A Statistical Evaluation of Inverse Iteration
The eigenvalues of a symmetric tridiagonal matrix can be computed by bisection using Sturm sequences and their corresponding eigenvectors found by inverse iteration. The independent computing tasks comprising this combination are especially appropriate for implementation on a distributed memory multiprocessor, but the parallel efficiency of bisection and inverse iteration is not enough to recommend their use in general. Bisection produces eigenvalues to high accuracy. Inverse iteration results in eigenvectors with quality dependent on the spacing of the eigenvalues and on the starting vector. This paper examines the factors influencing the accuracy of inverse iteration. The effects of random starting vectors are discussed, and some uses of statistical analysis in the design of an inverse iteration algorithm are presented.
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