Fast and Stable Pascal Matrix Algorithms

TL;DR

This paper introduces fast, stable algorithms for multiplying and inverting Pascal matrices with $O(n log^2 n)$ complexity, improving stability over previous FFT-based methods and relating to Bézier curve evaluation.

Contribution

The paper presents a novel recursive factorization approach for Pascal matrices that enhances computational speed and numerical stability compared to prior methods.

Findings

Algorithms run in $O(n log^2 n)$ time

Demonstrated improved stability over Toeplitz-based algorithms

Established practical speed through numerical experiments

Abstract

In this paper, we derive a family of fast and stable algorithms for multiplying and inverting $n \times n$ Pascal matrices that run in $O(n log^2 n)$ time and are closely related to De Casteljau's algorithm for B\'ezier curve evaluation. These algorithms use a recursive factorization of the triangular Pascal matrices and improve upon the cripplingly unstable $O(n log n)$ fast Fourier transform-based algorithms which involve a Toeplitz matrix factorization. We conduct numerical experiments which establish the speed and stability of our algorithm, as well as the poor performance of the Toeplitz factorization algorithm. As an example, we show how our formulation relates to B\'ezier curve evaluation.