On numerical improvement of the first kind Gauss–Chebyshev quadrature rules

摘要

One of the integration methods of the equality type is Gauss–Chebyshev quadrature rule, which is in the following form:∫-11f(x)1-x2dx=πn∑k=1nfcos(2k-1)π2n+2π22n(2n)!f(2n)(η),-1<η<1.According to Gauss quadrature rules, the precision degree of above formula is the highest, i.e. 2n − 1. Hence, it is not possible to increase the precision degree of Gauss–Chebyshev integration formulas anymore. In this way, we present a matrix proof for this matter. But, on the other hand, we claim that we can improve the above formula numerically. To do this, we consider the integral bounds as two unknown variables. This causes to numerically be extended the monomial space f(x) = xj from j = 0, 1, … , 2n − 1 to j = 0, 1, … , 2n + 1. This means that we have two monomials more than Gauss–Chebyshev integration method. In other words, we give an approximate formula as:∫abf(x)1-x2dx≃∑i=1nwif(xi),in which a, b and w1, w2, … , wn and x1, x2, … , xn are all unknowns and the formula is almost exact for the monomial basis f(x) = xj, j = 0, 1, … , 2n + 1. Some important examples are finally given to show the excellent superiority of the proposed nodes and weights with respect to the usual Gauss–Chebyshev nodes and weights. Let us add that in this part we have also some wonderful 2-point formulas that are comparable with 71-point formulas of Gauss–Chebyshev quadrature rules in average.