Lecture 3 Taylor Series Expansion

數值方法2008, Applied Mathematics NDHU

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Inline function Taylor series Matlab codes for Taylor series expansion

Lecture 3 Taylor Series Expansion


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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 expressionApply inline.m to translate a string to an inline

function


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>> g=inline('x^2+1')

g =

Inline function: g(x) = x^2+1

>> g(2)

ans =

5


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>> g=inline('x^2+1')

g =

Inline function: g(x) = x^2+1

>> g(2)

ans =

5


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Function plot

1. Get a string for representing an expression2. Apply inline.m to translate a string to an

inline function3. Apply linspace.m to form a linear partition

to [-5 5]4. Determine function outputs of all elements

in a partition vector5. Plot a figure for given function


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Input a string, fs

f=inline(fs)

x=linspace(-5,5)

y=f(x)plot(x,y)

Use ‘inline’ to generate an inline function

form a uniform partition to [-5 5]


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Demo

% This example illustrates creation% of an inline functionfprintf('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);

Source codes


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cos(x)


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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, '. ')


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A sample from function surface

f(x,y)=x.^2+3*y.^2


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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,'.');


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3D plot

3d plot

t = 0:pi/50:10*pi; plot3(sin(t),cos(t),t);


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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.^2mesh(x,y,Z)


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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 interplightlighting phong


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demo mesh

demo_mesh.m

Key in a 2D function:cos(x+y)


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Taylor series

0,1,kfor exists )(

)(!

)(

)(!2

)(''))((')()(

near Expand

tionRepresenta Polynomial

0

2

cfif

cxk

cf

cxcf

cxcfcfxf

cxf

k

k

k

k


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Taylor’s Theorem

c)(x, )(

b], [a cx,

b][a, intervalat exists

)()!1(

))(()(

!

)(

)(!2

)(''))((')()(

1

1)1(

0

2

x

fif

cxn

xfcx

k

cf

cxcf

cxcfcfxf

n

nn

kn

k

k


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Taylor’s Theorem

Approximate f(x) by first n+1 terms of Taylor series

Truncating error:

kn

k

k

cxk

cf)(

!

)(

0

1)1(

)()!1(

))((

nn

cxn

xf


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Taylor series expansion

)812)((

)(8)(8)(4)(

)24)((

)(4)(2)(2)(2)(

)(2)(

)(

)exp(2

1)(

3

3'''

2

2'''

'

2

xxxf

xfxxxfxxfxf

xxf

xfxxfxxfxfxf

xxfdx

xdfxf

xxf


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c=1,n=3

k

k

k

cxk

cfxf )(

!

)()(

3

0

3'''

2 )(!3

)()(

!2

)(''))((')( cx

cfcx

cfcxcfcf

)exp(2

1)( 2xxf


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Normal function

source codefunction y=f_normal(x) y=1/sqrt(2*pi)*exp(-x.^2);return

x=linspace(-5,5);>> plot(x,f_normal(x))


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Taylor expansion at c=1 with n=0


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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;endplot(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');


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Symbolic differentiation

Use diff.m to find the derivative of a function

diff Input: a string for representing a valid

expressionOutput: the derivative of a given expression


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>> x=sym('x');>> diff(x.^2) ans = 2*x


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>> ss='x.^2';>> inst=['diff(' ss ')'];>> eval(inst) ans = 2*x

Form a string

Form an instruction

Evaluate the Instruction by

calling eval

Derivative


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eval

Execute a string that collects MATLAB instructions

Ex: >> ss='x=linspace(-2*pi,2*pi);plot(x,cos(x))';>> eval(ss)


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diff

source codes f_diff

% input a string to specify a function% find its derivativess=input('function:','s');fx=inline(ss);ss=['diff(' ss ')'];ss1=eval(ss);fx1=inline(ss1)


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Example

>> f_difffunction:cos(x)


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Derivative

% input a string to specify a function% plot its derivativess=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


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2nd derivative

source codes f_diff3.m


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Example

>> f_diff3function of x:x.^3-2*x+1

fx1 =

Inline function: fx1(x) = 3.*x.^2-2

fx2 =

Inline function: fx2(x) = 6.*x


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Example

>> f_diff3function of x:tanh(x)

fx1 =

Inline function: fx1(x) = 1-tanh(x).^2

fx2 =

Inline function: fx2(x) = -2.*tanh(x).*(1-tanh(x).^2)


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Exercise

ex3.pdf

Documents

Lecture 3 Taylor Series Expansion