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Chapter 5Numerical Integration

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A Review of the Definite Integral

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Riemann Sum A summation of the

form

is called a Riemann

sum.

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5.2 Improving the Trapezoid Rule The trapezoid rule for computing integrals:

The error:

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5.2 Improving the Trapezoid Rule

So that

Therefore,

Error estimation:

Improvement of the approximation:

the corrected trapezoid rule

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Example 5.1

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Example 5.2

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h4

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Approximate Corrected Trapezoid Rule

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5.3 Simpson’s Rule and Degree of Precision

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let

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Example 5.3

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The Composite RuleAssume

Example 5.4

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Example 5.5

h4

It’s OK!!

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Discussion From our experiments:

From the definition of Simpson’s rule:

Why? Why Simpson’s rule is “more accurate than it ought to be”?

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Example 5.6

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5.4 The Midpoint Rule Consider the integral:

And the Taylor approximation:

The midpoint rule:

Its composite rule:

because

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Example 5.7

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5.5 Application: Stirling’s Formula Stirling’s formula is an interest

ing and useful way to approximate the factorial function, n!, for large values of n.

Use Stirling’s formula to show that

for all x.

0!

lim

n

xn

n

Example

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5.6 Gaussian Quadrature Gaussian quadrature is a very powerful tool for approximati

ng integrals. The quadrature rules are all base on special values of weigh

ts and abscissas (called Gauss points) The quadrature rule is written in the form

n

i

ni

nin dxxfxfwfG

1

1

1

)()( )()()(

weights Gauss points

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Example 5.8

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Question

k

×

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Discussion The high accuracy of Gaussian quadrature then comes from the f

act that it integrates very-high-degree polynomials exactly.

We should choose N=2n-1, because a polynomial of degree 2n-1 has 2n coefficients, and thus the number of unknowns (n weights plus n abscissas) equals the number of equations.

Taking N=2n will yield a contradiction.

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Only to find an example

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Finding Gauss Points

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Theorem 5.3

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Theorem 5.4

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Other Intervals, Other Rules

)1)((2

1 zabax

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Example 5.9

Table 5.5

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Error estimation!! See exercise!!

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Example 5.10

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5.7 Extrapolation Methods One of the most important ideas in

computational mathematics is— We can take the information from a few

approximations and Use that to both estimate the error in the

approximation and generate a significantly improved approximation

In this section we will embark on a more detailed study of some of these ideas.

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Approximation

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Estimating p

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Example 5.11

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Example 5.12

)(",)(',0 when4

1)(",

2

1)(' ,)( 2/32/12/1

xfxfx

xxfxxfxxf

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Error Estimation and an Improved Approximation

I2n-In

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Example 5.13

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Example 5.14

O(h2): Error of trapezoid ruleO(h4): Error of Richardson extrapolation