Matrix transformations for computing rightmost eigenvalues of large sparse non-symmetric eigenvalue problems

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Abstract
This paper gives an overview of matrix transformations for finding rightmost eigenvalues of Ax = λx and Ax = λBx with A and B real non-symmetric and B possibly singular. The aim is not to present new material. but to introduce the reader to the application of matrix transformations to the solution of large-scale eigenvalue problems. The paper explains and discusses the use of Chebyshev polynomials and the shift–invert and Cayley transforms as matrix transformations for problems that arise from the discretization of partial differential equations. A few other techniques are described. The reliability of iterative methods is also dealt with by introducing the concept of domain of confidence or trust region. This overview gives the reader an idea of the benefits and the drawbacks of several transformation techniques. We also briefly discuss the current software situation.