數值方法2008, Applied Mathematics NDHU 1

Numerical Integration

數值方法2008, Applied Mathematics NDHU 2

Four 2-variate Gaussians

數值方法2008, Applied Mathematics NDHU 3

Four 2-variate Gaussians

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Gaussian pdf

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Weight sum of Gaussian pdfs

k

kkk pq ),|()( yAxx

數值方法2008, Applied Mathematics NDHU 6

myfx4.m

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Plot 4Gplot_4G.m

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Integration of 4Gdemo_int_4G.m

Double integration

數值方法2008, Applied Mathematics NDHU 9

Numeric calculus for double integrals

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D

yx dxdyedsyxf1

1

1

1

)( ),(22 2)1(erf

Plot of domain D(x, y)

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Example

Approx. integral 2.230985141404140

True Integral 2.230985141404130

demo_ex2_4G.m

2.230985172585607 by dblquard.m

數值方法2008, Applied Mathematics NDHU 11

exp(cos(x))plot_expcos.m

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Definite Integrationdemo_quad.m

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Symbolic integration

demo_int.m

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Example

-5 -4 -3 -2 -1 0 1 2 3 4 5-10

0

10

20

30

40

50

function of x:x.^2+2*x-5

fx1 =

Inline function: fx1(x) = 1./3.*x.^3+x.^2-5.*x

數值方法2008, Applied Mathematics NDHU 15

Numerical integration - quadrature

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Counter example

>> demo_intfunction of x:(1-sin(x.^2)).^(1/3)Warning: Explicit integral could not be found.

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-5 -4 -3 -2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

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plot_sin13.m

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Numerical integration

demo_quad2.m

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Mesh

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Lower and upper sum

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Lower sum : lower bound

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Upper sum : upper bound

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Composite Trapezoid rule

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Uniform mesh

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Error Analysis

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Partition size

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Composite Trapezoid rule

input s, a and b f = inline(s); Set n h = (b-a)/n; ans = 1/2*(f(a)+f(b)) for i=1:n-1 add f(a+i*h) to ans ans= ans*h

數值方法2008, Applied Mathematics NDHU 29

f = inline(s); Set nh = (b-a)/n; ans = 1/2*(f(a)+f(b))

Input s,a,b

Flow Chart

for i=1:n-1

add f(a+i*h) to ansans=

ans*h

EXIT

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Simpson rule for numerical integration

b

aabfbfbafafabdxxf

baCf

5)4(

4

)(2880

)()()2

(4)(6

)(

withexists b)(a,in number a then ],,[ If

數值方法2008, Applied Mathematics NDHU 31

Exercise

Draw a flow chart to illustrate integration by the composite Trapezoid rule

Implement the composite Trapezoid rule for numerical integration, including flow chart and Matlab codes

Test your matlab function with the following integration

* Test your matlab function with definite integration of the weight sum of four Gaussian pdfs

* Compare your results with those obtained by using quad.m

f(x)=exp(cos(x))

0)(xf