Decompositional quadrature formulas used to calculate integrals of high information complexity (Q1282573)

Let functions \(f(x)\) and \(g(x)\) be defined on the interval \([a,b]\) and let \(g\) be an computational model of the function \(f\). The authors assume that the calculation of function \(f\) is very complex and the same calculation by the function \(g\) is sufficiently (many times) simpler. So, for calculation the integral \[ \int_a^bf(x)dx \] the decompositional quadrature formula \[ \int^b_af(x)dx= \sum^N_{k=0} p_kf(x_k)+ \sum^M_{k=0} q_k\bigl[g(y_k)+ \Delta(y_k,x_L, \dots, x_N)\bigr], \tag{1} \] \(a\leq x_1<\cdots <x_N\leq b\), \(a\leq y_1<\cdots <y_M\leq b\), is used. Here \(\Delta(y_k,x_1, \dots,x_N)\) is a correction of the \(g(y_k)\) value. Optimal decompositional quadrature formulas of the type (1) in the classes of Hölder and \(W^r\) functions are constructed.