Factorization of selfadjoint quadratic matrix polynomials with real spectrum

Factorization theorems, and properties of sets of eigenvectors, are established for regular selfadjoint quatratic matrix polynomials L(λ) whose leading coefficeint is indefinite or possibly singular, and for which all eigenvalues are real of definite type. The two linear factors obtained have spectra which are just the eigenvalues of L(λ) of positive and negative types, respectively.     

Computing multiple roots of inexact polynomials

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials.

Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and it calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations. To fulfill the input requirement of Algorithm I, we develop a numerical GCD-finder containing a successive singular value updating and an iterative GCD refinement as the main engine of Algorithm II that calculates the multiplicity structure and the initial root approximation. While limitations exist in certain situations, the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic, even if the coefficients are inexact. This is perhaps the first blackbox-type root-finder with such capabilities.

To measure the sensitivity of the multiple roots, a structure-preserving condition number is proposed and error bounds are established. According to our computational experiments and error analysis, a polynomial being ill-conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved, and its multiple roots can be computed with high accuracy.

     

Construction of unital matrix polynomials with mutually distinct characteristic roots

We investigate the construction of unital matrix polynomials with mutually distinct characteristic roots, namely, their similarity and reducibility by the similarity transformation to block-triangular, block-diagonal, and, in particular, to triangular and diagonal forms. We also study the problem of extracting linear factors.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 69–77, January, 1993.     

Separation of roots of matrix and operator polynomials

In a complex Hilbert spaceX for an arbitrary operator polynomialL() ( C) of degreem the following theorem is proved. If the equation (L()x, x)=0 hasm distinct roots at every pointx X, x=1, then there existm pairwise disjoint connected sets in C such that each set contains a root at everyx. The minimal distance between the roots is separated from zero under the same assumption on the discriminant and the leading coefficient of that equation.     

Factorization of monic polynomials

We prove a uniqueness result about the factorization of a monic polynomial over a general commutative ring into comaximal factors. We apply this result to address several questions raised by Steve McAdam. These questions, inspired by Hensel's Lemma, concern properties of prime ideals and the factoring of monic polynomials modulo prime ideals.

     

The Durand-Kerner polynomials roots-finding method in case of multiple roots

The convergence of the Durand-Kerner algorithm is quadratic in case of simple roots but only linear in case of multiple roots. This paper shows that, at each step, the mean of the components converging to the same root approaches it with an error proportional to the square of the error at the previous step. Since it is also shown that it is possible to estimate the multiplicity order of the roots during the algorithm, a modification of the Durand-Kerner iteration is proposed to preserve a quadratic-like convergence even in case of multiple zeros.This work is supported in part by the Research Program C3 of the French CNRS and MEN, and by the Direction des Recherches et Etudes Techniques (DGA).     

Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

An asymptotic expansion for large sample size n is derived by a partial differential equation method, up to and including the term of order n^?2, for the ₀F₀ function with two argument matrices which arise in the joint density function of the latent roots of the covariance matrix, when some of the population latent roots are multiple. Then we derive asymptotic expansions for the joint and marginal distributions of the sample roots in the case of one multiple root.     

The distance from a matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue

For a matrix polynomial P(λ) and a given complex number μ, we introduce a (spectral norm) distance from P(λ) to the matrix polynomials that have μ as an eigenvalue of geometric multiplicity at least κ, and a distance from P(λ) to the matrix polynomials that have μ as a multiple eigenvalue. Then we compute the first distance and obtain bounds for the second one, constructing associated perturbations of P(λ).     

Factorization of multivariate positive Laurent polynomials

Recently Dritschel proved that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Fejér–Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. Finally we discuss how to compute nonnegative Laurent polynomial factorizations in the multivariate setting.     

Pseudospectra, critical points and multiple eigenvalues of matrix polynomials

We develop a general framework for perturbation analysis of matrix polynomials. More specifically, we show that the normed linear space L_m(C^n×n) of n-by-n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in L_m(C^n×n). We present a family of natural norms on the space L_m(C^n×n) and show that the norms on the spaces C^m+1 and C^n×n play a crucial role in the perturbation analysis of matrix polynomials. We define pseudospectra of matrix polynomials in the general framework of the normed space L_m(C^n×n) and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases. We analyze various properties of pseudospectra in the unified framework of the normed space L_m(C^n×n). We analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial. We show that common boundary points of components of pseudospectra of matrix polynomials are critical points. As a consequence, we show that a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials.     

On roots of polynomials with positive coefficients

We prove that an algebraic number α is a root of a polynomial with positive rational coefficients if and only if none of its conjugates is a nonnegative real number. This settles a recent conjecture of Kuba.     

Smooth roots of hyperbolic polynomials with definable coefficients

We prove that the roots of a definable C ^∞ curve of monic hyperbolic polynomials admit a definable C ^∞ parameterization, where ‘definable’ refers to any fixed o-minimal structure on (ℝ,+, ·). Moreover, we provide sufficient conditions, in terms of the differentiability of the coefficients and the order of contact of the roots, for the existence of C ^p (for p ∈ ℕ) arrangements of the roots in both the definable and the non-definable case. These conditions are sharp in the definable and, under an additional assumption, also in the non-definable case. In particular, we obtain a simple proof of Bronshtein’s theorem in the definable setting. We prove that the roots of definable C ^∞ curves of complex polynomials can be desingularized by means of local power substitutions t ↦ ±t ^N . For a definable continuous curve of complex polynomials we show that any continuous choice of roots is actually locally absolutely continuous.