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.