Korea UniversityComputer Networks Research Lab. 5. Nonlinear Functions of Several Variables Jieun Yu ([email protected])[email protected] Kyungsun

Korea UniversityComputer Networks Research Lab.

5. Nonlinear Functions of Several Variables

Jieun Yu ([email protected]) Kyungsun Kwon ([email protected])

Kyunghwi Kim ([email protected])Sungjin Kim ([email protected])Computer Networks Research Lab.

Dept. of Computer Science and EngineeringKorea University

2

Contents Introduction Newton’s Method for Nonlinear Systems Fixed-Point Iteration for Nonlinear Systems Minimum of a Nonlinear Function of Several Variables MATLAB’s Methods Case Study

3

Introduction

Chapter 2

Chapter 3,4

Chapter 5

Chapter 1

4

Introduction A zero of a nonlinear function of a single variable Nonlinear functions of several variables The latter problem is much more difficult

Newton’s method is applicable to this problem as well as that of a single variable


5.1 Newton’s Method for Nonlinear Systems

6

Newton’s Method Tangent line function

Iteration form

Tangent plane function

Iteration using r, s

)(')()()( 000 xfxxxfxf

)('

)(

0

001 xf

xfxx

),()(

),()(),(),(

),()(

),()(),(),(

000

00000

000

00000

yxgyy

yxgxxyxgyxg

yxfyy

yxfxxyxfyxf

y

x

y

x

syy

rxx

0

0

7

Newton’s Method (cont’) Step1

Step2

),(),(),(

),(),(),(

000000

000000

yxgyxgsyxgr

yxfyxfsyxfr

yx

yx

syy

rxx

01

01

8

5.1.1 Matrix-Vextor Notation Step1

Step2

),(

),(

),(),(

),(),(

00

00

0000

0000

yxg

yxf

s

r

yxgyxg

yxfyxf

yx

yx

)(),()( oldnewoldold xxywherexFyxJ

s

r

y

x

y

x

0

0

1

1

yxx oldnew

9

Ex5.1 Intersection of a Circle and a Parabola

yxyxg

yxyxf

2

22

),(

1),( ( Circle )

( Parabola )

10


Step1– Partial derivative

– Initial estimate (x0,y0) = (1/2, 1/2)

),(

),(

12

22

00

00

yxg

yxf

s

r

x

yx

4/1

2/1

11

11

s

r

8/1,8/3 sr

11


Step2– The second approximate solution (x1,y1)

8/5

8/7

8/1

8/3

2/1

2/1

0

0

1

1

s

r

y

x

y

x

12


iteration

x y | |

0 0.5 0.5

1 0.875 0.625 0.39528

2 0.79067 0.61806

0.084611

The true solution is (x, y) = (0.78615, 0.61803)

• Initial estimate and 2 iterations of Newton’s method

13

5.1.2 MATLAB Function for Newton’s Method for Nonlinear

Systems

'__

)(\)(

)()(

yoldxnewx

xFxJFy

xFyxJF

\ operation -> 100p

14

Example 5.2 a System of Three Equations

04),,(

,04/1),,(

,01),,(

322

213213

23

213212

23

22

213211

xxxxxxf

xxxxxf

xxxxxxf

322

21

23

21

23

22

21

4

4/1

1

)(

xxx

xx

xxx

xF

422

202

222

)(

21

31

321

xx

xx

xxx

xJ

15

Example 5.2 a System of Three Equations

16

Example 5.3Intersection of a Circle and an

Ellipse

02/3

0343

22

22

yx

yx

yx

yxyxJ

22

86),(

x

y

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

]5.05.0[0 x

17

Example 5.3Intersection of a Circle and an

Ellipse

18

Example 5.4Positioning a Robot Arm

• the end of the first link(x1,y1)

• the end of the second link(x2,y2)

We need to solve

)sin(

),cos(

112

212

dyy

dxx

0 2 4 6 80

2

4

6

8

10

10

α

β

(x1,y1)

(x2,y2)

(10,4)

)sin()sin(

),cos()cos(

212

211

ddp

ddp

)sin(),cos( 1111 dydx

19

Example 5.4Positioning a Robot Arm

Find the angles so that the arm will move to the point(10,4)

– The length of link (d1,d2) = (5, 6)

– initial angles(α , β) = (0.7,0.7)

– The system of equations

04)sin(6)sin(6

,010)cos(6)cos(5


5.2 Fixed-Point Iteration For Nonlinear Systems

21


Ex 5.5 Fixed-Point Iteration for a System of two Nonlinear Function– Consider the problem of finding a zero of the system

