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
數值方法2008, Applied Mathematics NDHU 4
Gaussian pdf
數值方法2008, Applied Mathematics NDHU 5
Weight sum of Gaussian pdfs
k
kkk pq ),|()( yAxx
數值方法2008, Applied Mathematics NDHU 6
myfx4.m
數值方法2008, Applied Mathematics NDHU 7
Plot 4Gplot_4G.m
數值方法2008, Applied Mathematics NDHU 8
Integration of 4Gdemo_int_4G.m
Double integration
數值方法2008, Applied Mathematics NDHU 9
Numeric calculus for double integrals
數值方法2008, Applied Mathematics NDHU 10
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
數值方法2008, Applied Mathematics NDHU 12
Definite Integrationdemo_quad.m
數值方法2008, Applied Mathematics NDHU 13
Symbolic integration
demo_int.m
數值方法2008, Applied Mathematics NDHU 14
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
數值方法2008, Applied Mathematics NDHU 16
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 17
-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 18
plot_sin13.m
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Numerical integration
demo_quad2.m
數值方法2008, Applied Mathematics NDHU 20
Mesh
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Lower and upper sum
數值方法2008, Applied Mathematics NDHU 22
Lower sum : lower bound
數值方法2008, Applied Mathematics NDHU 23
Upper sum : upper bound
數值方法2008, Applied Mathematics NDHU 24
Composite Trapezoid rule
數值方法2008, Applied Mathematics NDHU 25
Uniform mesh
數值方法2008, Applied Mathematics NDHU 26
Error Analysis
數值方法2008, Applied Mathematics NDHU 27
Partition size
數值方法2008, Applied Mathematics NDHU 28
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
數值方法2008, Applied Mathematics NDHU 30
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