ROMBERG RULE

INTERACTIVE E-BOOK

INTEGRATION

MAJOR

GENERAL

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Title	An interactive e-book for illustrating Romberg Rule of integration
Creator	Autar K Kaw
Subject and Keywords	Romberg Rule, Numerical Integration, Mathcad, Maple, Mathematica, Matlab, Simulations.
Description	This is an interactive E-book for illustrating Romberg Rule of integration. It includes links to simulations in Mathcad, Maple, Mathematica and Matlab for the algorithm; multiple choice quizzes, problem set, and a power point presentation.
Publisher	Holistic Numerical Methods Institute, College of Engineering, University of South Florida, Tampa, FL 33620-5350.
Contributors	Autar Kaw
Format	Text/HTML
Last Revised	January 22, 2006
Identifier	http://numericalmethods.eng.usf.edu/ebooks/romberg_07int_ebook.pdf
Language	English
Rights	http://numericalmethods.eng.usf.edu/rights.htm

Table of Contents

Background

What is integration

Error in Multiple-Segment Trapezoidal Rule

Richardson’s Extrapolation Formula

Romberg Integration

PowerPoint presentation

Example 1

Example 2

Simulation

Romberg Method (Mathcad Maple Mathematica Matlab)

Convergence Simulation of Romberg Method (Mathcad Maple Mathematica Matlab)

Problem Set

Multiple choice question examination

Background

This is an example of how an instructor of a Numerical Methods course can develop an E-book on a single topic by using resources that are only available at the Holistic Numerical Methods Institute (HNMI) Website.

Although one of the primary advantages of a web-based resource is that one can use documents outside of its own domain, we are particularly using this example as a way to illustrate the holistic nature of the resources available at HNMI. You are welcome to modify and edit this e-book to suit your purpose.

Using the textbook document written in MS Word 2000 as a foundation, it was made interactive within MS Word 2000 by putting bookmarks and hyperlinks to background information, PowerPoint presentations, Mathcad simulations, historical anecdotes, multiple choice tests, problem sets, etc. Then the MS Word 2000 file was saved as a webpage and that is what you are about to read here.

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What is integration?

Integration is the process of measuring the area under a function plotted on a graph. Why would we want to do so? Among the most common examples are finding the velocity of a body from acceleration functions, and displacement of a body from velocity data. Throughout the engineering fields, there are (what sometimes seems like) countless applications for integral calculus. Sometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. For this reason, a wide variety of numerical methods have been developed to find the integral.

Here, we will discuss the Romberg rule of approximating integrals of the form

(1)

where

is called the integrand,

lower limit of integration, and

upper limit of integration

Figure 1: Integration of a function

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Error in Multiple-Segment Trapezoidal Rule

The true error in a multiple segment Trapezoidal Rule with n segments for an integral

is given by

(2)

where for each i, is a point somewhere in the domain , and

the term can be viewed as an approximate average value of in . This leads us to say that the true error, E_t in Equation (2)

(3)

in estimate of the integral using the n-segment Trapezoidal Rule.

Table 1: Values obtained using multiple segment Trapezoidal rule for

n	Value	E_t
1	11868	807	7.296	---
2	11266	205	1.854	5.343
3	11153	91.4	0.8265	1.019
4	11113	51.5	0.4655	0.3594
5	11094	33.0	0.2981	0.1669
6	11084	22.9	0.2070	0.09082
7	11078	16.8	0.1521	0.05482
8	11074	12.9	0.1165	0.03560

Table 1 shows the results obtained for the integral using multiple-segment Trapezoidal rule

The true error for the 1-segment Trapezoidal rule is -807, while for the 2-segment rule, the true error is -205. The true error of -205 is approximately a quarter of -807. The true error gets approximately quartered as the number of segments is doubled from 1 to 2. Same trend is observed when the number of segments is doubled from 2 to 4 (true error for 2-segments is -205 and for four segments is -51.5). This follows Equation (3).

This information, although interesting, can also be used to get better approximation of the integral. That is the basis of Richardson’s extrapolation formula for integration by Trapezoidal Rule.

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Richardson’s Extrapolation Formula for Trapezoidal Rule

The true error, , in the n-segment Trapezoidal rule is estimated as

(4)

where

C is an approximate constant of proportionality.

Since

(5)

where

= true value

= approximate value using n-segments.

Then from equations (4) and (5),

(6)

If the number of segments is doubled from n to 2n in the Trapezoidal rule,

(7)

Equations (1) and (2) can be solved simultaneously to get

. (8)

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Example 1

The vertical distance covered by a rocket from to seconds is given by

a) Use Romberg’s rule to find the distance covered. Use the 2-segment and 4-segment Trapezoidal rule results given in Table 1.

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

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

Solution

Using Richardson’s extrapolation formula for Trapezoidal rule

and choosing n=2,

b) The exact value of the above integral is

so the true error

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

Table 2 shows the Richardson’s extrapolation results using 1, 2, 4, 8 segments. Results are compared with those of Trapezoidal rule.

Table 2: Values obtained using Richardson’s extrapolation formula for Trapezoidal rule for

Trapezoidal Rule

for Trapezoidal Rule

Richardson’s Extrapolation

for Richardson’s Extrapolation

11868

11266

11113

11074

7.296

1.854

0.4655

0.1165

11065

11062

11061

0.03616

0.009041

0.0000

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Romberg Integration

Romberg integration is same as Richardson’s extrapolation formula as given by equation (8). However, Romberg used a recursive algorithm for the extrapolation as follows.

The estimate of the true error in trapezoidal rule is given by (See Equation ( ) ),

Since the segment width, h

equation (2) can be written as

(9)

The estimate of true error is given by

(10)

It can be shown that the exact true error could be written as

(11)

and for small h,

(12)

Since we used in the formula (Equation (8)), the result obtained from Equation (8) has an error of and can be written as

(13)

where the variable TV is replaced by as the value obtained using Richardson’s extrapolation formula. Note also that the sign is replaced by = sign.

Hence the estimate of the true value now is

Determine another integral value with further halving the step size (doubling the number of segments),

(14)

then

From Equation (13) and (14),

(15)

The above equation now has the error of . The above procedure can be further improved by using the new values of the estimate of true value that has the error of to give an estimate of .

Based on this procedure, a general expression for Romberg integration can be written as

(16)

The index k represents the order of extrapolation. For example, represents the values obtained from the regular Trapezoidal rule, represents values obtained using the true error estimate as . The index j represents the more and less accurate estimate of the integral. The value of the integral with j+1 index is more accurate than with j index.