A Matrix Problem with Application to Rapid Solution of Integral Equations

Abstract

Let $\{ z_j \} _{j = 0}^{n - 1} $ be a set of distinct points in the complex plane $\mathbb{C}$, and introduce the $n \times n$ matrix $A_n = [a_{jk} ]_{j,k = 0}^{n - 1} $, $a_{jk} = (z_j - z_k )^{ - 1} ,\, j \ne k$ and $a_{jj} = 0$. Recently Golub and Trummer raised the question of whether or not, for an arbitrary vector $x \in \mathbb{C}^n $, $A{\bf x}$ can be computed in fewer than $O(n^2 )$ arithmetic operations by using the structure of $A_n $. In this paper it is assumed that there is a smooth $2\pi $-periodic bijective function $z(t)$ such that $z_j = z(2\pi (j - 1)/n),\, j = 1(1)n$, and shown that when n increases, there is a sequence of matrices of low rank $\tilde A_n ,\, n = 1,2,3, \cdots $ such that $\tilde A_n \to A_n $ as $n \to \infty $ and $\tilde A_n {\bf x}$ can be computed in $O(n\log n)$ arithmetic operations. The method to construct the matrices $\tilde A_n $ is then used in a fast solution scheme for Fredholm integral equations of the second kind with smooth periodic kernels. The integral equations are discretized by the trapezoidal rule using the nodes $z_j = z(2\pi j/n)$, $0 \leqq j < n$, and it is shown that arbitrarily accurate approximate solutions can be computed in $O(n\log n)$ arithmetic operations for large n, provided that $z(t)$ is sufficiently smooth. When the asymptotic analysis is not applicable, fast iterative $O(n^2 )$ solution methods are obtained. The scheme is applied to the solution of a Fredholm integral equation of the second kind of plane potential theory and Cauchy singular integral equations.