數值方法 2008, Applied Mathematics NDHU1 An iterative approach for root finding Newton method Lecture 3II Root Finding

數值方法 2008, Applied Mathematics NDHU 1

An iterative approach for root finding Newton method

Lecture 3II Root Finding


Problem statement

• Find x such that f(x)=0 for a given f• f denotes an arbitrary single-variable function

Root finding


1102)(2

xxf x >> x=linspace(-1,3);>> plot(x,2.^x.^2-10*x+1)


Root finding

Input: an expressionPlot given functionFind a root


Tangent line

xxn

y=f(x)

yn=f(xn)

xn+1

)('

)(

)('

)('

1

1

n

nn

n

nnn

nn

nn

xf

xfx

xf

yxx

xx

yxfslope

Taylor series


)('/)(

)('/)()(

0))((')(

)(

))((')()(

at

nnn

nnn

nnn

nnn

n

xfxfxx

xfxfxx

xxxfxf

xfzero toSet

xxxfxfxf

xxExpand


An iterative approach

Start at some guessRefine current guess by

Find a tangent line that passes (xn,f(xn))

Set xn to a point at intersection of the tangent line to horizontal axis


Updating rule

)('

)(1

n

nnn xf

xfxx


Newton method

An Iterative approach

Random guess Refine current guess by an updating rule If a halting condition holds, go to step 2

otherwise exit



n=0 ; Set xn

Determine xn+1

Increase n by one

Halting condition

T

F



Set x randomly

Apply an updating rule

to refine x

~( halting condition)exit


Halting condition

Let x denote the current guessThe absolute value of f(x) is expected as

close as possible to zeroHalting Condition

abs(f(x)) < epslonepslon denotes a predefined small positive

number


Initialization

Random initialization A guess by the user



x = rand*2-1; s=sym(ss); ep=10^-6f=inline(s); s1=diff(s); f1=inline(s1)

x=x-f(x)/f1(x);fprintf('%f\n' ,x)

~( abs(f(x)) < ep)exit

input an expression, ss


Demo_newton

source code demo_newton

fstr=input('input a function:','s');x_ini=input('guess its zero:');x_zero=newton(fstr,x_ini);

newton.m


Main Program


Newton method


>> demo_newtoninput a function:cos(x)guess its zero:2 iter=1 x=1.542342 fx=0.028450 iter=2 x=1.570804 fx=-0.000008 iter=3 x=1.570796 fx=0.000000>>


Example

demo_newtoninput a function:2.^x.^2-10*x+1guess its zero:1.5


Example

>> demo_newtoninput a function:2.^x.^2-10*x+1guess its zero:0.5

Second order expansion


2

22

2

2

!2

)('')(')(

),(')('',!2

)(''2

4 ,0

0)(!2

)(''))((')(

)(

)(!2

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

at

nn

nnn

nnn

nn

nnn

nn

nnn

n

xxf

xfxxfc

xfxfbxf

a

a

acbbxcbxax

xxxf

xxxfxf

xfzero toSet

xxxf

xxxfxfxf

xxExpand

Second order method

Update rule


2

2

1

!2

)('')(')(

),(')('',!2

)(''2

4

nn

nnn

nnn

n

xxf

xfxxfc

xfxfbxf

a

a

acbbx



x = rand*2-1; s=sym(ss); ep=10^-6f=inline(s); s1=diff(s); f1=inline(s1)

s2=diff(s1); f2=inline(s2)

a=f2(x)/2;b=-f2(x)+f1(x);c=f(x)-x*f1(x)+f2(x)*x^2/2;

x=…fprintf('%f\n' ,x)



2

2

1

!2

)('')(')(

),(')('',!2

)(''2

4

nn

nnn

nnn

n

xxf

xfxxfc

xfxfbxf

a

a

acbbx


Unconstrained Optimization

Problem statement

Given a differentiable function, y=g(x), unconstrained optimization aims to find the minimum of g(x)

Let x denote a minimum anddx

xdgxf

)()(

0 that suchx

Then

f(x)find


Optimization by Newton method

Use symbolic differentiation to find the first derivative of a given function

Apply the Newton method to find zeros of the first derivative



x = rand*2-1; s=sym(ss); ep=10^-6g=inline(s); s1=diff(s); f=inline(s1);

s2=diff(s1); f1=inline(s2)

x=x-f(x)/f1(x);fprintf('%f gx=%f\n' ,x, g(x))




First derivative

ss=input('input a function:','s');s=sym(ss);f=inline(s);s1=diff(s);f1=inline(s1);x=linspace(-5,5);plot(x,f(x));hold on;plot(x,f1(x));


>> x=linspace(-2*pi,2*pi);>> plot(x,(x-tanh(2*x+10)).^2)


>> demo_mininput a function:(x-tanh(2*x+10)).^2guess its minimum:4

Local minimum


>> demo_mininput a function:(x-tanh(2*x+10)).^2guess its minimum:-4

Local Minimum


Global minimum


>> demo_mininput a function:0.2*x.^3-3*x+cos(x)guess its minimum:2


Exercise

ex3II.pdf

Documents

數值方法 2008, Applied Mathematics NDHU1 An iterative approach for root finding Newton method Lecture 3II Root Finding