Question

Please use matlab for the following problem

I. For the function. /(x) -X-4X+2XY+2Y+2Y+14 a. Plot the surface function for X over [5 6]. and Y over [-4, -2]. ecolour plot for X over [O 101 and Yover [43] and values for the contours of V [1 1.25 1.5 22.5 3] c. Write an m-file to find the minimum of the function using the gradient descent method Use a starting value of [4, -4].

Answer 1

a.Matlab Code

[X,Y] = meshgrid(5:0.1:6,-4:.1:-2);
Z = X.^2-4*X+2*X.*Y+2*Y.^2+2*Y+14;
surf(X,Y,Z)

Output

6 5 3 654321-2

b. Matlab Code

x = linspace(0,10);
y = linspace(-4,-2);
[X,Y] = meshgrid(x,y);
Z = X.^2-4*X+2*X.*Y+2*Y.^2+2*Y+14;
v = [1 1.25 1.5 2 2.5 3];

Output

2.2 -2.4 -2.6 -2.8 -3.2 -3.4 -3.6 -3.8 0 1 234 5 6 78 9 10

(c)Gradient Descent Method

Matlab Code

function file steep.m code

function [f]=steep(x)
f=(x(1))^2-4*x(1)+2*x(1)*x(2)+2*(x(2))^2+2*(x(2))+14;
end

Steepest descent main m file

clc
clear
n=input('input n');

y=input('y');
tic

B=zeros(n,n);
x=sym('x', [n 1],'real');
f=steep(x);
J=jacobian(f,x);
fun_ev=1;
NORM=1;
k=0;
%H=eye(n);
time=0;
while(1~=2)
L1=vpa(subs(J,x,y),7);
NORM=vpa(norm(L1),7);
if(NORM<10^-6)
display('flag NORM');
break;
end
if(time>3000)
display('flag TIME');
break;
end
% p=-H*vpa((subs(J,x,y))',7);
p=-vpa((subs(J,x,y))',7);
alpha=1;
beta=0.5; %sigma in given problem
c1=0.1;%s in given problem
%c2=0.9;
fun_ev=fun_ev+1;
while(vpa(subs(f,x,y+alpha*p),7)> vpa(subs(f,x,y),7)+c1*alpha*p'*vpa(subs(J,x,y)',7))
alpha=beta*alpha;
end
z=vpa(y+alpha*p,7);
fun_ev=fun_ev+1;

k=k+1;

y=z;
time=toc;

end
k
toc
NORM
y
f1=steep(y)
fun_ev

Output

input n2
y[4;-4]
flag NORM

k =

38

Elapsed time is 16.544480 seconds.

NORM =

0.0000009536743


y =

4.999999 (Minimizer)
-3.0


f1 =

1.000000000000454747350886464119 (Minimum Value)

fun_ev =

77