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23
An extension of matlab to continuous functions and operators
 SIAM J. SCI. COMPUT
, 2004
"... An objectoriented 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 “chebf ..."
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Cited by 66 (11 self)
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An objectoriented 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.
Optimization Strategies for the Approximate GCD Problem
 IN PROC. ISSAC'98
, 1998
"... 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 SVDbased algorithm), we formulate and solve a nonlinear optimization problem in order to d ..."
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Cited by 33 (3 self)
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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 SVDbased 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.
More on Pseudospectra for Polynomial Eigenvalue Problems and Applications in Control Theory
, 2002
"... 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 ..."
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Cited by 24 (5 self)
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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 statespace 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.
SPECTRAL PROPERTIES OF SCHRÖDINGER OPERATORS ARISING IN THE STUDY OF QUASICRYSTALS
, 2012
"... We survey results that have been obtained for selfadjoint operators, and especially Schrödinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus our attention on the onedimensional case, and in particula ..."
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Cited by 14 (7 self)
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We survey results that have been obtained for selfadjoint operators, and especially Schrödinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus our attention on the onedimensional case, and in particular on several key examples. The most prominent of these is the Fibonacci Hamiltonian, for which much is known by now and to which an entire section is devoted here. Other examples that are discussed in detail are given by the more general class of Schrödinger operators with Sturmian potentials. We put some emphasis on the methods that have been introduced quite recently in the study of these operators, many of them coming from hyperbolic dynamics. We conclude with a multitude of numerical calculations that illustrate the validity of
Tompaidis: Computation of domains of analyticity for some perturbative expansions from mechanics
 Physica D
, 1994
"... 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 nonperturb ..."
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Cited by 5 (2 self)
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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 nonperturbative 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.
Helmut Wielandt's Contributions To The Numerical Solution Of Complex Eigenvalue Problems
 in Helmut Wielandt, Mathematische Werke, Mathematical
, 1994
"... 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]. ..."
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Cited by 3 (2 self)
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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].
Towards an Implementation of a Computer Algebra System in a Functional Language
 In Intelligent Computer Mathematics, AISC
, 2008
"... Abstract. This paper discusses the pros and cons of using a functional language for implementing a computer algebra system. The contributions of the paper are twofold. Firstly, we discuss some language–centered design aspects of a computer algebra system — the “language unity” concept. Secondly, we ..."
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Abstract. This paper discusses the pros and cons of using a functional language for implementing a computer algebra system. The contributions of the paper are twofold. Firstly, we discuss some language–centered design aspects of a computer algebra system — the “language unity” concept. Secondly, we provide an implementation of a fast polynomial multiplication algorithm, which is one of the core elements of a computer algebra system. The goal of the paper is to test the feasibility of an implementation of (some elements of) a computer algebra system in a modern functional language.
Is the Polynomial So Perfidious?
, 1994
"... 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 f ..."
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Cited by 1 (0 self)
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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: 225238 (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