Clenshaw algorithm

About: Clenshaw algorithm is a research topic. Over the lifetime, 106 publications have been published within this topic receiving 1914 citations.

A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions

Abstract: Spherical harmonic expansions form partial sums of fully normalised associated Legendre functions (ALFs). However, when evaluated increasingly close to the poles, the ultra-high degree and order (e.g. 2700) ALFs range over thousands of orders of magnitude. This causes existing recursion techniques for computing values of individual ALFs and their derivatives to fail. A common solution in geodesy is to evaluate these expansions using Clenshaw's method, which does not compute individual ALFs or their derivatives. Straightforward numerical principles govern the stability of this technique. Elementary algebra is employed to illustrate how these principles are implemented in Clenshaw's method. It is also demonstrated how existing recursion algorithms for computing ALFs and their first derivatives are easily modified to incorporate these same numerical principles. These modified recursions yield scaled ALFs and first derivatives, which can then be combined using Horner's scheme to compute partial sums, complete to degree and order 2700, for all latitudes (except at the poles for first derivatives). This exceeds any previously published result. Numerical tests suggest that this new approach is at least as precise and efficient as Clenshaw's method. However, the principal strength of the new techniques lies in their simplicity of formulation and implementation, since this quality should simplify the task of extending the approach to other uses, such as spherical harmonic analysis.

Fast algorithms for discrete polynomial transforms

Abstract: Consider the Vandermonde-like matrix P:= (P k (cos jπ/N)) j,k=0 N , where the polynomials P k satisfy a three-term recurrence relation. If P k are the Chebyshev polynomials T k , then P coincides with C N+1 := (cos jkπ/N) j,k=0 N . This paper presents a new fast algorithm for the computation of the matrix-vector product Pa in O(N log 2 N) arithmetical operations. The algorithm divides into a fast transform which replaces Pa with C N+1 ā and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes Pa with almost the same precision as the Clenshaw algorithm, but is much faster for N ≥ 128.