Numerical Differentiation 2008 Applied Mathematics NDHU 1 Composite

Numerical Differentiation 數值方法 2008, Applied Mathematics NDHU 1

Composite Simpson rule h=(b-a)/2 n Simpson's Rule for Numerical Integration 數值方法 2008, Applied Mathematics NDHU 2

A procedure for CSR ► ► ► function Q=CSR(fs, a, b, n) fx=inline(fs); h=(b-a)/(2*n); ind=0: 1: 2*n; Q=fx(a)+fx(b); % Red ind_odd=1: 2: (2*n-1); % white ind_even=2: 2: 2*(n-1); Q=Q+sum(4*fx(a+ind_odd*h)); Q=Q+sum(2*fx(a+ind_even*h)); Q=Q/3*h; 數值方法 2008, Applied Mathematics NDHU 3

Symbolic Differentiation demo_diff. m 數值方法 2008, Applied Mathematics NDHU 4

Example function of x: tanh(x) fx 1 = Inline function: fx 1(x) = 1 -tanh(x). ^2 數值方法 2008, Applied Mathematics NDHU 5

Numerical differentiation 數值方法 2008, Applied Mathematics NDHU 6

Truncation error The truncation error linearly depends on h 數值方法 2008, Applied Mathematics NDHU 7

Better approximation 數值方法 2008, Applied Mathematics NDHU 8

Truncation error O(h 2) : big order of h square The truncation error linearly depends on h 2 數值方法 2008, Applied Mathematics NDHU 9

Richardson extrapolation is with O(h 3) Start at the formula that is with O(h 2) Strategy: elimination of h 2 term 數值方法 2008, Applied Mathematics NDHU 10

Halving step-size 數值方法 2008, Applied Mathematics NDHU 11

Richardson extrapolation 數值方法 2008, Applied Mathematics NDHU 12

Richardson extrapolation 數值方法 2008, Applied Mathematics NDHU 13

Demo_Richardson demo_Richardson. m 數值方法 2008, Applied Mathematics NDHU 14

Example >>demo_Richardson function of x: sin(x) fx 1 = Inline function: fx 1(x) = cos(x) h: 0. 01 數值方法 2008, Applied Mathematics NDHU 15

Example: differentiation of sin [-5 5] h=0. 01 數值方法 2008, Applied Mathematics NDHU 16

Exercise Due to 12/26 ► Implement the Richardson extrapolation method for numerical differentiation ► Give two examples to verify your matlab functions for Richardson extrapolation implementation 數值方法 2008, Applied Mathematics NDHU 17

Line fitting 數值方法 2008, Applied Mathematics NDHU 18

Problem statement ► S={(ui vi)}i ► Find a line to fit a set of 2 D points, 數值方法 2008, Applied Mathematics NDHU 19

Paired data m=100; u=rand(1, m); v=1. 5*u+2+rand(1, m)*0. 1 -0. 05; plot(u, v, '. ') 數值方法 2008, Applied Mathematics NDHU 20

Linear model 數值方法 2008, Applied Mathematics NDHU 21

Over-determined Linear system 數值方法 2008, Applied Mathematics NDHU 22

Form A and b A=[u' ones(100, 1)]; b=v'; 數值方法 2008, Applied Mathematics NDHU 23

Line fitting >> x=inv(A'*A)*A'*b x= 1. 4816 2. 0113 數值方法 2008, Applied Mathematics NDHU 24

Demo_line_fitting demo_line_fitting. m 數值方法 2008, Applied Mathematics NDHU 25

Stand alone executable file mcc -m demo_line_fitting. m 數值方法 2008, Applied Mathematics NDHU 26