Chapter 11

Numerical Integration Gerald Recktenwald Portland State University Mechanical Engineering Department [email protected]

These slides are a supplement to the book Numerical Methods with Matlab: Implementations and Applications, by Gerald W. Recktenwald, c 2002, Prentice-Hall, Upper Saddle River, NJ. These slides are
c
2002 Gerald W. Recktenwald. The PDF version of these slides may be downloaded or stored or printed only for noncommercial, educational use. The repackaging or sale of these slides in any form, without written consent of the author, is prohibited. The latest version of this PDF file, along with other supplemental material for the book, can be found at www.prenhall.com/recktenwald.

Version 0.01

March 9, 2002

Primary Topics

• Basic concepts • Newton Cotes Rules
Trapezoid rule
Simpson’s rule • Gaussian Quadrature • Adaptive Quadrature • Improper Integrals

NMM: Numerical Integration

page 1

Figure 11.2

a

b

L

NMM: Numerical Integration

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Figure 11.3

f (x )

a

NMM: Numerical Integration

b

page 3

Figure 11.4

Composite Rule: N panels f (x )

1 a

2

...

N b

Basic Rule: m nodes

1

2 ...

NMM: Numerical Integration

m

page 4

Figure 11.5

f(x) f2 f1

P1(x) h = x2 –x1 x1

NMM: Numerical Integration

x2

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Figure 11.6

fn–1

fn–2

fn

f3

f2 f1 h x1 a

NMM: Numerical Integration

h x2

h x3

xn–2

h xn–1

xn b

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Figure 11.7

0.4 3 panels error = -0.208654 0.2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.4 4 panels error = -0.124097 0.2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.4 5 panels error = -0.081554 0.2

0 0

0.5

1

NMM: Numerical Integration

1.5

2

2.5

3

3.5

4

4.5

5

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Figure 11.8

P2(x) f(x)

f3

f2

f1 h x1

NMM: Numerical Integration

h x2

x3

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Figure 11.9

f5

fn–2

f3

f1 h x1

fn–3

fn h

x3

h x5

NMM: Numerical Integration

h

xn–3 xn–2 xn

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Figure 11.10

0.4 3 panels error = -0.007070 0.2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.4 4 panels error = -0.002368 0.2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.4 5 panels error = -0.000997 0.2

0 0

0.5

1

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1.5

2

2.5

3

3.5

4

4.5

5

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Figure 11.11

Three point closed rule

h = b 2– a

f3

f2

h=b–a 4

Three point open rule

f3

f2

f1

f1 h x1 a

h x2

NMM: Numerical Integration

h x3 b

a

h x1

h x2

h x3

b

page 11

Figure 11.12

Order 2 1.0000

1.0000

Order 3 0.5556

0.8889

0.5556

Order 4 0.3479

0.6521

0.6521

0.3479

Order 5 0.2369

0.4786

0.5689

0.4786

0.2369

Order 6 0.1713

-1

0.3608

0.4679

-0.5

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0.4679

0

0.3608

0.5

0.1713

1

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Figure 11.13

f(x)

f(z)

H 2 xi

x*1

H 2 x*2

xm,i

x*3

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1 2 x*4 xi+1

–1

z1

1 2 z2

0

z3

z4

1

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Figure 11.14

0

10

trapezoid simpson GL 4 node

-5

Truncation error

10

-10

10

-15

10

0

10

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1

2

10 10 Number of function evaluations

10

3

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Figure 11.15

H for level 1 level 1 a

d

c

e

left

b

right

level 2 a

d

c left

e

b

a

d

c

e

b

right

level 3 a

d

c

e

b

a

d

c

e

b

level 4 a

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b

a

b

page 15

Figure 11.16

100

y = humps(x)

80 60 40 20 0

Space between f(x) evaluations

-20

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1 x

1.2

1.4

1.6

1.8

2

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

NMM: Numerical Integration

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Figure 11.17

0

10

quad quad8 -2

tol

10

-4

10

-6

10 -10 10 10

-8

10

-6

10

-4

10 Absolute error

-2

10

10

0

5

quad quad8 4

flops

10

10

3

2

10 -10 10

-8

10

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-6

10

-4

10 Absolute error

-2

10

10

0

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