1. Lecture 3 Taylor Series Expansion
Inline function
Taylor series
Matlab codes for Taylor series expansion
數值方法2008, Applied Mathematics NDHU

2. 數值方法2008, Applied Mathematics NDHU
inline function
How to create a function ?
Such as
f(x)=sin(x)*cos(x)
Two steps for creation of an inline function
Get a string for representing a valid expression
Apply inline.m to translate a string to an inline function
數值方法2008, Applied Mathematics NDHU

3. 數值方法2008, Applied Mathematics NDHU
>> g=inline('x^2+1')
g =
Inline function:
g(x) = x^2+1
>> g(2)
ans = 5
數值方法2008, Applied Mathematics NDHU

4. 數值方法2008, Applied Mathematics NDHU
>> g=inline('x^2+1')
g =
Inline function:
g(x) = x^2+1
>> g(2)
ans = 5
數值方法2008, Applied Mathematics NDHU

5. 數值方法2008, Applied Mathematics NDHU
Function plot
Get a string for representing an expression
Apply inline.m to translate a string to an inline function
Apply linspace.m to form a linear partition to [-5 5]
Determine function outputs of all elements in a partition vector
Plot a figure for given function
數值方法2008, Applied Mathematics NDHU

6. 數值方法2008, Applied Mathematics NDHU
Input a string, fs
f=inline(fs)
Use ‘inline’ to generate an inline function
x=linspace(-5,5)
form a uniform partition to [-5 5]
y=f(x)
plot(x,y)
數值方法2008, Applied Mathematics NDHU

7. 數值方法2008, Applied Mathematics NDHU
Demo
Source codes
% This example illustrates creation
% of an inline function
fprintf('input a string to specify a desired fun\n');
fs=input(‘ex. x.^2+2*x+1:','s');
f=inline(fs);
range=5;
x=-range:0.01:range;
y=f(x);
plot(x,y);
數值方法2008, Applied Mathematics NDHU

8. 數值方法2008, Applied Mathematics NDHU
cos(x)
數值方法2008, Applied Mathematics NDHU

9. 數值方法2008, Applied Mathematics NDHU
Input a string, fs
f=inline(fs)
N=800;d=2;
A=rand(2,N);
x=A(1,:); y=A(2,:);
z=f(x1,x2)
plot3(x,y, z, '. ')
數值方法2008, Applied Mathematics NDHU

10. A sample from function surface
f(x,y)=x.^2+3*y.^2
數值方法2008, Applied Mathematics NDHU

11. 數值方法2008, Applied Mathematics NDHU
Demo
Source codes
% Inline function creation
% The created inline function has two input arguments
% Plot a sample from the function surface
fs='x.^2+3*y.^2';
f=inline(fs);
a=rand(2,800)*2-1;
x=a(1,:);y=a(2,:);
z=f(x,y);
plot3(x,y,z,'.');
數值方法2008, Applied Mathematics NDHU

12. 數值方法2008, Applied Mathematics NDHU
3D plot
3d plot
t = 0:pi/50:10*pi;
plot3(sin(t),cos(t),t);
數值方法2008, Applied Mathematics NDHU

13. 數值方法2008, Applied Mathematics NDHU
mesh
x=linspace(0,1);
x=linspace(-1,1);
y=linspace(-1,1)';
X=repmat(x,100,1);
Y=repmat(y,1,100);
Z=X.^2+Y.^2
mesh(x,y,Z)
數值方法2008, Applied Mathematics NDHU

14. 數值方法2008, Applied Mathematics NDHU
surface
x=linspace(-1,1);
y=linspace(-1,1)';
X=repmat(x,100,1);
Y=repmat(y,1,100);
surface(x,y,X.^2+Y.^2)
shading interp
light
lighting phong
數值方法2008, Applied Mathematics NDHU

15. 數值方法2008, Applied Mathematics NDHU
demo mesh
demo_mesh.m
Key in a 2D function:cos(x+y)
數值方法2008, Applied Mathematics NDHU

16. 數值方法2008, Applied Mathematics NDHU
Taylor series
數值方法2008, Applied Mathematics NDHU

17. 數值方法2008, Applied Mathematics NDHU
Taylor’s Theorem
數值方法2008, Applied Mathematics NDHU

18. 數值方法2008, Applied Mathematics NDHU
Taylor’s Theorem
Approximate f(x) by first n+1 terms of Taylor series
Truncating error:
數值方法2008, Applied Mathematics NDHU

19. Taylor series expansion
數值方法2008, Applied Mathematics NDHU

20. 數值方法2008, Applied Mathematics NDHU
c=1,n=3
數值方法2008, Applied Mathematics NDHU

