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
數值方法 2008, Applied Mathematics NDHU 20
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
數值方法 2008, Applied Mathematics NDHU 25
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
數值方法 2008, Applied Mathematics NDHU 28
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))
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