I. S. Duff
Abstract
This paper discusses the design and development of a code to calculate the eigenvalues of a large sparse real unsymmetric matrix that are the rightmost, leftmost, or are of the largest modulus. A subspace iteration algorithm is used to compute a sequence of sets of vectors that converge to an orthonormal basis for the invariant subspace corresponding to the required eigenvalues. This algorithm is combined with Chebychev acceleration if the rightmost or leftmost eigenvalues are sought, or if the eigenvalues of largest modulus are known to be the rightmost or leftmost eigenvalues. An option exists for computing the corresponding eigenvectors. The code does not need the matrix explicitly since it only requires the user to multiply sets of vectors by the matrix. Sophisticated and novel iteration controls, stopping criteria, and restart facilities are provided. The code is shown to be efficient and competitive on a range of test problems.
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Index Terms
- Computing selected eigenvalues of sparse unsymmetric matrices using subspace iteration
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Reviews
Charles Raymond Crawford
The authors discuss the design of a code, EB12, to calculate a certain subset of the eigenvalues of a real matrix that are the rightmost or leftmost in the complex plane or are largest in modulus. Techniques for finding other subsets of eigenvalues by transforming the problem are also discussed. The algorithm used is based on subspace iteration combined with Chebys hev acceleration. The paper gives an excellent description of the development of the algorithm from previous theoretical and applied work. It also includes a complete account of the various user-specified, problem-specific parameters such as the dimension of the iteration subspace, which is usually marginally larger than the number of eigenvalues required. Recommendations for parameter settings are based on both theory and numerical experience. EB12 is especially designed for sparse matrices or matrices defined implicitly. The only direct references to the matrix are in the computation of matrix-vector products. EB12 returns control to the user to compute these products, allowing the user to take advantage of any efficiencies offered by the form of the matrix or local hardware or software. Although this technique may seem more awkward than calling a user-written procedure, for this problem it offers more flexibility since, for example, the user has the option of stopping and restarting the iteration.
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