Term Paper of Numerical Analysis

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  • Topic: Integral, Trapezoidal rule, Numerical analysis
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TERM PAPER OF 5TH SEMESTER 2010

NUMERICAL ANALYSIS

MTH204

TOPIC: Using trapezoidal rule and simpson’s rule, evaluate the integral [pic]

DOA:

DOS: 12th November, 2010

Submitted to:   Submitted by:

Ms.Nitika Chugh                        Mr. William Anthony Deptt. Of Mathematics Regd no:10805460

Rollno:RD1803B30

ACKNOWLEDGEMENT

I would like to express my gratitude for the many helpful comment and suggestions .I have received over the last few days regarding the expository and critical expects of my term work and especially for those comments which bear directly or may various argument for the center thesis of term work. Most importantly I would like to thank my HOD (head of department) and my teacher Ms. Nitika Chugh for her days of supervision. Her critical commentary on my work has played a major role in both the content and presentation of our discussion and arguments. I have extend my appreciation to the several sources which provided various kinds of knowledge base support for me during this period. Last and not the least my God and my parent helped me to overcome the problem while preparing this project.

WILLIAM ANTHONY

TABLE OF CONTENTS

INTRODUCTION TO TRAPEZOIDAL RULE AND SIMPSON'S RULE

EVALUATION OF THE INTEGRAL

• APPLICATION OF TRAPEZOIDAL RULE AND SIMPSON’S RULE • BIBLIOGRAPHY

INTRODUCTION TO TRAPEZOIDAL RULE AND SIMPSON'S RULE

Two widely used rules for approximating areas are the trapezoidal rule and Simpson's rule. To motivate the new methods, we recall that rectangular rules approximated the function by a horizontal line in each interval. It is reasonable to expect that if we approximate the function more accurately inside each interval then a more efficient numerical scheme will follow. This is the idea behind the trapezoidal and Simpson's rules. Here the trapezoidal rule approximates the function by a suitably chosen (not necessarily horizontal) line segment. The function values at the two points in the interval are used in the approximation. While Simpson's rule uses a suitably chosen parabolic shape (see Section 4.6 of the text) and uses the function at three points. The Maple student package has commands trapezoid and simpson that implement these methods. The command syntax is very similar to the rectangular approximations. See the examples below. Note that an even number of subintervals is required for the simpson command and that the default number of subintervals is n=4 for both trapezoid and simpson. > with(student):

> trapezoid(x^2,x=0..4);
[pic]
> evalf(trapezoid(x^2,x=0..4));
[pic]
> evalf(trapezoid(x^2,x=0..4,10));
[pic]
> simpson(x^2,x=0..4);
[pic]
> evalf(simpson(x^2,x=0..4));
> evalf(simpson(x^2,x=0..4,10));

Simpson's rule
In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation: [pic].
The method is credited to the mathematician Thomas Simpson (1710–1761) of Leicestershire, England.

Derivation

Simpson's rule can be derived in various ways.

Quadratic interpolation

One derivation replaces the integrand f(x) by the quadratic polynomial P(x) which takes the same values as f(x) at the end points a and b and the midpoint m = (a+b) / 2. One can use Lagrange polynomial interpolation to find an expression for this polynomial, [pic]

An easy (albeit tedious) calculation shows that
[pic][1]
This calculation can be carried out more easily if one first observes that (by scaling) there is no loss of generality in assuming that...
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