[Paper Review] A new and efficient method for the computation of Legendre coefficients

Enrico De Micheli, G. A. Viano|arXiv (Cornell University)|Jun 2, 2011

Digital Filter Design and Implementation7 references4 citations

TL;DR

This paper presents a novel O(N log N) algorithm for computing Legendre coefficients by transforming the problem into computing Fourier coefficients of an Abel-type integral transform of the function f(x). By leveraging the Dirichlet–Murphy integral representation of Legendre polynomials and applying a single Fast Fourier Transform, the method achieves high efficiency and accuracy, significantly outperforming standard O(N²) quadrature approaches.

ABSTRACT

An efficient procedure for the computation of the coefficients of Legendre expansions is here presented. We prove that the Legendre coefficients associated with a function f(x) can be represented as the Fourier coefficients of an Abel-type transform of f(x). The computation of N Legendre coefficients can then be performed in O(N log N) operations with a single Fast Fourier Transform of the Abel-type transform of f(x).

Motivation & Objective

• To develop a faster and more accurate algorithm for computing Legendre coefficients, which are essential in approximation theory, PDEs, and spectral methods.
• To overcome the O(N²) computational cost of standard quadrature methods for high-index Legendre coefficients with rapidly oscillating integrands.
• To exploit the connection between Legendre expansions and Fourier series through an Abel-type integral transform to enable efficient computation.
• To achieve O(N log N) complexity using a single Fast Fourier Transform after computing the Abel transform via standard quadrature.

Proposed method

• The method uses the Dirichlet–Murphy integral representation of Legendre polynomials to relate Legendre coefficients to Fourier coefficients of an Abel-type transform of f(x).
• It defines a transformed function f̂(y) via an integral involving f(x) and a square-root singularity, which is computed using nonlinear variable transformations to handle weak singularities.
• The coefficients cₙ are expressed as cₙ = (2n+1) × aₙ, where aₙ are the non-negative Fourier coefficients of f̂(y) on [−π, π].
• The Fourier coefficients aₙ are computed efficiently using a single Fast Fourier Transform, reducing the overall complexity to O(N log N).
• The Abel-type integral is evaluated with high accuracy using Gauss-Legendre quadrature and nonlinear variable substitution to manage the algebraic singularity.
• The algorithm is implemented in double precision using the GNU Scientific Library (GSL), with validation against high-precision reference values.

Experimental results

Research questions

• RQ1 Can Legendre coefficients be computed more efficiently than O(N²) using spectral methods?
• RQ2 Is there a connection between Legendre expansions and Fourier series that enables faster computation?
• RQ3 Can the Abel-type integral transform of f(x) be computed accurately and efficiently to enable O(N log N) algorithms?
• RQ4 How does the proposed method compare in accuracy and speed to standard quadrature and existing O(N(log N)²) algorithms?

Key findings

• The method achieves O(N log N) computational complexity, representing a significant improvement over standard O(N²) quadrature methods.
• For f(x) = |x|³/², the computed Legendre coefficients matched reference values to within 10⁻¹⁰ to 10⁻¹³, confirming high accuracy.
• The algorithm demonstrated a performance improvement ratio proportional to N/log N, with speedups becoming increasingly crucial for large N.
• The use of a single FFT after accurate evaluation of the Abel transform enables both speed and precision, validated across multiple test functions including polynomials, exponentials, and rational functions.
• The method is general and scalable, with potential for extension to multivariate Legendre transforms and Gegenbauer polynomial expansions.