Simple Simpson

Mathematics 1502 Honors Calculus II Fall, 2000

Here is a really, really simple derivation of Simpson's rule for numerical integration.

The problem is to integrate f(x), which has been "sampled" at the values x_k and also at the midpoints (x_k + x_k+1)/2.

Simpson's idea was to fit a curve by the parabola which passes through each set of three consecutive data, i.e.,

x₁ y₁ := f(x₁)

(x₁+x₂)/2 y_1.5 := f ((x₁+x₂)/2)

x₂ y₂ := f(x₂)

,

x₁	y₁ := f(x₁)
(x₁+x₂)/2	y_1.5 := f ((x₁+x₂)/2)
x₂	y₂ := f(x₂)

x₂ y₂ := f(x₂)

(x₂+x₃)/2 y_2.5 := f ((x₂+x₃)/2)

x₃ y₃ := f(x₃)

,

x₂	y₂ := f(x₂)
(x₂+x₃)/2	y_2.5 := f ((x₂+x₃)/2)
x₃	y₃ := f(x₃)

etc. The point being that it is easy to integrate a quadratic, and it gives a better fit than the straight-line trapezoid rule. As we have seen in class, Simpson's rule is equivalent to taking (1/3) of the trapezoid rule and adding (2/3) of the midpoint rule.

If you try to derive Simpson's rule directly, there is a considerable amount of algebra. Instead, let us consider the simplest situation, but rewrite everything so that we express the integral directly in terms of three sets of (x,y) data.

The simplest situation would be where the values of x are -1, 0, and +1, and the quadratic is

Let's call the y-values y_-1, y₀, and y₁. The value y₀ corresponds to the value of the function at the midpoint of (-1,1), which is x=0.

Now, the integral of y from -1 to +1 is easily calculated:

I = 2a/3 + 2 c. Let's write a and c directly in terms of the values of y - notice that we don't even need b. We find

_-1

₀

₁

Since we don't need b, we eliminate it by

_-1

₁

As for a and c, we now find:

₀

_-1

₁

₀

Hence the integral is:

_-1

₁

₀

This has to be the general rule for the integral of a quadratic, since we have written it in a way that does not use specific values of a,b,c, or the fact that x runs from -1 to 1.

If you have Excel or a compatible spreadsheet, you can test out the different methods with the spreadsheet shown in class.
(Here are some alternative formats, in case you have trouble with the one above: 1, 2, 3, 4,

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