2 - California State Polytechnic University, Pomona



EGR 511 NUMERICAL METHODS _______________________

LAST NAME, FIRST

Problem set #2

1. Use Newton’s method with x(0) = 0 to compute x(2) for each of the following nonlinear systems:

a. 4[pic]- 20x1 + [pic][pic]+ 8 = 0 b. sin(4(x1x2) – 2x2 – x1 = 0

[pic]x1[pic] + 2x1 – 5x2 + 8 = 0 [pic]([pic]- e) + 4e[pic] - 2ex1 = 0

>> s2p1a

x = 0.4958936 1.9834235

>> s2p1b

x = -0.5131616 -0.0183762

2. Use the method of Steepest Descent with x(0) = 0 to approximate the solutions for

4[pic]- 20x1 + [pic][pic]+ 8 = 0

[pic]x1[pic] + 2x1 – 5x2 + 8 = 0

Iterating until g = [pic]< 0.05

% Steepest Descent Method

% Set 2, problem 2

%

f1='4*x(1)^2-20*x(1)+.25*x(2)^2+8';

f2='.5*x(1)*x(2)^2+2*x(1)-5*x(2)+8';

% Initial guess

%

x=[0 0];

f=[eval(f1) eval(f2)];

g1=f*f';

for i=1:50

Jt=[8*x(1)-20 .5*x(2)

.5*x(2)^2+2 x(1)*x(2)-5];

% Gradient vector

%

delg=2*Jt*f';

zo=sqrt(delg'*delg);

% Unit vector in the steepest descent

%

z=delg/zo;

xs=x;a3=1;x=x-a3*z';

f=[eval(f1) eval(f2)];

g3=f*f';

while g3>g1

a3=a3/2;

x=xs-a3*z';

f=[eval(f1) eval(f2)];

g3=f*f';

end

a2=a3/2;x=xs-a2*z';

fsave3=f;

f=[eval(f1) eval(f2)];

g2=f*f';h1=(g2-g1)/a2;h2=(g3-g2)/(a3-a2);h3=(h2-h1)/a3;

% Choose a0 so that g will be minimum in z direction

%

a0=.5*(a2-h1/h3);

x=xs-a0*z';

f=[eval(f1) eval(f2)];

g0=f*f';fsave0=f;

if g0 s2p2b

x = -0.357926 0.000000 , g = 0.139390

x = -0.361009 0.057884 , g = 0.003316

3. Find the directional derivative of f(x, y) = 2x2 + y2 at x = 2 and y = 2 in the direction of h = 3i + 2j.

Ans: 8.875

4. Find the gradient vector and Hessian matrix for each of the following functions:

a) f(x, y) = 2xy2 + 3exy b) f(x, y, z) = x2 + y2 + 2z2

c) f(x, y) = ln(x2 + 2xy + 3y2)

Ans:

(a) (f = [pic], H = [pic]

(b) (f = [pic], H = [pic]

(c) (f = [pic][pic], H = [pic][pic]

5. a) Find the first two iterations of the Gauss-Seidel method for the following linear system using x(0) = 0;

i) 3x1 - x2 + x3 = 1, ii) 10x1 - x2 = 9,

3x1 + 6x2 + 2x3 = 0, - x1 + 10x2 - 2x3 = 7,

3x1 + 3x2 + 7x3 = 4. - 2x2 + 10x3 = 6.

b) Repeat Exercise (a) using the SOR method with relaxation factor = 1.1.

Gauss Seidel 0.1111111 -0.2222222 0.6190476

Gauss Seidel 0.9790000 0.9495000 0.7899000

SOR method 0.0541008 -0.2115435 0.6477159

SOR method 0.9876790 0.9784934 0.7899328

6. Solve the system

x2 + y2 + z2 = 9,

xyz = 1,

x + y - z2 = 0

by Newton's method to obtain the solution near (2.5, 0.2, 1.6).

% Newton Method

%

f1='x(1)^2+x(2)^2+x(3)^2-9';

f2='x(1)*x(2)*x(3)-1';

f3='x(1)+x(2)-x(3)^2';

% Initial guess

%

x=[2.5 0.2 1.6];

for i=1:20

f=[eval(f1) eval(f2) eval(f3)];

Ja=[2*x(1) 2*x(2) 2*x(3)

x(2)*x(3) x(1)*x(3) x(1)*x(2)

1 1 -2*x(3)];

%

dx=Ja\f';

x=x-dx';

if max(abs(dx)) ................
................

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