By using the quadratic interpolating spline a new class of the quadrature rules was obtained. Those formulas are modifications of the well known trapezoidal rule. The basic characteristic of those formulas is a free parameter. With appropriate choice of that parameter, accuracy of the trapezoidal rule can be improved up to $O(h^4).$ Besides this, by using this nonstandard techniques some well known quadrature rules were also obtained.   2000 Mathematics Subject Classification. 65D32, 65D07

In this paper we describe five methods for the calculation of the moments\begin{equation*}\label{Momenti}\mathbb{M}_{n,m}=\int_{0}^{m}\varphi_{m}(t)t^{n}dt,n\in\mathbb{N}_{0},\end{equation*}where weight function $\varphi_{m}(\cdot)$ is the cardinal B-spline of order $m,m\in\mathbb{N}.$   2000 Mathematics Subject Classification. 65D07, 41A15

Abstract We are considering the quadrature formulas of “practical type” (with five knots) for approximate computation of integral [xxx] where w(·) denotes (even) Tchebychev weight function. We prove that algebraic degree of exactness of those formulas can not be greater than five. We also determined some admissible nodes and compared proposed formula with some other quadrature formulas.

We gave a short review of several results which are related to the role of splines (cardinal, centered or interpolating) in numerical integration. Results deal with the problem of approximate computation of the integrals with spline as a weight function, but also with the problem of approximate computation of the integrals without weight function. Besides, we presented an algorithm for calculation of the coefficients of the polynomials which correspond to the cardinal B-spline of arbitrary order and described five methods for calculation of the moments in the case when cardinal B-spline of order m,m ∈ N, is a weight function.

We consider the quadrature rules of “practical type” (with five knots) for approximately computation of the integral ∫ 2

In this article we consider stability of nonlinear equations which have the following form: Ax + F (x) = b, (1) where F is any function, A is a linear operator, b is given and x is an unknown vector. We give (under some assumptions about function F and operator A) a generalization of inequality: ‖X1 − X2‖ ‖X1‖ ≤ ‖A‖ ∥∥A−1∥∥ ‖b1 − b2‖ ‖b1‖ (2) (equation (2) estimates the relative error of the solution when the linear equation Ax = b1 becomes the equation Ax = b2) and a generalization of inequality: ‖X1 − X2‖ ‖X1‖ ≤ ∥∥∥A−1 1 ∥∥∥ ‖A1‖ (‖b1 − b2‖ ‖b1‖ + ‖A1‖ ∥∥∥A−1 2 ∥∥∥ ‖b2‖ ‖b1‖ · ‖A1 − A2‖ ‖A1‖ ) (3) (equation (3) estimates the relative error of the solution when the linear equation A1x = b1 becomes the equation A2x = b2). 1. Basic results Teorem 1. Let V be a normed space, let the linear operator A : V → V be invertible and bounded, let the inverse operator of the operator A be also bounded, let b1, b2 ∈ V and let the functions F1, F2 : V → V and the set S ⊆ V have the following properties: 1. the function F1 is Lipschitz on S, i.e., (∃L > 0) (∀x1, x2 ∈ S) ‖F1 (x1) − F1 (x2)‖ ≤ L ‖x1 − x2‖ , and the constant L is such that the inequality 1 − L∥∥A−1∥∥ > 0, holds; 2. (∃M > 0) (∀x ∈ S) ‖F1 (x)‖ ≤ M ‖x‖; and 3. (∃ε ≥ 0) (∀x ∈ S) ‖F1 (x) − F2 (x)‖ ≤ ε. AMS Subject Classification: 65J15