The Ehrlich-Aberth method for palindromic matrix polynomials represented in the Dickson basis

An algorithm based on the Ehrlich-Aberth root-finding method is presented for the computation of the eigenvalues of a T-palindromic matrix polynomial. A structured linearization of the polynomial represented in the Dickson basis is introduced in order to exploit the symmetry of the roots by halving the total number of the required approximations. The rank structure properties of the linearization allow the design of a fast and numerically robust implementation of the root-finding iteration. Numerical experiments that confirm the effectiveness and the robustness of the approach are provided.Comment: in press in Linear Algebra Appl. (2011

Polynomial Eigenproblems: a Root-Finding Approach

A matrix polynomial, also known as a polynomial matrix, is a polynomial whose coefficients are matrices; or, equivalently, a matrix whose elements are polynomials. If the matrix polynomial P(x) is regular, that is if p(x):=det(P(x)) is not identically zero, the polynomial eigenvalue problem associated with P(x) is equivalent to the computation of the roots of the polynomial p(x); such roots are called the eigenvalues of the regular matrix polynomial P(x). Sometimes, one is also interested in computing the corresponding (left and right) eigenvectors. Recently, much literature has been addressed to the polynomial eigenvalue problem. This line of research is currently very active: the theoretical properties of PEPs are studied, and fast and numerically stable methods are sought for their numerical solution. The most commonly encountered case is the one of degree 2 polynomials, but there exist applications where higher degree polynomials appear. More generally, PEPs are special cases belonging to the wider class of nonlinear eigenvalue problems. Amongst nonlinear eigenvalue problems, rational eigenvalue problems can be immediately brought to polynomial form, multiplying them by their least common denominator; truly nonlinear eigenvalue problems may be approximated with PEPs, truncating some matrix power series, or with rational eigenproblems, using rational approximants such as Padé approximants. To approximate numerically the solutions of PEPs, several algorithms have been introduced based on the technique of linearization where the polynomial problem is replaced by a linear pencil with larger size and the customary methods for the generalised eigenvalue problem, like for instance the QZ algorithm, are applied. This thesis is addressed to the design and analysis of algorithms for the polynomial eigenvalue problem based on a root-finding approach. A root-finder is applied to the characteristic equation p(x)=0. In particular, we discuss algorithms based on the Ehrlich-Aberth iteration. The Ehrlich-Aberth iteration (EAI) is a method that simultaneously approximates all the roots of a (scalar) polynomial. In order to adapt the EAI to the numerical solution of a PEP, we propose a method based on the Jacobi formula; two implementation of the EAI are discussed, of which one uses a linearization and the other works directly on the matrix polynomial. The algorithm that we propose has quadratic computational complexity with respect to the degree k of the matrix polynomial. This leads to computational advantage when the ratio k^2/n, where n is the dimension of the matrix coefficients, is large. Cases of this kind can be encountered, for instance, in the truncation of matrix power series. If k^2/n is small, the EAI can be implemented in such a way that its asymptotic complexity is cubic (or slightly supercubic) in nk, but QZ-based methods appear to be faster in this case. Nevertheless, experiments suggest that the EAI can improve the approximations of the QZ in terms of forward error, so that even when it is not as fast as other algorithms it is still suitable as a refinement method. The EAI does not compute the eigenvectors. If they are needed, the EAI can be combined with other methods such as the SVD or the inverse iteration. In the experiments we performed, eigenvectors were computed in this way, and they were approximated with higher accuracy with respect to the QZ. Another root-finding approach to PEPs, similar to the EAI, is to apply in sequence the Newton method to each single eigenvalue, using an implicit deflation of the previously computed roots of the determinant in order to avoid to approximate twice the same eigenvalue. Our numerical experience suggests that in terms of efficiency the EAI is superior with respect to the sequential Newton method with deflation. Specific attention concerns structured problems where the matrix coefficients have some additional feature which is reflected on structural properties of the roots. For instance, in the case of T-palindromic polynomials, the roots are encountered in pairs (x,1/x). In this case the goal is to design algorithms which take advantage of this additional information about the eigenvalues and deliver approximations to the eigenvalues which respect these symmetries independently of the rounding errors. Within this setting, we study the case of polynomials endowed with specific properties like, for instance, palindromic, T-palindromic, Hamiltonian, symplectic, even/odd, etc., whose eigenvalues have special symmetries in the complex plane. In general, we may consider the case of structures where the roots can be grouped in pairs as (x,f(x)), where f(x) is any self-inverse analytic function such that. We propose a unifying treatment of structured polynomials belonging to this class and show how the EAI can be adapted to deal with them in a very effective way. Several structured variants of the EAI are available to this goal: they are described in this thesis. All of such variants enable one to compute only a subset of eigenvalues and to recover the remaining part of the spectrum by means of the symmetries satisfied by the eigenvalues. By exploiting the structure of the problem, this approach leads to a saving on the number of floating point operations and provides algorithms which yield numerical approximations fulfilling the symmetry properties. Our research on the structured EAI can of course be applied also to scalar polynomials: in the next future, we plan to exploit our results and design new features for the software MPSolve. When studying the theoretical properties of the change of variable, useful to design one of the structured EAI methods, we had the chance to discover some theorems on the behaviour of the complete eigenstructure of a matrix polynomial under a rational change of variable. Such results are discussed in this thesis. Some, but not all, of the different structured versions of the EAI algorithm have a drawback: accuracy is lost for eigenvalues that are close to a finite number of critical values, called exceptional eigenvalues. On the other hand, it turns out that at least for some specific structures the versions that suffer from this problem are also the most efficient ones: thus, it is desirable to circumvent the loss of accuracy. This can be done by the design of a structured refinement Newton algorithm. Besides its application to structured PEPs, this algorithm can have further application to the computation of the roots of scalar polynomials whose roots appear in pairs. In this thesis, we also present the results of several numerical experiments performed in order to test the effectiveness of our approach in terms of speed and of accuracy. We have compared the Ehrlich-Aberth iteration with the Matlab functions polyeig and quadeig. In the structured case, we have also considered, when available, other structured methods, say, the URV algorithm by Schroeder . Moreover, the different versions of our algorithm are compared one with another