MATRIX TRANSFORMATIONS FOR COMPUTING RIGHTMOST EIGENVALUES OF LARGE SPARSE NONSYMMETRICAL EIGENVALUE PROBLEMS

This paper gives an overview of matrix transformations for finding rig htmost eigenvalues of Ax = lambda x and Ax = lambda Bx with A and B re al non-symmetric and B possibly singular. The aim is not to present ne w material, but to introduce the reader to the application of matrix t ransformations 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 prob lems that arise from the discretization bf partial differential equati ons. A few other techniques are described. The reliability of iterativ e methods is also dealt with by introducing the concept of domain of c onfidence or trust region. This overview gives the reader an idea of t he benefits and the drawbacks of several transformation techniques, We also briefly discuss the current software situation.