Perhaps the most successful numerical algorithm for
computing the complete eigensystem of a general square matrix
is the implicitly shifted QR algorithm. One of the keys to the
success of this method is its relationship to the
The QR algorithm produces a sequence of unitary similarity transformations that iteratively reduce to upper triangular form (see §7.3). In other words, it computes a Schur decomposition. A practical implementation of the QR algorithm begins with an initial unitary similarity transformation of to the condensed form where is upper Hessenberg (``almost upper triangular'') and is unitary. Then the following iteration is performed.
SHIFTED QR METHOD
for until convergence
select a shift
In this scheme, is unitary and is upper triangular (i.e., the QR factorization of ). It is easy to see that is unitarily similar to throughout the course of this iteration. The iteration is continued until the subdiagonal elements of converge to zero, i.e., until a Schur decomposition has been (approximately) obtained.
If represents the leading columns of , and the
leading principal submatrix of in (7.11),
We are going to exploit this aspect of the Schur decomposition. We shall also exploit the fact that a variant of the QR iteration can be devised that will force eigenvalues of interest to appear in this leading block of as the iteration proceeds. Our goal will be to develop a truncated form of the QR method that is suitable for computing a selected set of eigenvalues of a large matrix . We call implicitly restarted Arnoldi method the IRAM.