How to find the minimum using Newton method (matlab)

I am given this function: f(x, y) = 1(x − 1)2 + 1(10(y − x2))2 + 1y2 and I tried to write the code below to find the minimum but it doesn't seem to work.

syms a;

syms b;

g=gradient(vrosenbrock(a,b));

x=-1;

y=-1;

z=0;

i=1;

H=hessian(vrosenbrock(a,b));

while norm(subs(g,[a,b],[x,y])) > 10e-3

    z=[x,y]-((subs(inv(H),[a,b],[x,y]))*((subs(g,[a,b],[x,y])))).' ;
    x=z(1); y=z(2);
    i=i+1;
end

[x y]

And as function I've saved this already: function z = vrosenbrock(x,y)

FUN = 1;

if FUN == 1

    z = 0.5*(10*(y - x.^2)).^2 + 0.5*(1-x).^2 + 0.5*y.^2;

elseif FUN == 2

    z = (10*(y - x.^2)).^2 + (1-x).^2;

end

end

Answer 1

[Ayush Modi]

Hi Nikoo,

I understand you are trying to implement the Newton's method for finding the minimum of the given function.

Here is the example code to get you started:

syms a b;

% Define the function - taking example function 
vrosenbrock = @(x, y) 0.5*(10*(y - x.^2)).^2 + 0.5*(1-x).^2 + 0.5*y.^2;

% Compute the gradient and Hessian
g = gradient(vrosenbrock(a, b), [a, b]);
H = hessian(vrosenbrock(a, b), [a, b]);

% Initial guess
x = -1;
y = -1;

% Iteration counter
i = 1;

while norm(double(subs(g, [a, b], [x, y]))) > 1e-3
    grad_val = double(subs(g, [a, b], [x, y]));
    hess_val = double(subs(H, [a, b], [x, y]));
    z = [x; y] - hess_val \ grad_val;;
    x = z(1);
    y = z(2);
    i = i + 1;
end

disp(['Minimum found at x = ', num2str(x), ', y = ', num2str(y)]);

Minimum found at x = 0.59108, y = 0.34592

Refer to the following MathWorks documentation for more information on (\) i.e. backslash operator in MATLAB:

https://www.mathworks.com/help/matlab/ref/mldivide.html