Quadrature – Concepts (numerical integration) Don Allen

Principles Quadrature Methods - Types – Points are given – Newton-Cotes – Points are selected – Gaussian – Romberg – To extrapolate to better estimates – Adaptive – Predicting exactness from error Error formulas Composite forms

Newton-Cotes Includes Trapezoidal and Simpson’s rules. Based on polynomial interpolation of select points. – Usually equally spaced points. – Can be as the data as determined from practice. Error computed from errors of approximation Composite forms involve repeated application over select series of points.

Gaussian Quadrature Based on the question: What points can be selected to make the method exact for the maximum order of polynomials. Points must be computed beforehand. These are the (numerical) solutions of certain polynomial equations. Composite forms involve repeated application over select series of points.

Romberg method Based on a precise knowledge of the error formula. Uses the ideas of Richardson’s extrapolation. The idea is that we use the method for step size h, h/2, h/4, etc until the approximation is quite good.

Adaptive Methods Two different methods are employed over the same interval. One method is of higher order than the other. The play is to use the higher order method as exact to determine the approximation value of the lower order method. Composite forms involve repeated application over select series of points.