On an approximation formula

We generalize an approximation formula which in some particular cases has been studied by [J.F. Traub 1964] and \ [R.M.Humel and C.S. Secbeck 1949]. Denote by \(I_{x}\) the closed interval determined by the distinct points \(x,x_{0} \in \mathbb{R}\). Consider the nonlinear mapping \(f:I_{x}\rightarrow \mathbb{R}\), which has derivatives up to the order \(2n+1\) on \(I_{x}\), and deonte by \(G\) the set of functions $$G=\big\{g:g(t) =f(x_0) + (t-x_0) \sum \limits_{i=1}^{n} a_i f'(x_0 + b_i(t-x_0) , \ a_i, b_i \in \mathbb{R}, i=1,n, t\in I_x\big\}$$ From the set \(G\) we determine a function \(\bar{g}\) with the properties \(f^{(i)}(x_0) = \bar{g}^{(i)}(x_0)\). We determine the coefficients \(a_{i},b_{i},\ i=1,\ldots,n\) and we also evaluate the remainder \(f(t) -\bar{g}(t)\), \(t\in I_{x}\).

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Scanned paper: on the journal website.

Latex version of the paper.

I. Păvăloiu, On an approximation formula, Rev. Anal. Numér. Théor. Approx., 26 (1997) nos. 1-2, pp. 179-184.

[1] C. I. Berezin and N. Jidkov, Metody vychisleny, Fizmatgiz, Moscow (1962).

[2] P. M. Humel and C. L. Seebeck Jr., A generalization of Taylor’s expansion, Amer. Math. Monthly 56 (1949), pp. 243-247.

[3] A. Lupas Calculul valorilor unor functii elementare, Gazeta Matematica (Ser. A) VII, l (1986), pp. 15-26.

[4] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Inc., Englowood Cliffs, N.J., 1964.