k
P (y(i)j )
−1
k2(1 + α(yj(i)))
−1
measures the backward error for the approximation y(i)j , and can be cheaply evaluated during the EAI. The iteration can be halted when η(yj(i)) is smaller than a given tolerance.
Numerical experiments showed that in the case of multiple eigenvalues the latter criterion lead to premature stops. For simple eigenvalues, no significant differences emerged. Therefore, our default choice was in favour of the criterion based on the condition number.
Similarly to what has been done in the implementation of the EAI via lin-earization, once again it is also convenient to add, with the “or” logic operator, the following condition:
|N (y(i)j )/(1− N (yj(i))Aj(y(i), y(i+1))| ≤ τ2|y(i)j |, (2.7) where τ2 is a given tolerance. This condition says that the computed correction is too tiny and would not change the significant digits of the current approxi-mation.
2.4 Initial approximations
In practice, the cost of each vector iteration of the algorithm is strongly de-pendent on the amount of early convergence occurring for a given problem. In other words, a critical point to assess the efficiency of the novel method is the evaluation of the total number t of calls of either the function horner trace or the function trace linearization (depending on which specific implementation of the EAI one is using), and of its dependence on the total number N := nk of the eigenvalues. When the Ehrlich-Aberth method is used to approximate scalar polynomials roots, experiments show that t depends on the choice of the starting points.
2.4.1 Rouch´ e theorem and starting points
As pointed out in [1, 8, 51], when there is not any a priori knowledge about the location of the roots, practically effective choices of initial approximations for the EAI are complex numbers equally displaced along circles. For instance, in [1]
it is proposed to choose initial approximations displaced along a circle centred at the origin of sufficiently large radius so that it contains all the roots. In [51] the radius of the circle is suitably chosen. This strategy does not work effectively for polynomials having zeros with very large and with very small moduli. In [8]
this drawback is overcome by considering different circles centred at the origin of suitable radii. The computation of these radii relies on the Rouch´e theorem.
Here we try to extend this technique to the case of matrix polynomials.
We recall that, according to the Rouch´e theorem, if s(x) and q(x) are two polynomials such that
|s(x)| > |q(x)|, for |x| = r,
2.4. INITIAL APPROXIMATIONS 37 then s(x) and s(x) + q(x) have the same number of roots in the open disk{z ∈ C : |x| < r}. Applying this property with s(x) = xm and q(x) = p(x)− s(x), for 0 ≤ m ≤ N, implies that if rm > PN
j=0,j6=m|aj|rj then the polynomial p(x) has m roots in the open disk of centre 0 and radius r. This property is at the basis of the criterion described in [8], based on the Newton polygon construction, for choosing initial approximations equidistributed along different circles centred in 0.
In order to extend this criterion to the case of polynomial eigenvalue prob-lems we need a generalization of the Rouch´e theorem to matrix polynomials.
We report the following result of [89] which we rephrase in a simpler way better suited for our problem.
Theorem 2.1. Let S(x) and Q(x) be matrix polynomials and let r be a posi-tive real. If S(x)HS(x)− Q(x)HQ(x) is positive definite for |x| = r, then the polynomials det S(x) and det(S(x) + Q(x)) have the same number of roots of modulus less than r.
The following result is an immediate consequence of the above theorem applied to the polynomial P (x) of (2.1) with S(x) = xmPm and Q(x) =
where A B means that A − B is positive definite. Then the matrix polynomial P (x) has mk eigenvalues in the open disk of centre 0 and radius r.
Observe that if det Pm= 0 then condition (2.8) cannot be verified. In fact, the vector v such that Pmv = 0 would be such that
In particular, if det Pk6= 0 the above corollary, applied with m = k, implies that all the eigenvalues of P (z) are included in the disk of centre 0 and radius r provided that
Observe that the latter condition is implied by
r2kPkHPk
Similarly, applying Corollary 2.1 with m = 0 provides a disk where P (x) has no eigenvalues.
As an example of application, consider the 5×5 quadratic matrix polynomial P (x) = Ax2+ Bx + AT, where B is the tridiagonal matrix defined by the en-tries [1, 2, 1], and A is the matrix with diagonal enen-tries 100, 1, 1/1000, 1/100000, superdiagonal entries equal to 1 and with zero entries elsewhere. The eigenval-ues of P (x) have approximate moduli 2.0050e+05, 1.4969e+03, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 6.6805e-04, 4.9874e-06. The cri-terion based on the above corollary in the form (2.8) yields the bound 4.4e− 6 <
|x| < 2.24e5 which is quite good. Applying condition (2.10) yields the bounds 1.96e− 6 < |x| < 5.1e5 which is still good.
Similar results have been obtained in [101] in the framework of tropical algebras; an ongoing future research project is to extend those results [13].
Another possibility is to choose the radii of the starting circles by applying the prescriptions of [8] to the scalar polynomial whose coefficients are the norms of the matrix coefficients of P (x). This option is cheaper, leads to similar results, and is the default choice in our implementations.
Other options that we considered and that will be analysed in deeper details in Section 2.6, are:
• to pick points lying on only one circle, for instance the circle |x| = 1;
• to pick points lying on a small number of circles, chosen according to some criterion;
• to start with randomly chosen points, or more precisely to start with points whose modulus is generated according to some prescribed proba-bility distribution (e.g. a log-normal distribution).
2.4.2 A posteriori error bounds
In the case of a scalar polynomial p(x) of degree N , given a set of approximations x1, . . . , xN to the roots of p(x) it is possible to prove that [102] the set of disks Di = D(xi, ri) of centre xi and radius ri = n
p(xi) (pN
QN
j=1, j6=i(xi− xj)) is such that
1. the union of the disks contains all the roots of p(x);
2. each connected component formed by the union of, say, c overlapping disks, contains c roots of p(x).
The set formed by Di, i = 1, . . . , N with the above properties is called set of inclusion disks.
In the case of a matrix polynomial P (x) where det Pk 6= 0, it is quite cheap to compute a set of inclusion disks. In fact, if P (x) = ΠLU is the PLU factorization of P (x), then |p(x)| = | det P (x)| =Qn
j=1|Ujj|. Moreover, the leading coeffi-cient pN of det P (x) coincides with det Pk which can be computed once for all.
Observe that the LAPACK routine zgesv which solves a linear system with the matrix P (x), is used to compute the Newton correction 1/trace(P (x)−1P0(x)) during the EAI. Such routine, applied with x = xi, provides at a negligible cost also the radius ri.
The availability of a set of inclusion disks enables us to perform a cluster analysis. In fact, once an isolated disk has been detected, we have isolated
2.5. MULTIPLE EIGENVALUES 39