Adaptive procedures for numerical methods (Q2739932)

Let on the interval \([a,b]\) the base function \(\phi(x,\alpha_1,\ldots, \alpha_{m})\in C[a,b]\) be given, and let \(\{p_0(x)\),NEWLINENEWLINENEWLINE\(p_1(x)\), \(\ldots, p_{n}(x)\}\), \(p_{i}(x)\in C[a,b]\) be given nodal functions. The polynomial \(g(x)=\sum_{i=0}^{s} C_{i} p_{i}(x)\)NEWLINENEWLINENEWLINE\(\phi(x,\alpha_{i1},\ldots, \alpha_{im})\) is called generalized interpolation polynomial with base function \(\phi(x,\alpha_1,\ldots,\alpha_{m})\) and nodal coefficients \(p_{i}(x)\). Here the system of functions \(\phi(x,\alpha_{i1},\ldots, \alpha_{im})\), \(i=0,1,\ldots,n\) forms the Chebyshev system on \([a,b]\). NEWLINENEWLINENEWLINEThis paper deals with adaptive quadrature formulas and the construction of piecewise smooth stable solutions of the Cauchy problem \(y'=f(x,y)\), \(x\in [a,b]\), \(y(a)=y_0\) on the base of generalized interpolation polynomials. For example, the following result is proposed. If the function \(g(x)\) has derivatives up to order \(n\) and the remainder term \(r(x)=(g(x)-P_{n-2}(x))/ (x-x_0)^{n-2}\) is a monotone convex function, then the error of a quadrature formula which is precise for polynomials \(P_{m}(x), m\leq n\), can be minimized by generalized interpolation polynomials.