Is it just a coincidence – true error in multiple segment Trapezoidal rule gets approximately quartered as the number of segments is doubled?


Look at the table below. This is a table that shows the approximate value of the integral

\int_{8}^{30} 2000 ln \frac{140000}{140000-2100t}-9.8t dt 

as a function of the number of segments used in the Trapezoidal rule and the corresponding true error.

n

Value

Et

1

11868

-807

2

11266

-205

3

11153

-91.4

4

11113

-51.5

5

11094

-33.0

6

11084

-22.9

7

11078

-16.8

8

11074

-12.9

The true error for n=1 is -807 and for n=2 is -205. As you can see the quarter of -807 is approximately -201.75 and close to the true error for n=2. Is this a coincidence?

Look at the true error for n=2 which is -205 and for n=4 is -51.5. As you can see the quarter of -205 is approximately -51.75 and close to the value of the true error for n=4. Is this a coincidence?

No. This is because the true error in a single segment trapezoidal rule is

\frac{(b-a)^3}{12} f^{\prime\prime} (c)

where c is some point not known but in the domain [a,b] of \int_{a}^{b} f(x) dx. It can be then shown (see page 14 of this pdf file for full proof) that for the multiple segment trapezoidal rule, the true error is

\frac{(b-a)^3}{12n^2} f^{\prime\prime}

where the f^{\prime\prime} is an average value of the second derivative of the function f(x) calculated at some point within each of the n segments. Since a and b are constant, and f^{\prime\prime} becomes almost a constant as n increases, the true error is approximately inversely proportional to the square of the number of segments.

Note to the reader: Develop a similar table as given above for an integral of your choice and see it for yourself if the true error gets approximately quartered as the number of segments is doubled.

_____________________________________________________

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu

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Published by

Autar Kaw

Autar Kaw (http://autarkaw.com) is a Professor of Mechanical Engineering at the University of South Florida. He has been at USF since 1987, the same year in which he received his Ph. D. in Engineering Mechanics from Clemson University. He is a recipient of the 2012 U.S. Professor of the Year Award. With major funding from NSF, he is the principal and managing contributor in developing the multiple award-winning online open courseware for an undergraduate course in Numerical Methods. The OpenCourseWare (nm.MathForCollege.com) annually receives 1,000,000+ page views, 1,000,000+ views of the YouTube audiovisual lectures, and 150,000+ page views at the NumericalMethodsGuy blog. His current research interests include engineering education research methods, adaptive learning, open courseware, massive open online courses, flipped classrooms, and learning strategies. He has written four textbooks and 80 refereed technical papers, and his opinion editorials have appeared in the St. Petersburg Times and Tampa Tribune.

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