A Chebyshev polynomial of a square matrix A is a monic polynomial p of specified degree that minimizes ‖p(A)‖2. The study of such polynomials is motivated by the analysis of Krylov subspace iterations in numerical linear algebra. An algorithm is presented for computing these polynomials based on reduction to a semidefinite program which is then solved by a primaldual interior point method. Exam...

............................................................................................................ iv Chapter 1 Introduction 1.1 Motivation ......................................................................................... 1 1.2 Preliminaries, notation, terminology and definitions ........................ 4 1.3 Quasi – biclique literature review....................................

We present in this paper several extremely efficient and accurate spectral-Galerkin methods for secondand fourth-order equations in polar and cylindrical geometries. These methods are based on appropriate variational formulations which incorporate naturally the pole condition(s). In particular, the computational complexities of the Chebyshev–Galerkin method in a disk and the Chebyshev–Legendre–...

This paper concerns with the reducibility loss of (periodic) invariant curves of quasi-periodically forced one dimensional maps and its relationship with the renormalization operator. Let gα be a one-parametric family of one dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, there exists a parameter value αn such that gαn has a superstable perio...

A classical theorem of Kuratowski says that every Baire one function on a Gδ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index β(f). We p...

In this paper, by introducing a class of relaxed filtered Krylov subspaces, we propose the subspace method for computing eigenvalues with largest real parts and corresponding eigenvectors non-symmetric matrices. As by-products, generalizations Chebyshev–Davidson solving eigenvalue problems are also presented. We give convergence analysis complex Chebyshev polynomial, which plays significant rol...

It is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required, the Conjugate Gradient method will compute the optimal approximate solution from the appropriate Krylov subspace, that is, it will implicitly compute the optimal polynomial. Henc...

In this chapter, we will prove that given a set P of n points in IR, one can reduce the dimension of the points to k = O(ε−2 log n) and distances are 1 ± ε reserved. Surprisingly, this reduction is done by randomly picking a subspace of k dimensions and projecting the points into this random subspace. One way of thinking about this result is that we are “compressing” the input of size nd (i.e.,...