The equal coefficients quadrature rules and their numerical improvement

One of the quadrature rules is the “Equal coefficients quadrature rules” represented by∫abw(x)f(x)dx≃Cn∑i=1nf(xi),where Cn is a constant number and w(x) is a weight function on [a, b]. In this work, we show that the precisian degree of above formula can be increased by taking the upper and lower bounds of the integration formula as unknowns. This causes to numerically be extended the monomial space {1, x, … , xn} to {1, x, … , xn+2}. We use a matrix proof to show that the resulting nonlinear system for the basis f(x) = xj, j = 0, … , n + 2 has no analytic solution. Thus, we solve this system numerically to find unknowns x1, x2, … , xn, Cn, a and b. Finally, some examples will be given to show the numerical superiority of the new developed method.