Abstract. An object-oriented MATLAB system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices. About eighty MATLAB functions from plot and sum to svd and cond have been overloaded so that one can work with our “chebfun ” objects using almost exactly the usual MATLAB syntax. All functions live on [−1, 1] and are represented by values at sufficiently many Chebyshev points for the polynomial interpolant to be accurate to close to machine precision. Each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation, differentiation, integration, zerofinding, or transforms. Applications in approximation theory and numerical analysis are explored, and possible extensions for more substantial problems of scientific computing are mentioned.

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We describe algorithms for computing the greatest common divisor (GCD) of two univariate polynomials with inexactlyknown coefficients. Assuming that an estimate for the GCD degree is available (e.g., using an SVD-based algorithm), we formulate and solve a nonlinear optimization problem in order to determine the coefficients of the "best" GCD. We discuss various issues related to the implementation of the algorithms and present some preliminary test results. 1 Introduction There are many applications in which it is necessary to compute the greatest common divisor (GCD) of two or more polynomials. For example, symbolic computation programs must be able to simplify rational functions, such as (x 2 + 4x + 4)=(x + 2). Sometimes, the coefficients may be inexact, due to the accumulation of floating-point errors or to imprecise input (e.g., the coefficients come from physical measurements). This situation can cause great difficulties in GCD computation. Suppose we have p(x) = x 2 +3:999x...

Definitions and characterizations of pseudospectra are given for rectangular matrix polynomials expressed in homogeneous form: P(#,#)= # 0 .It is shown that problems with infinite (pseudo)eigenvalues are elegantly treated in this framework. For such problems stereographic projection onto the Riemann sphere is shown to provide a convenient way to visualize pseudospectra. Lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system (with equality for standard state-space systems) are obtained in terms of pseudospectra, showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory. How and why to incorporate linear structure into pseudospectra is also discussed by example.

To the memory of J.L. Tennyson Abstract. We compute the domain of analyticity of some perturbative expansions for invariant circles appearing in mechanics. We use Padé approximants for the perturbative expansions and introduce methods to ascertain their domain of convergence. We also use non-perturbative methods based on direct computation of the invariant circles and, in analogy with Greene’s criterion, approximation by circles with rational rotation. We find that the domains computed by all the methods agree within the limits of accuracy. We also study rigorously the nature of the singularities when the frequency is rational.

this technical report was apparently not published, Wielandt's method must have gone unnoticed. The same method is described, for instance, by Barnett [1, Theorem 3.15], yet Barnett attributes its origin [1, p 253] to a 1963 paper by Parks [13].

this paper we shall be principally concerned with deflation (Sects. 3 and 4) although we shall carry out analyses of other problems (Sects. 2 and 5). The classical result of Jim Wilkinson is that the standard method of deflation can be unstable if the polynomial is divided by the factor (z \Gamma ff) first where ff is the computed value of the largest root in modulus. If, on the other hand, the polynomial is divided by Numerische Mathematik Electronic Edition -- page numbers may differ from the printed version page 225 of Numer. Math. 68: 225--238 (1994) A.M. Cohen the factor (z \Gamma fi), where fi is the computed value of the smallest root in modulus then the deflation process is satisfactory. In 1980 I found that by a process of `backward recursion' one could deflate a polynomial in a stable way by dividing out by the factor (x \Gamma ff). I communicated the algorithm to Jim Wilkinson but I have no confirmation that he received my letter. In [1] Wilkinson tackles the problem by noting the correspondence between the polynomial equations

In this paper we propose a new deflation algorithm for line spectral pair (LSP) computation. This algorithm is much more reliable than other methods based on deflation. 1. THE PROBLEM Assuming that the reader is familiar with the statement of the problem of LSP computation, we introduce directly the notions and notations necessary to our presentation. Let Pm denote a polynomial of degree m Pm (x) = p0;mx m +p1;mx m 1 +: : :+pm 1;mx+pm;m ; (1) and Qm (x) another polynomial, with the properties: (1) the roots of Pm and Qm are distinct and real, and (2) an interlacing relation holds: if the roots of Pm and Qm are denoted x i , i = 1 : n (where n = 2m) and arranged in decreasing order, 1 > x1 > x2 > : : : > xn > 1; (2) then the odd roots belong to Pm and the even roots belong to Qm . In speech coding, the polynomials Pm and Qm are obtained by transforming a linear prediction model. Their roots, x i , are called LSPs (line spectral pairs). Our problem is the computation of the LSP s...

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conditioning of polynomial pole placement problems with application to low-order controller design for a flexible beam