Coba jelasin Error estimates dari Gaussian quadrature dalam bahasa sunda dong

The error of a Gaussian quadrature rule can be stated as follows (théorem 3.6.24 in Stoer and Bulirsch).

For an integrand which has 2n continuous derivatives,

for some ξ in (a, b), where pn is the orthogonal polynomial of order n.

Stoer and Bulirsch remark that this error estimate is inconvenient in practice,

since it may be difficult to estimate the 2n'th derivative,

and furthermore the actual error may be much less than a bound established by the derivative.

Another approach is to use two Gaussian quadrature rules of different orders,

and to estimate the error as the difference between the two results.

Gauss-Kronrod rules can be useful.

If the interval [a, b] is subdivided,

the evaluation points of the new subintervals generally do not coincide with the previous evaluation points,

and thus the integrand must be evaluated at every point.

Gauss-Kronrod rules are Gaussian quadrature rules that are modified to maké some of the evaluation points coincide after subdivision.

The difference between the results before and after subdivision can be taken as an estimate of the error of approximation,

so such an approach can incréase the accuracy achieved for a given number of function evaluations.

The algorithms in QUADPACK (see below) are based on Gauss-Kronrod rules.