only if GCDF[y](Φdeg p(p), Φdeg q(q)) = κ· Φdeg r(r), where κ ∈ F is such that κ· Φdeg r(r) is monic.
Proof. Let α, β be two suitable elements of F and let us write the prime factor decompositions p = α·Q
(x− pi)πi, q = β·Q
(x− qi)θi, r =Q
(x− ri)ρi. Of course we have that (x− ri)ρi|r if and only if (x − pi)πi|p, (x − qi)θi|q and ρi= min(πi, θi). Applying (3.1), we have that Φdeg p(p) = α·Q
(n(y)−pid(y))πi, Φdeg q(q) = β·Q
(n(y)−qid(y))θi and Φdeg r(r) =Q
(n(y)−rid(y))ρi. The thesis follows by invoking Corollary 3.1.
Lemma 3.4 implies condition (a). This follows from the equation GCDi(ξi(y)) = GCDi([d(y)]B−γi)· GCDi(Φγi(ζi(x))) = 1F[y]· 1F[y]. The first 1F[y] comes from the fact that maxi(γi) = B, while the second 1F[y]
comes by applying the previous lemma to
GCD(ξ1(y), . . . , ξs(y)) = GCD(GCD(. . . GCD(ξ2(y), ξ1(y)) . . . )) and from the identity Φ0(1F[x]) = 1F[y].
3.5.2 Sufficiency
To complete the proof, suppose now that Q(y) = Φg(P (x)) for some P (x)∈ F[x]
and that ˆW (y) is a minimal basis for ker Q(y), with minimal indices 1≤ · · · ≤
s. The other implication that we proved in the previous subsection implies that G|i ∀ i, so define βi = Gi. Suppose that there exists a minimal basis ˆV (x) = (ˆv1(x), . . . , ˆvs(x)) for ker P (x); suppose moreover that an index i0 ∈ {1, . . . , s}
exists such that deg ˆvi0 6= βi0. Applying the reverse implication, this would imply that there is a minimal basis ˜W (y) = ( ˜w1(y), . . . , ˜ws(y)) for ker Q(y) whose i0th right minimal index is not equal to i0. This is absurd because every minimal basis has the same minimal indices.
3.6 Extension to more relaxed hypotheses
For the sake of convenience in exposition, we have so far assumed that F is algebraically closed. This assumption is heavily used in the construction of Smith forms. Nevertheless, it is possible to state analogous results for fields that are not algebraically closed: to see it, letK be the algebraic closure of F.
Then (F[x])m×p⊆ (K[x])m×p, so we can use Theorem 3.1 to identify the Smith forms of P (x) and Q(y) = Φg(P (x)) over the polynomial rings K[x] and K[y].
We can then join back elementary divisors inK[x] and K[y] to form elementary divisors in F[x] and F[y]. Of course, in this case an elementary divisor is no more necessarily associated with a characteristic value in F. For instance, if F = Q, then the elementary divisor x2+ 2 is not associated with any rational characteristic value, but if we consider the field of complex algebraic numbers K = Q then we can split it as (x −√
2i)(x +√
2i) and associate it to the characteristic values±√
2i. We have used again the fact thatF is algebraically closed somewhere else (e.g., Lemma 3.4), but once again it is straightforward to extend those results to a generic fieldF via an immersion into its algebraic closureK.
Chapter 4
The Ehrlich-Aberth method for structured polynomial eigenproblems
In the present chapter, that relies on the papers [12, 42, 92], we analyse various possible strategies for the adaptation of the EAI to the solution of structured PEPs. There are three main strategies that we used to exactly extract the structure in the spectrum: one is the very simple solution to force the approxi-mation of the EAI to preserve the structure; the second is to explicitly build a new matrix polynomials with less distinct eigenvalues than the original one, and to obtain the sought eigenvalues from such new polynomial; finally, the third method is to make an implicit change of variable.
The second and the third approach deserve more attention, because they are more sophisticated and because from experiments they seem to achieve, in some cases, a better efficiency. We will review the application of both methods to several kinds of structured PEPs.
4.1 Introduction to structured PEPs
The EAI is particularly suited to deal with matrix polynomials endowed with specific structures of the matrix coefficients. We are interested in matrix struc-tures which induce particular symmetries on the location of the eigenvalues.
Polynomials of this kind are encountered in the applications and include, for instance, palindromic, T-palindromic, even/odd, skew-Hamiltonian and Hamil-tonian/skew-Hamiltonian polynomials.
Customary PEP-solving algorithms, such as the application of the QZ algo-rithm to any suitable linearization of the polynomial, are not able to fully catch these symmetries of the spectrum. In the literature, there are specific matrix methods that achieve this goal. The EAI enables to exploit the additional in-formation both in the computation of the Newton correction and in the choice and in the management of the (initial) approximation of the roots in view of the structure-induced symmetries. We will often refer to the resulting struc-tured variants of the Ehrlich-Aberth method as the strucstruc-tured Ehrlich-Aberth
71
iteration (SEAI).
Assume that the structured PEP is such that the eigenvalues appear in pairs {x, f(x)}, with f(f(x)) = x ∀x. A naive adaptation of the EAI to this property would be to apply (2.2) or (2.3) updating only the first half of the components of the vector y and simultaneously imposing y(i) = f (y(i−nk/2)), i = nk/2 + 1, . . . , nk. From numerical experiments we see that this approach is working well sometimes, while in other instances it does not seem to be very efficient in terms of number of scalar iterations needed for numerical convergence. This motivates the design of more sophisticated structured variants of the EAI, that we are going to describe in the following. A more detailed comparison of the efficiencies of the “naive” approach and the “sophisticated” methods is reported in Section 4.6.
In the following sections, we will analyse various classes of structured matrix polynomials and describe the design of different adaptations of the EAI that aim to solve them. Before doing that, let us recall some basic definitions of special matrices.
An n × n square matrix A ∈ Cn×n is said to be symmetric if AT = A and skew-symmetric if AT = −A. Let n = 2m. The matrix A is said to be Hamiltonian if it is such that ATJ = −JA where J is the matrix 0 Im
−Im 0
; A is said to be skew-Hamiltonian if it is such that ATJ = J A; A is said to be symplectic if it is such that ATJ A = J .
Remark 4.1. In general, it is possible to give two different definitions [31, 79]
of complex Hamiltonian, skew-Hamiltonian and symplectic matrices, one with respect to transposition and the other one with respect to conjugate transposition.
In this thesis, we are mainly interested in the former case, and therefore we always refer to the definitions given above.
Every skew-Hamiltonian matrix can be obtained as the square of a Hamilto-nian matrix, and conversely the square of a HamiltoHamilto-nian matrix is always skew-Hamiltonian[32]. Symplectic matrices are exponentials of Hamiltonian matrices.