Abstract
This paper presents a compensated algorithm for accurate evaluation of a polynomial in Legendre basis. Since the coefficients of the evaluated polynomial are fractions, we propose to store these coefficients in two floating point numbers, such as double-double format, to reduce the effect of the coefficients’ perturbation. The proposed algorithm is obtained by applying error-free transformation to improve the Clenshaw algorithm. It can yield a full working precision accuracy for the ill-conditioned polynomial evaluation. Forward error analysis and numerical experiments illustrate the accuracy and efficiency of the algorithm.
Highlights
Legendre polynomial is often used in numerical analysis [1,2,3], such as approximation theory and quadrature and differential equations
The Clenshaw algorithm [4, 5] is usually used to evaluate a linear combination of Chebyshev polynomials, but it can apply to any class of functions that can be defined by a three-term recurrence relation
We consider the polynomials in Legendre basis with real coefficients and floating point entry x
Summary
Legendre polynomial is often used in numerical analysis [1,2,3], such as approximation theory and quadrature and differential equations. The Clenshaw algorithm [4, 5] is usually used to evaluate a linear combination of Chebyshev polynomials, but it can apply to any class of functions that can be defined by a three-term recurrence relation. For ill-conditioned problems, several researches applied error-free transformations [11] to propose accurate compensated algorithms [12,13,14,15] to evaluate the polynomials in monomial, Bernstein, and Chebyshev bases with Horner, de Casteljau, and Clenshaw algorithms, respectively.
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