0110),(

,0510),(

2321212

2131211

xxxxxf

xxxxxf

– Coverting ),,( 2111 xxgx ),( 2122 xxgx form

1.01.01.0

,51.01.03212

2311

xxx

xxx

– The graph of the equations and01 f 02 f

22


1x

2x

0.2 0.4 0.6 0.8 100

0.2

0.4

0.6

101 f

02 f

0.8

Iteration0 0.60000 0.60000

1 0.53840 0.18160 0.42291

2 0.50255 0.15444 0.044975

3 0.50275 0.15062 0.0038204

4 0.50235 0.15062 0.00039661

5 0.50238 0.15058 4.9381e-05

1x 2x x

23


24

Using MATLAB


% Actually using function : Fixed_pt_sys >> x=Fixed_pt_sys(@ex5_5, [0.6 0.6], 0.00001, 5)

function G = ex5_5(x)

G = [ (-0.1*x(1)^3 + 0.1*x(2) + 0.5)

(0.1*x(1) + 0.1*x(2)^3 + 0.1)];

25

Result


26


Ex 5.5 Fixed-Point Iteration for a System of Three Nonlinear Function

07540),,(

,05020),,(

,020050),,(

322

213213

232

213212

23

221

213211

xxxxxxf

xxxxxxf

xxxxxxxf

– Coverting this system to a fixed point iteration form

7540

,5020

,20050

22

213

23

212

23

22

211

xxx

xxx

xxxx

)(xgx

27


875.1025.0025.0),,(

,5.205.005.0),,(

,402.002.002.0),,(

22

2132133

23

2132122

23

22

2132111

xxxxxgx

xxxxxgx

xxxxxxgx

– Using the preceding MATLAB function and a starting estimate of (2, 2, 2)

Iteration0 2.0000 2.0000 2.00001 3.7600 2.1000 -1.6750 4.07592 3.5729 1.6528 -1.4113 0.55183 3.6502 1.7621 -1.4876 0.154034 3.6272 1.7232 -1.4643 0.0509035 3.6346 1.7350 -1.4719 0.01158996 3.6323 1.7312 -1.4695 0.00506967 3.6330 1.7324 -1.4702 0.00160318 3.6328 1.7320 -1.4700 0.000508669 3.6328 1.7321 -1.4701 0.0001611710 3.6328 1.7321 -1.4701 5.1098e-05

1x 2x x3x

28

5.2 Fixed-Point Iteration For Nonlinear Systems Using MATLAB

function G = ex5_6(x)G = [ (-0.02*x(1)^2 - 0.02*x(2)^2 - 0.02*x(3)^2 + 4) (-0.05*x(1)^2 - 0.05*x(3)^2 + 2.5) (0.025*x(1)^2 + 0.025*x(2)^2 - 1.875)];

% Actually using function : Fixed_pt_sys >> x=Fixed_pt_sys(@ex5_6, [2 2 2], 0.00001, 10)

29


Result

30


If g(x) maps D into D, then g has a fixed point in D. In other words, if g(x) is in D whenever x is in D, then there is some point p in D such that p=g(p)

If sequence of approximations to the fixed point1)(

pG

),( )()1( kk xgx

intial point )0(x is sufficiently close to the fixed point p

If there is a constant K<1 such that for every x in D

n

K

x

xg

i

i

)( For each i= 1,….,n and each j=1,…,n,

)0()1()(

1xx

K

Kpx

mm

A bound on the error at the mth step is given by

31


Example 5.5

),(1.01.01.0

),,(51.01.0

2123212

2112311

xxgxxx

xxgxxx

First check to make sure that g(x) maps the retangle 10,10 21 xx

That is, for 1,0 21 xx we have 1,0 21 gg

In fact 6.04.0 1 g and 3.01.0 2 g

22

2

2

1

2

2

121

1

1 3.0)(

,1.0)(

,1.0)(

,3.0)(

xx

xg

x

xg

x

xgx

x

xg

Which are all less than 0.5 (for ), as is required for the corollary1,0 21 xx


5.3 Minimum of a Nonlinear Function of Several

Variables

33

Example 5.7 and , we define

The minimum value of h(x,y) is 0, which occurs when f(x,y)=0 and g(x,y)=0.

