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2014•Numerical AlgorithmsOpen access

The three-term recursion for Chebyshev polynomials is mixed forward-backward stable

Alicja Smoktunowicz, Agata Smoktunowicz, Agata Smoktunowicz, Agata Smoktunowicz, Ewa Pawelec

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Abstract

This paper provides an error analysis of the three-term recurrence relation (TTRR) T n+1(x)=2x T n (x)−T n−1(x) for the evaluation of the Chebyshev polynomial of the first kind T N (x) in the interval [−1,1]. We prove that the computed value of T N (x) from this recurrence is very close to the exact value of the Chebyshev polynomial T N of a slightly perturbed value of x. The lower and upper bounds for the function $C_{N}(x)= |T_{N}(x)| + |x T_{N}^{\prime }(x)|$ are also derived. Numerical examples that illustrate our theoretical results are given.

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What this paper is about

This paper provides an error analysis of the three-term recurrence relation (TTRR) T n+1(x)=2x T n (x)−T n−1(x) for the evaluation of the Chebyshev polynomial of the first kind T N (x) in the interval [−1,1]. We prove that the computed value of T N (x) from this recurrence is very close to the exact value of the Chebyshev polynomial T N of a slightly perturbed value of x. The lower and upper bounds for the function $C_{N}(x)= |T_{N}(x)| + |x T_{N}^{\prime }(x)|$ are also derived. Numerical examples that illustrate our theoretical results are given.

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Available abstract

This paper provides an error analysis of the three-term recurrence relation (TTRR) T n+1(x)=2x T n (x)−T n−1(x) for the evaluation of the Chebyshev polynomial of the first kind T N (x) in the interval [−1,1]. We prove that the computed value of T N (x) from this recurrence is very close to the exact value of the Chebyshev polynomial T N of a slightly perturbed value of x. The lower and upper bounds for the function $C_{N}(x)= |T_{N}(x)| + |x T_{N}^{\prime }(x)|$ are also derived. Numerical examples that illustrate our theoretical results are given.

Key concepts: Mathematics, Chebyshev polynomials, Recurrence relation, Term (time), Theory of computation, Chebyshev nodes, Prime (order theory), Combinatorics

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