Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method

Given the $n\times n$ matrix polynomial $P(x)=\sum_{i=0}^kP_i x^i$, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial $\det P(x)$, is treated in polynomial form rather than in matrix form by means of the Ehrlich-Aberth iteration. The main computational issues are discussed, namely, the choice of the starting approximations needed to start the Ehrlich-Aberth iteration, the computation of the Newton correction, the halting criterion, and the treatment of eigenvalues at infinity. We arrive at an effective implementation which provides more accurate approximations to the eigenvalues with respect to the methods based on the QZ algorithm. The case of polynomials having special structures, like palindromic, Hamiltonian, symplectic, etc., where the eigenvalues have special symmetries in the complex plane, is considered. A general way to adapt the Ehrlich-Aberth iteration to structured matrix polynomial is introduced. Numerical experiments which confirm the effectiveness of this approach are reported.Comment: Submitted to Linear Algebra App

The location of roots of equations with particular reference to the generalized eigenvalue problem

A survey is presented of algorithms which are in current use for the solution of a single algebraic or transcendental equation in one unknown, together with an appraisal of their practical performance. The first part of the thesis consists of an account of the theoretical basis of a number of iterative methods and an examination of the problems to be overcome in order to achieve a successful computer implementation. In the selection of specific programs for testing, the emphasis has been placed on methods which are suitable for use, in conjunction with determinant evaluation, for the solution of standard eigenvalue problems and generalized problems of the form A(λ)x = O, where the elements of A are linear or non-linear functions of λ. The principal requirements for such purposes are that: 1. the algorithm should not be restricted to polynomial equations 2. derivative evaluation should not be required. Examples of eigenvalue problems arising from engineering applications illustrate the potential difficulties of determining roots. Particular attention is given to the problem of calculating a number of roots in cases where a priori estimates for each root are not available. The discussion is extended to give a brief account of possible approaches to the problem of locating complex roots. Interpolation methods are found to be particularly versatile and can be recommended for their accuracy and efficiency. It is also suggested that such algorithms may often be employed as search strategies in the absence of good initial estimates of the roots. Mention is also made of those features of practical implementation which were found to be particularly useful, together with a list of some outstanding difficulties, associated principally with the automatic computation of several roots of an equation

Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving

Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but typically nonreal roots are much more numerous than the real ones. We dramatically accelerate the known algorithms in this case by exploiting the correlation between the computations with matrices and polynomials, extending the techniques of the matrix sign iteration, and exploiting the structure of the companion matrix of the input polynomial. We extend some of the proposed techniques to the approximation of the real eigenvalues of a real nonsymmetric matrix.Comment: 17 pages, added algorithm

Solving secular and polynomial equations: A multiprecision algorithm

We present an algorithm for the solution of polynomial equations and secular equations of the form $S(x)=0$ for $S(x)=sum_{i=1}^nrac{a_i}{x-b_i}-1=0$, which provides guaranteed approximation of the roots with any desired number of digits. It relies on the combination of two different strategies for dealing with the precision of the floating point computation: the strategy used in the package MPSolve of D.~Bini and G.~Fiorentino [Numer.~Algo.~23, 2000] and the strategy used in the package Eigensolve of S.~Fortune [J. Symb. Comput.~33, 2002]. The algorithm is based on the Ehrlich-Aberth (EA) iteration, and on several results introduced in the paper. In particular, we extend the concept and the properties of root-neighborhoods from polynomials to secular functions, provide perturbation results of the roots, obtain an effective stop condition for the EA iteration and guaranteed a posteriori error ounds. We provide an implementation, released in the package MPSolve 3.0, based on the GMP library. From the many numerical experiments it turns out that our code is generally much faster than MPSolve 2.0 and of the package Eigensolve. For certain polynomials, like the Mandelbrot or the partition polynomials the acceleration is dramatic. The algorithm exploits the parallel architecture of the computing platform

Orthogonal polynomials on generalized Julia sets

We extend results by Barnsley et al. about orthogonal polynomials on Julia sets to the case of generalized Julia sets. The equilibrium measure is considered. In addition, we discuss optimal smoothness of Green functions and Parreau-Widom criterion for a special family of real generalized Julia sets.Comment: We changed the second part of the article a little bit and gave sharper results in this versio