The derivatives for the gradient are

1),( 22 yxyxf yxyxg 2),(

22222 )()1(),( yxyxyxh

).1)((22)1(2),(

,2)(22)1(2),(222

222

yxyyxyxh

xyxxyxyxh

y

x

34

Table 5.8 Result of ffmin

Table 5.8 Approximate zeros at each step of iteration

Step x y Change

0 0.5 0.5

1 0.875 0.625 -0.26831

2 0.74512 0.61133 -0.035987

3 0.79251 0.61902 -0.0079939

4 0.78446 0.6178 -0.00019414

5 0.78657 0.6181 -1.3631e-05

35

5.3.1 MATLAB Function for Minimization by Gradient

Descent

36

5.3.1 MATLAB Function for Minimization by Gradient

Descent

function dh = ex_min_g(x)dh = [ -(4*(x(1)^2 + x(2)^2 - 1)*x(1) + 4*(x(1)^2 - x(2))*x(1)) -(4*(x(1)^2 + x(2)^2 - 1)*x(2) - 2*(x(1)^2 - x(2)))]';

function h = ex_min(x)h = (x(1)^2 + x(2)^2 - 1)^2 + (x(1)^2 - x(2) )^2;

xmin = ffmin(@ex_min, @ex_min_g, [0.5 0.5], 0, 5) 0 0.5000 0.5000

1.0000 0.8750 0.6250 -0.2683

2.0000 0.7451 0.6113 -0.0360

3.0000 0.7925 0.6190 -0.0080

4.0000 0.7845 0.6178 -0.0002

5.0000 0.7866 0.6181 -0.0000

37

Example 5.8 Given three points in the plane, we wish to find the locati

on of the point P=(x,y) sp that the sum of the squares of the distances from P to the three given points, is as small as possible.

),(),,(),,( 332211 yxandyxyx

.2226)(2)(2)(2),(

,2226)(2)(2)(2),(

.)()()()()()(),(

321321

321321

23

23

22

22

21

21

yyyyyyyyyyyxf

xxxxxxxxxxyxf

yyxxyyxxyyxxyxf

y

x

38

Figure 5.7


5.4 MATLAB’s Methods

40

MATLAB’s METHODS FMINS

– finds the minimum of a scalar function of several variables, starting at an initial estimate

– Note• The fmins function was replaced by fminsearch in Release 11 (MAT

LAB 5.3).• In Release 12 (MATLAB 6.0), fmins displays a warning message an

d calls fminsearch – Syntax

• x = fmins('fun',x0) • x = fminsearch(fun,x0)

– starts at the point x0 and finds a local minimum x of the function described in fun.

– x0 can be a scalar, vector, or matrix.

41

MATLAB’s METHODS Examples

– A classic test example for multidimensional minimization is the Rosenbrock banana function

– The traditional starting point is (-1.2,1). The M-file banana.m defines the function.

– The minimum is at (1,1) and has the value 0.

a


5.5 Nonlinear system: Case Study

43

Nonlinear system: Case Study

The analytical model to compute the 802.11 DCF throughput

44


The stationary probability τ – The station transmits a packet in a generic slot time

The conditional collision probability p

aa

45


Two Equations represent a nonlinear system in the two unknowns τ and p

W = Cwmin

m = Maximum backoff stageN = The number of stations

46


Solve the nonlinear problem of two variables– Assumptions

• W = 32• m = 3 • N = 3, 10, 50

– Newton’s Method and fminsearch(fun,x0)

Problem1 by Newton’s Method (W=32, m = 3, N =3)

64τ p^4 + 34pτ - 33τ - 4p +2 =0 (1)

p = 1-(1- τ )^2 (2)

*Equations (1) and (2) is nonlinear system

47


• function f = newton_case1(x)f = [ (64*x(2)*x(1)^4 + 34*x(1)*x(2) -33*x(2) -4*x(1) +2) (x(2)^2 - 2*x(2) + x(1)) ];

• function df = newton_case1_j(x)df = [ (256*x(2)*x(1)^3 + 34*x(2) - 4) (64*x(1)^4 + 34*x(1)-33) 1 ( 2*x(2) -2) ];

τ = 0.4165 p = 0.6595

aa

48


Problem1 by fminsearch(fun,x0) (W=32, m = 3, N =3)

– function h = fminsearch_case1(x)h = (64*x(2)*x(1)^4 + 34*x(1)*x(2) -33*x(2) -4*x(1) +2)^2 + ((1-x(2))^2 -1 +x(1))^2;

49


Problem2 by Newton’s Method (W=32, m = 3, N =10)64τ p^4 + 34pτ - 33τ - 4p +2 =0 (1)

p = 1-(1- τ )^9 (2) *Equations (1) and (2) is nonlinear system

– τ = 0.1259– p = 0.7021

– dd

50




51


Problem3 by Newton’s Method (W=32, m = 3, N =50)64τ p^4 + 34pτ - 33τ - 4p +2 =0 (1)

p = 1-(1- τ )^49 (2) *Equations (1) and (2) is nonlinear system

– τ = 0.0427– p = 0.8823

– dd

52




Documents

Korea UniversityComputer Networks Research Lab. 5. Nonlinear Functions of Several Variables Jieun Yu ([email protected])[email protected] Kyungsun