21. 數值方法2008, Applied Mathematics NDHU
Normal function
function y=f_normal(x)
y=1/sqrt(2*pi)*exp(-x.^2);
return
source code
x=linspace(-5,5);
>> plot(x,f_normal(x))
數值方法2008, Applied Mathematics NDHU

22. Taylor expansion at c=1 with n=0
數值方法2008, Applied Mathematics NDHU

23. Taylor expansion at c=1 with n=1
數值方法2008, Applied Mathematics NDHU

24. Taylor expansion at c=1 with n=2
數值方法2008, Applied Mathematics NDHU

25. 數值方法2008, Applied Mathematics NDHU
Taylor expansion at c=1 with n=3
數值方法2008, Applied Mathematics NDHU

26. 數值方法2008, Applied Mathematics NDHU
Taylor expansion
Source code Taylor_nor
% Taylor expansion
% f(x)=1/sqrt(2pi)*exp(-x^2);
x=linspace(0.6,1.4,500);
n=length(x);
c=1;
for i=1:n
y1(i)=f_nm(c);
y2(i)=-2*c*f_nm(c)*(x(i)-c);
y3(i)=(4*c^2-2)*f_nm(c)/2*(x(i)-c)^2;
y4(i)=(12*c-8*c^3)*f_nm(c)/6*(x(i)-c)^3;
end
plot(x,f_nm(x));hold on;plot(x,y1,'r'); title('one term');
figure; plot(x,f_nm(x));hold on;plot(x,y1+y2,'r');title('two terms');
figure; plot(x,f_nm(x));hold on;plot(x,y1+y2+y3,'r');title('three terms');
figure; plot(x,f_nm(x));hold on;plot(x,y1+y2+y3+y4,'r');title('four terms');
數值方法2008, Applied Mathematics NDHU

27. Symbolic differentiation
Use diff.m to find the derivative of a function
diff
Input: a string for representing a valid expression
Output: the derivative of a given expression
數值方法2008, Applied Mathematics NDHU

28. 數值方法2008, Applied Mathematics NDHU
>> x=sym('x');
>> diff(x.^2)
ans =
2*x
數值方法2008, Applied Mathematics NDHU

29. 數值方法2008, Applied Mathematics NDHU
Derivative
Form a string
>> ss='x.^2';
>> inst=['diff(' ss ')'];
>> eval(inst)
ans =
2*x
Form an instruction
Evaluate the
Instruction by
calling eval
數值方法2008, Applied Mathematics NDHU

30. 數值方法2008, Applied Mathematics NDHU
eval
Execute a string that collects MATLAB instructions
Ex:
>> ss='x=linspace(-2*pi,2*pi);plot(x,cos(x))';
>> eval(ss)
數值方法2008, Applied Mathematics NDHU

31. 數值方法2008, Applied Mathematics NDHU
diff
source codes f_diff
% input a string to specify a function
% find its derivative
ss=input('function:','s');
fx=inline(ss);
ss=['diff(' ss ')'];
ss1=eval(ss);
fx1=inline(ss1)
數值方法2008, Applied Mathematics NDHU

32. 數值方法2008, Applied Mathematics NDHU
Example
>> f_diff
function:cos(x)
數值方法2008, Applied Mathematics NDHU

33. 數值方法2008, Applied Mathematics NDHU
Derivative
% input a string to specify a function
% plot its derivative
ss=input('function of x:','s');
fx=inline(ss);
x=sym('x');
ss=['diff(' ss ')'];
ss1=eval(ss);
fx1=inline(ss1)
x=linspace(-1,1);
plot(x,fx(x),'b');hold on;
plot(x,fx1(x),'r');
source code : f_diff2
數值方法2008, Applied Mathematics NDHU

34. 數值方法2008, Applied Mathematics NDHU
2nd derivative
source codes f_diff3.m
數值方法2008, Applied Mathematics NDHU

35. 數值方法2008, Applied Mathematics NDHU
Example
>> f_diff3
function of x:x.^3-2*x+1
fx1 =
Inline function:
fx1(x) = 3.*x.^2-2
fx2 =
fx2(x) = 6.*x
數值方法2008, Applied Mathematics NDHU

36. 數值方法2008, Applied Mathematics NDHU
Example
>> f_diff3
function of x:tanh(x)
fx1 =
Inline function:
fx1(x) = 1-tanh(x).^2
fx2 =
fx2(x) = -2.*tanh(x).*(1-tanh(x).^2)
數值方法2008, Applied Mathematics NDHU

37. 數值方法2008, Applied Mathematics NDHU
Exercise
ex3.pdf
數值方法2008, Applied Mathematics NDHU