Fast Approximate Computations with Cauchy Matrices and Polynomials

TL;DR

This paper introduces nearly linear time numerical algorithms for multipoint polynomial evaluation and interpolation by transforming Vandermonde matrices into structured Cauchy matrices and applying hierarchical and fast multipole techniques.

Contribution

The authors develop a novel approach that transforms Vandermonde matrices into structured Cauchy matrices and accelerates computations using hierarchical and fast multipole methods, achieving nearly linear time complexity.

Findings

Algorithms run in nearly linear arithmetic time.

Effective approximation of Cauchy matrices by hierarchically semiseparable matrices.

Applicable to a broad class of polynomial and matrix computations.

Abstract

Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to quadratic for numerical solution. We fix this discrepancy: our new numerical algorithms run in nearly linear arithmetic time. At first we restate our goals as the multiplication of an n-by-n Vandermonde matrix by a vector and the solution of a Vandermonde linear system of n equations. Then we transform the matrix into a Cauchy structured matrix with some special features. By exploiting them, we approximate the matrix by a generalized hierarchically semiseparable matrix, which is a structured matrix of a different class. Finally we accelerate our solution to the original problems by applying Fast Multipole Method to the latter matrix. Our resulting numerical algorithms run in nearly optimal arithmetic time when they perform the above fundamental computations with polynomials, Vandermonde matrices, transposed Vandermonde matrices, and a large class of Cauchy and Cauchy-like matrices. Some of our techniques may be of independent interest.