ClenshawCurtisin

ClenshawCurtisin is a numerical method used for the efficient evaluation of a polynomial in Chebyshev series form. It is named after Charles Clenshaw and Curtis A. Smith, who independently developed the algorithm in the 1960s. The method is particularly useful in numerical analysis and computational mathematics for minimizing rounding errors and computational costs when evaluating polynomials.

A Chebyshev series is a form of expressing a polynomial as a sum of Chebyshev polynomials of the first kind, which are a sequence of orthogonal polynomials. The ClenshawCurtisin algorithm takes advantage of the recursive properties of these polynomials to evaluate the series efficiently. The algorithm works by recursively computing the coefficients of the series in a backward manner, starting from the highest degree and moving towards the lowest.

The primary advantage of the ClenshawCurtisin algorithm is its computational efficiency. It reduces the number of operations required to evaluate a polynomial from O(n^2) to O(n), where n is the degree of the polynomial. This makes it particularly useful for evaluating high-degree polynomials. Additionally, the method minimizes rounding errors, making it more accurate than direct evaluation methods, especially for high-degree polynomials.

The algorithm is widely used in various fields of numerical analysis, including numerical integration, solution of differential equations, and approximation theory. It is also used in computer algebra systems and numerical libraries for polynomial evaluation. The ClenshawCurtisin algorithm is a fundamental tool in the field of numerical mathematics and continues to be an active area of research and development.