Abstract
This paper provides an error analysis of the three-term recurrence relation (TTRR) Tn+1(x)=2xTn(x)−Tn−1(x) for the evaluation of the Chebyshev polynomial of the first kind TN(x) in the interval [−1,1]. We prove that the computed value of TN(x) from this recurrence is very close to the exact value of the Chebyshev polynomial TN 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.
Highlights
This paper proves the numerical stability of the three-term recurrence relation (TTRR) of Chebyshev polynomials of the first kind (Tn(x)): Tn(x) = 2xTn−1(x) − Tn−2(x), n = 2, 3, . . . , (1)
In this paper we study the mixed forward-backward stability of the TTRR
The computed value of TN (x) by mixed forward-backward stable algorithm is very close to the exact value of the Chebyshev polynomial TN of a slightly perturbed value of x
Summary
A desirable property for algorithms is numerical stability (see [12, 17]). The term stability is sometimes used to refer to the forward or backward analysis of the algorithm. In this paper we study the mixed forward-backward stability of the TTRR (see [12], Section 1.5). A precise definition of mixed forward-backward stability is given. The computed value of TN (x) by mixed forward-backward stable algorithm is very close to the exact value of the Chebyshev polynomial TN of a slightly perturbed value of x. Where UN−1(x) denotes the Chebyshev polynomial of the second kind. We prove that the TTRR is mixed forward-backward stable in the sense of (3) (which is equivalent to (2).
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