Simpson's 1/3rd Rule Textbook Notes: Integration: Computer Engineering

Chapter 5

Simpsons Rule

5.2 Simpsons 1/3rd Rule5.2.1 Simpsons Rule:

A wide variety of numerical methods have been developed to simplify the integral. Here, we will discuss Simpsons 1/3rd Rule of integral approximation, which improves upon the accuracy of the Trapezoidal Rule.

Trapezoidal rule was based on approximating the integrand by a first order polynomial, and then integrating the polynomial in the interval of integration. Simpsons 1/3rd rule is an extension of Trapezoidal rule where the integrand is non-approximated by a second order polynomial.

5.2.2 Deriving Simpson's RuleHence

where is a second order polynomial.

Choose

EMBED Equation.3 and as the three points of the function to evaluate and .

Solving the above three equations for unknowns, and give

Then

Substituting values of and give

Since for Simpsons 1/3rd Rule, the interval is broken into 2 segments, the segment width

Hence the Simpsons 1/3rd rule is given by

Since the above form has 1/3 in its formula, it is also called Simpsons 1/3rd Rule.

Example 1:

Human vision has the remarkable ability to infer 3D shapes from 2D images. The intriguing question is: can we replicate some of these abilities on a computer? Yes, it can be done and to do this, integration of vector fields is required. The following integral needs to integrated.

where

a) Use Simpsons 1/3rd Rule to find the integration.b) Find the true error,

c) Find the absolute relative true error, .

Solution:

a)

b) The exact value of the above integral is found using Maple for calculating the true error and relative true error.

so the true error is

c) The absolute relative true error, , would then be

5.2.3 Multiple Segment Simpsons 1/3rd Rule

Just like in multiple-segment Trapezoidal Rule, one can subdivide the interval into segments and apply Simpsons 1/3rd Rule repeatedly over every two segments. Note that needs to be even. Divide interval into equal segments, hence the segment width .

where

Apply Simpsons 1/3rd Rule over each interval,

Since

then

EMBED Equation.3 H.W: Draw a flow chart to calculate the integral of the function f(x) from a to b using n equal intervals by Simpson Rule.

Example 2:

Human vision has the remarkable ability to infer 3D shapes from 2D images. The intriguing question is: can we replicate some of these abilities on a computer? Yes, it can be done and to do this, integration of vector fields is required. The following integral needs to integrated.

where

a) Use four segment Simpsons 1/3rd Rule to find the value of the integral.

b) Find the true error, Et for part (a).

c) Find the absolute relative true error for part (a).Solution:

a) Using segment Simpsons 1/3rd Rule,

So

b) The exact value of the above integral is found using Maple for calculating the true error and relative true error.

so the true error is

c) The absolute relative true error, , would then be

Table 1: Values of Simpsons 1/3rd Rule for Example 2 with multiple segments

Approximate Value

2

4

6

884.9466726.9166766.606362.3177

-24.15433.875-5.8137-1.5251

39.731%

55.723%

9.5632%

2.5088%

89

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