Error Analysis of Newton-Cotes formulas
The Newton-Cotes formulas are a group of formulas for evaluating numeric integration at equally spaced points.
Let , , be equally spaced points, and be the corresponding values. Let be the space , and let be the interpolation variable . Thus to interpolate at x,
A polynomial of degree can be derived to pass through these points and approximate the function . Using divided differences and Newton polynomial, can be obtained as
From the general form of polynomial interpolation error, the error of using to interpolate can be obtained as
Since , the error term of numerical integration is
Error terms for different rules
The Trapezoid Rule
Let's consider the trapezoid rule in a single interval. In each interval, the integration uses two end points. Thus . Then . Applying (1
), we get
Thus the local error is . Consider the composite trapezoid rule. Given that , the global error is
where , .
To justify (2
), we can need the theorem below in page 345:
If is continuous and the , then for some value in the interval of all the arguments
The Simpson's 1/3 Rule
Consider Simpson's 1/3 rule. In this case, three equally spaced points are used for integration. Thus . Applying (1
), we get
This doesn't mean that the error is zero. It simply means that the cubic term is identically zero. The error term can be obtained from the next term in the Newton polynomial, obtaining
Thus the local error is and the global error is .
The Simpson's 3/8 Rule
Consider Simpson's 3/8 rule. In this case, since four equally spaced points are used. Applying (1
), we get
Both the Simpon's 1/3 rule and the 3/8 rule have error terms of order . With smaller coefficient, the 1/3 rule seems more accurate. Then why do we need the 3/8 rule? The 3/8 rule is useful when the total number of increments is odd. Three increments can be used with the 3/8 rule, and then the rest even number of increments can be used with 1/3 rule.
A Numerical Example
Given the set of data points, solve the numerical integration
Use the trapezoid rule. First try . That is, use only the two end points. We can get
Compared with the exact solution we have
Using all three points with we can get
Thus the error ratio is . This is close to what we can get by inspecting
Using the data given below, find the maximum error incurred in using Newton's forward interpolation formula to approximate .
According the general error formula of polynomial interpolation
Given that , and , we can get
When using Simpson's 1/3, what is the error ratio supposed to be?