Simpson's 3/8 Rule for Numerical Integration

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Module

for

Simpson's 3/8 Rule for Numerical Integration

The numerical integration technique known as "Simpson's 3/8 rule" is credited to the mathematician Thomas Simpson (1710-1761)

of Leicestershire, England. His also worked in the areas of numerical interpolation and probability theory.

Theorem (Simpson's 3/8 Rule) Consider over , where , , and . Simpson's

3/8 rule is

.

This is an numerical approximation to the integral of over and we have the expression

.

The remainder term for Simpson's 3/8 rule is , where lies somewhere between , and have the

equality

.

Proof Simpson's 3/8 Rule Simpson's 3/8 Rule

Composite Simpson's 3/8 Rule

Our next method of finding the area under a curve is by approximating that curve with a series of cubicsegments that lie above the intervals . When several cubics are used, we call it the composite Simpson's3/8 rule.

Theorem (Composite Simpson's 3/8 Rule) Consider over . Suppose that the interval is subdivided

into subintervals of equal width by using the equally spaced sample points

Simpson's 3/8 Rule for Numerical Integration

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for . The composite Simpson's 3/8 rule for subintervals is

.

This is an numerical approximation to the integral of over and we write

.

Proof Simpson's 3/8 Rule Simpson's 3/8 Rule

Remainder term for the Composite Simpson's 3/8 Rule

Corollary (Simpson's 3/8 Rule: Remainder term) Suppose that is subdivided into subintervals

of width . The composite Simpson's 3/8 rule

.

is an numerical approximation to the integral, and

.

Furthermore, if , then there exists a value with so that the error term has the form

.

This is expressed using the "big " notation .

Remark. When the step size is reduced by a factor of the remainder term should be reduced by approximately

.

Algorithm Composite Simpson's 3/8 Rule. To approximate the integral

Simpson's 3/8 Rule for Numerical Integration

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,

by sampling at the equally spaced sample points for , where . Notice that

and .

Animations (Simpson's 3/8 Rule Simpson's 3/8 Rule). Internet hyperlinks to animations.

Computer Programs Simpson's 3/8 Rule Simpson's 3/8 Rule

Mathematica Subroutine (Simpson's 3/8 Rule). Object oriented programming.

Example 1. Numerically approximate the integral by using Simpson's 3/8 rule with m = 1, 2, 4.

Solution 1.

Example 2. Numerically approximate the integral by using Simpson's 3/8 rule with m = 10, 20, 40, 80, and 160.

Solution 2.

Example 3. Find the analytic value of the integral (i.e. find the "true value").

Solution 3.

Example 4. Use the "true value" in example 3 and find the error for the Simpson' 3/8 rule approximations in example 2.

Simpson's 3/8 Rule for Numerical Integration

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Solution 4.

Example 5. When the step size is reduced by a factor of the error term should be reduced by approximately

. Explore this phenomenon.

Solution 5.

Example 6. Numerically approximate the integral by using Simpson's 3/8 rule with m = 1, 2, 4.

Solution 6.

Example 7. Numerically approximate the integral by using Simpson's 3/8 rule with m = 10, 20, 40, 80, and 160.

Solution 7.

Example 8. Find the analytic value of the integral (i.e. find the "true value").

Solution 8.

Example 9. Use the "true value" in example 8 and find the error for the Simpson's 3/8 rule approximations in example 7.

Solution 9.

Example 10. When the step size is reduced by a factor of the error term should be reduced by approximately

. Explore this phenomenon.

Solution 10.

Various Scenarios and Animations for Simpson's 3/8 Rule.

Example 11. Let over . Use Simpson's 3/8 rule to approximate the value of the integral.

Solution 11.

Simpson's 3/8 Rule for Numerical Integration

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Animations (Simpson's 3/8 Rule Simpson's 3/8 Rule). Internet hyperlinks to animations.

Research Experience for Undergraduates

Simpson's Rule for Numerical Integration Simpson's Rule for Numerical Integration Internet hyperlinks to web sites and a bibliography of

articles.

Download this Mathematica Notebook Simpson 's 3/8 Rule for Numerical Integration

Return to Numerical Methods - Numerical Analysis

Simpson's 3/8 Rule for Numerical Integration

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(c) John H. Mathews 2004