Error of Analysis of Newton-Cotes formulas

Error Analysis of Newton-Cotes formulas edit

The Newton-Cotes formulas are a group of formulas for evaluating numeric integration at equally spaced points.

The Methods[1] edit

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


where  .

Since  , the error term of numerical integration is







Error terms for different rules edit

The Trapezoid Rule edit

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


where  . 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[2] 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 edit

Consider Simpson's 1/3 rule. In this case, three equally spaced points are used for integration. Thus  . Applying (1), we get


where  .

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 edit

Consider Simpson's 3/8 rule. In this case,   since four equally spaced points are used. Applying (1), we get


where  .

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 edit

Given the set of data points, solve the numerical integration  

3.1 -0.32258065
3.5 -0.28571429
3.9 -0.25641026

Solution edit

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


and so


Thus the error ratio is  . This is close to what we can get by inspecting


Exercises edit

Exercise 1[3] edit

Using the data given below, find the maximum error incurred in using Newton's forward interpolation formula to approximate  .

0.1 1.10517
0.2 1.22140
0.3 1.34986
0.4 1.49182
0.5 1.64872

Exercise 2 edit

When using Simpson's 1/3, what is the error ratio supposed to be?

References edit

  1. Hoffman, Joe D. (2001). Numerical Methods for Engineers and Scientists (2nd ed.). Marcel Derkker, INC. ISBN 0-8247-0443-6. 
  2. Hamming, R. W. (1986). Numerical Methods for Scientists and Engineers (2nd ed.). New York: Dover Publications. ISBN 0-486-65241-6. 
  3. Tenenbaum, Morris; Pollard, Harry (1985). Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences. New York: Dover Publications. ISBN 9780486649405.