The Laurent expansion for a nearly singular matrix

A Method for Computing Nearly Singular Integrals

We develop a method for computing a nearly singular integral, such as a double layer potential due to sources on a curve in the plane, evaluated at a point near the curve. The approach is to regularize the singularity and obtain a preliminary value from a standard quadrature rule. We then add corrections for the errors due to smoothing and discretization, which are found by asymptotic analysis....

Evaluating Singular and Nearly Singular Integrals Numerically

Euclidean Algorithm for Laurent Polynomial Matrix Extension

In this paper, we develop a novel and effective Euclidean algorithm for Laurent polynomial matrix extension (LPME), which is the key of the construction of perfect reconstruction filter banks (PRFBs). The algorithm simplifies the dual-chain approach to the construction of PRFBs in the paper [5]. The algorithm in this paper can also be used in the applications where LPME plays a role.

Beta-expansion and continued fraction expansion over formal Laurent series

Let x ∈ I be an irrational element and n 1, where I is the unit disc in the field of formal Laurent series F((X−1)), we denote by kn(x) the number of exact partial quotients in continued fraction expansion of x, given by the first n digits in the β-expansion of x, both expansions are based on F((X−1)). We obtain that lim inf n→+∞ kn(x) n = degβ 2Q∗(x) , lim sup n→+∞ kn(x) n = degβ 2Q∗(x) , wher...

Gmres on (nearly) Singular Systems

We consider the behavior of the GMRES method for solving a linear system Ax = b when A is singular or nearly so, i.e., ill conditioned. The (near) singularity of A may or may not affect the performance of GMRES, depending on the nature of the system and the initial approximate solution. For singular A, we give conditions under which the GMRES iterates converge safely to a least-squares solution...