Chapter 6 Enotes

8/13/2019 Chapter 6 Enotes

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Copyright 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

NumericalDifferentiation and

IntegrationIntroduction

Newton-Cotes Integration FormulasIntegration of EquationsNumerical Differentiation


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Introduction

Calculus is the mathematics of change. Because engineersmust continuously deal with systems and processes thatchange, calculus is an essential tool of engineering.

Standing in the heart of calculus are the mathematicalconcepts of differentiation and integration :

b

a

ii x

ii

dx x f I

x x f x x f

dxdy

x

x f x x f

x

y

)(

)()(lim

)()(

0


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NumericalDifferentiation

and Integration

Newton-CotesIntegrationFormulas

Integration ofEquations

NumericalDifferentiation

High-accuracyformulas

Richardsonextrapolation

Unequal-spaced data

GaussQuadrature

Trapezoidal ruleSimpsons rules


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Graphical definition of a derivative


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Graphical representation of the integral


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Noncomputer Methods forDifferentiation and Integration

The function to be differentiated or integrated willtypically be in one of the following three forms: A simple continuous function such as polynomial, an

exponential, or a trigonometric function. A complicated continuous function that is difficult or

impossible to differentiate or integrate directly.

A tabulated function where values of x and f(x) are given ata number of discrete points, as is often the case withexperimental or field data.


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Newton-Cotes IntegrationFormulas

The Newton-Cotes formulas are the most commonnumerical integration schemes.

They are based on the strategy of replacing acomplicated function or tabulated data with anapproximating function that is easy to integrate:

n is order of polynomial.

nn

nnn

b

an

b

a

xa xa xaa x f

dx x f dx x f I

1110)(

)()(


8/29Copyright 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Figure 21.1



Figure 21.2 integral can be approximated using a seriesof polynomials applied piecewise to the function or dataover segments of constant length.

3 straightlinesegmentsare used



The Trapezoidal Rule

The Trapezoidal rule is the first of the Newton-Cotes closed integration formulas, correspondingto the case where the polynomial is first order :

The area under this first order polynomial is an

estimate of the integral of f(x) between the limitsof a and b :>>>equation 21.3

b

a

b

adx x f dx x f I )()( 1

Derivation of Trapezoidal Rule >> Box 21.1



I width x average height

Try out Problem 21.10 for first 2 data

Try out Example 21.1



The Multiple Application Trapezoidal Rule One way to improve the accuracy of the

trapezoidal rule is to divide the integrationinterval from a to b into n number of segmentsand apply the method to each segment.

The areas of individual segments can then beadded to yield the integral for the entire interval.

General form:>>>equation 21.10

n

n

x

x

x

x

x

x

n

dx x f dx x f dx x f I

xb xan

abh

1

2

1

1

0

)()()(

0



General format for multiple-application integrals

Try out Problem 21.10 a

Try out Example 21.2



Simpsons Rules

More accurate estimate of an integral is obtained if ahigh-order polynomial is used to connect the points.The formulas that result from taking the integrals

under such polynomials are called Simpsons rules .

Simpsons 1/3 Rule

Results when a second-order Lagrange interpolating polynomial is used.

>>>equation 21.15



Simpsons 1/3 rule (parabola)



The Multiple- Application Simpsons 1/3 Rule

Just as the trapezoidal rule, Simpsons rulecan be improved by dividing the integrationinterval into a number of segments of equalwidth.

Yields accurate results and considered superiorto trapezoidal rule for most applications.

Can be employed only if the number ofsegments is even>>>equation 21.18



Simpsons 3/8 Rule Results when a third-order Lagrange

interpolating polynomial is used.>>>equation 21.20>>>n? data points?

Estimate five segments?? Use trapezoidal rule (large truncation error)

Alternative : apply Simpsons 1/3 rule to the

first 2 segments & Simpsons 3/8 rule to thelast three.

Try out Problem 21.10 & Problem 21.11



Simpsons 3/8 rule (cubic)



Integration of Equations Functions to be integrated

numerically are in two forms: A table of values . We are limited by the

number of points that are given. A function . We can generate as many

values of f(x) as needed to attain

acceptable accuracy.



Gauss Quadrature Gauss quadrature implements a

strategy of positioning any twopoints on a curve to define a straightline that would balance the positiveand negative errors.

Hence the area evaluated under thisstraight line provides an improvedestimate of the integral.



Two-point formula General form

>>>equation 22.12

Yield to>>>equation 22.17

Equation 22.23 & 22.24 can be substitutedfor x & dx in the equation to be integrated

Substitution transform the integrationinterval

Try out Problem 22.8



Figure 22.7



Higher-point formula General form

>>>equation 22.25

>>>refer Table 22.1




Numerical Differentiation

Notion of numerical differentiationhas been introduced in Chapter 1. Inthis chapter more accurate formulasthat retain more terms will bedeveloped.


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High Accuracy DifferentiationFormulas

High-accuracy divided-differenceformulas can be generated by includingadditional terms from the Taylor series

expansion.>>>Figure 23.1 to 23.3



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Richardson Extrapolation There are two ways to improve derivative

estimates when employing finite divideddifferences: Decrease the step size, or Use a higher-order formula that employs more

points.

A third approach, based on Richardsonextrapolation, uses two derivativeestimates to compute a third, moreaccurate approximation.>>>equation 23.8



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Derivatives of Unequally Spaced Data

Data from experiments or field studiesare often collected at unequal intervals.One way to handle such data is to fit a

second-order Lagrange interpolatingpolynomial.>>>equation 23.9


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