If not, the algorithm is recursively applied to the two halves of the subinterval. If the values obtained from the 3-point Simpson's rule and the 5-point composite Simpson's rule agree to within a specified tolerance (which they clearly do not for these figures), then an extrapolation taken from the two values is accepted as the value for the integral over the subinterval. The algorithm works recursively on subintervals of the original interval. These two figures illustrate the basis for adaptive Simpson's method. It was based on adaptive Simpson's method, pioneered by my grad school buddy Bill McKeeman to show off recursion in Algol. Here is a coarse picture.įor most of the time since the beginning of MATLAB, up until when I wrote the chapter on quadrature twelve years ago, the MATLAB function for evaluating definite integrals was one named quad. One of the first things I have to do in that chapter is explain that quadrature is an old-fashioned term derived from the notion that the value of a definite integral can be approximated by plotting the function on graph paper and then counting the number of little squares that fall underneath the graph.
I wrote a chapter on the subject in my textbook Numerical Computing with MATLAB. We seek to compute the value of the definite integral, The task involves a real-valued function $f(x)$ of a real variable $x$, defined on an interval $a \le x \le b$. So this is an opportunity to get reacquainted. But then I stopped paying attention for a while.
Several years ago I regarded myself as an expert on numerical methods for approximating integrals.
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