues of the Hermitian
matrix will approach the eigenvalues of that are
largest in absolute value.

Several modifications are needed to make the simple
subspace iteration an efficient and practically applicable code.
First, it is natural to orthonormalize as infrequently as
possible, i.e., to perform several iterations before
performing an orthogonalization. Second, we may choose
to operate on a subspace whose dimension is larger than ,
the number of eigenvalues wanted, and use
a Rayleigh-Ritz process to get eigenvalue approximations.
Third, some eigenvalues will converge faster than others, and if this happens it
is a good idea to lock these and let the matrix operate only
on those vectors that have not yet converged.

In addition, the method is rarely used without some
form of acceleration; we describe some of those techniques at the end of this
section.

**Subsections**

** Next:** Subspace Dimension.
** Up:** Single- and Multiple-Vector Iterations
** Previous:** Rayleigh Quotient Iteration
** Contents**
** Index**
Susan Blackford
2000-11-20