The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

General Topology-I

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

A new approximation algorithm for maximum weighted matching in general edge-weighted graphs is presented. It calculates a matching with an edge weight of at least of the edge weight of a maximum weighted matching. Its time complexity is O(|E|), with |E| being the number of edges in the graph. This improves over the previously known -approximation algorithms for maximum weighted matching which require O(|E| log(|V|)) steps, where |V| is the number of vertices.

Linear Time 1/2-Approximation Algorithm for Maximum Weighted Matching in General Graphs

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by A.E. Pellet [Bulletin des Sciences Math\'ematiques, (2), vol 5 (1881), pp.393-395], some results of D.A. Bini [Numer. Algorithms 13:179-200, 1996] based on the Newton polygon technique, and recent results of M. Akian, S. Gaubert and M. Sharify (see in particular [LNCIS, 389, Springer p.p.291-303] and [M. Sharify, Ph.D. thesis, \'Ecole Polytechnique, ParisTech, 2011]). These extensions are applied for determining effective initial approximations for the numerical computation of the eigenvalues of matrix polynomials by means of simultaneous iterations, like the Ehrlich-Aberth method. Numerical experiments that show the computational advantage of these results are presented.

Locating the eigenvalues of matrix polynomials

We revisit the landmark paper [D S Mackey et al SIAM J Matrix Anal Appl, 28 (2006), pp 971--1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials We also show that every pencil in the double ansatz space is intrinsically connected to a Bezout matrix, which we use to prove the eigenvalue exclusion theorem In addition our exposition allows for any polynomial basis and for any field The new viewpoint also leads to new results We generalize the double ansatz space by exploiting its algebraic interpretation as a space of Bezout pencils to derive new linearizations with potential applications in the theory of structured matrix polynomials Moreover, we analyze the conditioning of double ansatz space linearizations in the important practical case of a Chebyshev basis

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Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by Pellet [Bull. Sci. Math. (2), 5 (1881), pp. 393--395], some results from Bini [Numer. Algorithms, 13 (1996), pp. 179--200] based on the Newton polygon technique, and recent results from Gaubert and Sharify (see, in particular, [Tropical scaling of polynomial matrices, Lecture Notes in Control and Inform. Sci. 389, Springer, Berlin, 2009, pp. 291--303] and [Sharify, Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues, Ph.D. thesis, Ecole Polytechnique, Paris, 2011]). These extensions are applied to determine effective initial approximations for the numerical computation of the eigenvalues of matrix polynomials by means of simultaneous iterations, like the Ehrlich--Aberth method. Numerical experiments that show the computational advantage of these results are presented.

/pdf/locating-the-eigenvalues-of-matrix-polynomials-2pakn6yvb8.pdf

Locating the Eigenvalues of Matrix Polynomials

The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bezout matrices with polynomial entries. Using techniques including domain subdivision, Bezoutian regularization, and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree ( $$\ge \!1{,}000$$ ? 1 , 000 ). We analyze the resultant method and its conditioning by noting that the Bezout matrices are matrix polynomials. Two implementations are available: one on the Matlab Central File Exchange and another in the roots command in Chebfun2 that is adapted to suit Chebfun's methodology.

/pdf/computing-the-common-zeros-of-two-bivariate-functions-via-2w9811g8yf.pdf

Vanni Noferini

Papers

Locating the eigenvalues of matrix polynomials

Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach

Locating the Eigenvalues of Matrix Polynomials

Computing the common zeros of two bivariate functions via Bézout resultants

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