**for
**

The adaptive Simpson's rule for quadrature uses the two subroutines "Simpson" and "Adapt." The program is "recursive". There is no brake available if something goes wrong, i.e. if a pathological "bad" function is thrown it's way it may proceed on a slippery path of infinite recursion.

**Proof ****Adaptive
Simpson's Rule** **Adaptive
Simpson's Rule**

**Animations (****Adaptive
Simpson's Rule** **Adaptive
Simpson's
Rule****).**

**Computer
Programs ****Adaptive
Simpson's Rule** **Adaptive
Simpson's Rule**

**Mathematica Subroutines (Adaptive
Simpson's Rule).**

**Example 1.** Use the
adaptive Simpson's rule to compute a numerical approximation to the
integral .

Use the tolerances . Compare
with the analytic or "true value" of the integral.

**Solution
1.**

**Example 2.** Use the
adaptive Simpson's rule to compute a numerical approximation to the
integral .

Use the tolerances . Compare
with the analytic or "true value" of the integral.

**Solution
2.**

**Example 3.** Use the
adaptive Simpson's rule to compute a numerical approximation to the
integral .

Use the tolerances . Compare
with the analytic or "true value" of the integral.

**Solution
3.**

Execute the following *Mathematica* subroutine, which is the
"long version" of the subroutine we have been using previously.

**Example 4.** Use the
adaptive Simpson's rule to compute a numerical approximation to the
integral that
we investigated in example 2.

The long solution is obtained if you add a print statement to
investigate the in between computations.

This subroutine is pedagogical and is intended to help us understand
what's happening in a recursive program.

You would probably not want to always print out the in between steps,
so you might want to re-execute the first version for some of your
work.

**Solution
4.**

**Example 5.** Use the
adaptive Simpson's rule to compute a numerical approximation to the
integral .

Use the tolerances . Compare
with the analytic or "true value" of the integral.

**Solution
5.**

**Example 6.** Use the
adaptive Simpson's rule to compute a numerical approximation to the
integral .

Use the tolerances . Compare
with the analytic or "true value" of the integral.

**Solution
6.**

**Various
Scenarios**** and
Animations for the Adaptive Simpson's Rule.**

**Example
7.** Let over . Use
the adaptive Simpson's rule to approximate the value of the
integral.

**Solution
7.**

**Animations (****Adaptive
Simpson's Rule** **Adaptive
Simpson's
Rule****).**

**Research Experience for
Undergraduates**

**Adaptive
Simpson's Rule** **Adaptive
Simpson's Rule** Internet hyperlinks to web
sites and a bibliography of articles.

**Download this
Mathematica Notebook**
**Adaptive
Simpson's Rule**

**Return
to Numerical Methods - Numerical Analysis**

(c) John H. Mathews 2004