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Power Method.

The direct iteration methods for the HEP can be generalized to the GHEP (5.1).

First we assume that a Cholesky factorization of $B = L \, L^{\ast}$ is easily obtainable. We then reformulate (5.1) as a standard Hermitian eigenvalue problem with the matrix $C$ as described in the previous subsection (5.5), possibly after making $B$ positive definite as noted in the introduction (5.2). The power method of §4.3 now takes the following form.

\begin{algorithm}{Power Method for GHEP}
... {\bf end if} \\
{\rm (9)} \> \> {\bf end for}

In step (3), the computation $ y := C \, z$ can be executed into three steps as

\begin{displaymath}y_1 = L^{-\ast}\, z, \quad y_2 = A \, y_1, \quad
\mbox{and}\quad y = L^{-1} \, y_2. \end{displaymath}

Hence, no explicit knowledge of the matrix $C$ is needed for the power method. The power method converges under conditions similar to those for the standard HEP.

Susan Blackford 2000-11-20