Clenshaw

Clenshaw is a surname associated with the British mathematician Charles W. Clenshaw, whose work in numerical analysis has led to important algorithms for evaluating polynomials and performing numerical integration. The name is most often encountered in two widely used methods that bear his imprint: Clenshaw’s algorithm and Clenshaw–Curtis quadrature.

Clenshaw’s algorithm is a stable technique for evaluating a finite sum in which the basis functions satisfy a linear recurrence, such as Chebyshev polynomials. Given a sum y(x) = sum_{k=0}^N a_k p_k(x) where the p_k follow a three-term recurrence, the algorithm uses a backward recurrence to compute y without explicitly forming each p_k. This approach reduces numerical instability and is efficient, making it a standard tool in the evaluation of Chebyshev series and in spectral methods for solving differential equations.

Clenshaw–Curtis quadrature is a numerical integration method that uses the Chebyshev points of the first kind as nodes on the interval [-1, 1]. The corresponding weights can be determined by a recurrence or via a fast transform, yielding an accurate quadrature rule for smooth integrands. The method is popular in situations where fast, stable computation is desirable and scales well to large node counts, often rivaling Gauss-type quadrature in practice for many applications.

Together, Clenshaw’s algorithm and Clenshaw–Curtis quadrature have had a lasting impact in numerical analysis, offering practical tools for polynomial approximation, spectral methods, and high-accuracy numerical integration. The surname appears primarily in these mathematical contexts, where it is recognized for its contributions to stable computation.