International Conference on Applied Mathematics edition:2012 location:Hong-Kong date:28 May - 1 June 2012
Polynomial eigenvalue problems are often solved by transforming them to a Companion-like form.
This is a linear eigenvalue problem of larger size that can be solved by methods for linear eigenvalue problems.
For the quadratic eigenvalue problem, this idea has led to the SOAR and Q-Arnoldi methods.
We will solve the non-linear eigenvalue problem A(λ) x = 0 by approximating A(λ) by an interpolating
When we choose the polynomial basis well, we can apply a Krylov method in a dynamic way, i.e., we can choose the
interpolation points during the Arnoldi run.
We will discuss four choices of polynomial bases: monomials, Newton, Lagrange and Chebyshev polynomials.
For each of these, we have a different method.
We will elaborate in more detail a rational Krylov method using Newton polynomials.
The Ritz values are computed from a small linear generalized eigenvalue problem using the QZ method. We will show a moment matching property, show a connection with Newton's method for the non-linear eigenvalue problem and discuss implementation details of the algorithm. Numerical examples will be given to illustrate the method.