Numerical Integration

數值方法 2008, Applied Mathematics NDHU 1

Numerical Integration

數值方法 2008, Applied Mathematics NDHU 2

Four 2-variate GaussiansFour 2-variate Gaussians

數值方法 2008, Applied Mathematics NDHU 3

Four 2-variate GaussiansFour 2-variate Gaussians

數值方法 2008, Applied Mathematics NDHU 4

Gaussian pdfGaussian pdf

數值方法 2008, Applied Mathematics NDHU 5

Weight sum of Gaussian pdfsWeight sum of Gaussian pdfs

數值方法 2008, Applied Mathematics NDHU 6

myfx4.m

數值方法 2008, Applied Mathematics NDHU 7

Plot 4GPlot 4Gplot_4G.m

數值方法 2008, Applied Mathematics NDHU 8

Integration of 4GIntegration of 4Gdemo_int_4G.m

Double integration

數值方法 2008, Applied Mathematics NDHU 9

exp(cos(x))exp(cos(x))plot_expcos.m

數值方法 2008, Applied Mathematics NDHU 10

Definite IntegrationDefinite Integrationdemo_quad.m

數值方法 2008, Applied Mathematics NDHU 11

Symbolic integrationSymbolic integration

demo_int.m

數值方法 2008, Applied Mathematics NDHU 12

ExampleExample

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

fx1 =

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

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

0

10

20

30

40

50

數值方法 2008, Applied Mathematics NDHU 13

Numerical integration - Numerical integration - quadrature quadrature

數值方法 2008, Applied Mathematics NDHU 14

Counter example Counter example

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

數值方法 2008, Applied Mathematics NDHU 15

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

0.2

0.4

0.6

0.8

1

1.2

1.4

數值方法 2008, Applied Mathematics NDHU 16

plot_sin13.m

數值方法 2008, Applied Mathematics NDHU 17

Numerical integrationNumerical integration

demo_quad2.m

數值方法 2008, Applied Mathematics NDHU 18

MeshMesh

數值方法 2008, Applied Mathematics NDHU 19

Lower and upper sumLower and upper sum

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Lower sum : lower boundLower sum : lower bound

數值方法 2008, Applied Mathematics NDHU 21

Upper sum : upper boundUpper sum : upper bound

數值方法 2008, Applied Mathematics NDHU 22

Composite Trapezoid ruleComposite Trapezoid rule

數值方法 2008, Applied Mathematics NDHU 23

Uniform meshUniform mesh

數值方法 2008, Applied Mathematics NDHU 24

Associate errorAssociate error

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Partition sizePartition size

數值方法 2008, Applied Mathematics NDHU 26

Procedure: Composite Trapezoid Procedure: Composite Trapezoid rulerule

1.1. Get ss, a and bGet ss, a and b

2.2. Set fx to inline(ss); Set nSet fx to inline(ss); Set n

3.3. h = (b-a)/n; result = 1/2*(fx(a)h = (b-a)/n; result = 1/2*(fx(a)+fx(b)); for i=1:n+fx(b)); for i=1:n

Add fx(a+i*h) to resultAdd fx(a+i*h) to result

4.4. Return result*hReturn result*h

數值方法 2008, Applied Mathematics NDHU 27

Simpson rule for numerical Simpson rule for numerical integrationintegration

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Exercise Exercise Due to 12/19Due to 12/19

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

► Implement the composite Sympson rule for numerical Implement the composite Sympson rule for numerical integration, including flow chart and Matlab codesintegration, including flow chart and Matlab codes

► Test your matlab function with the following integrationTest your matlab function with the following integration

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

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

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