Consider the quadrature rule on the interval a b given by th

Consider the quadrature rule on the interval [a, b] given by the formula sigma_j = 0^3 alpha f(x_j) How should alpha and the x_j be chosen to provide the highest degree of precision possible? What is this degree?

Solution

The degree of accuracy or precision of a quadrature formula is the largest positive integer n such that the formula is exact for x_j foe each j=0 to n

Therefore, the highest degree of precision possible is 3.

Also, the degree of precision of a quadrature formula is n if and only if the error is zero for all polynomials of degree j= 0, 1, ...n , but is NOT zero for some polynomial of degree n+1.