Newton's method for two variable functions

I have a problem in which I'm supposed to solve a system using Newton's method, but my function gives the same x and y as an output as I give to it as an input. How do I fix this?

function [x, y] = newton3(x,y)

  for N = 1:30

  D = inv([4*x^3-2*y^5 4*y^3-10*x*y^4; 6*x^5+2*x 4*y^3]);

  f = x^4+y^4-2*x*y^5 ;
  g = x^6+x^2+y^4-4 ;
  
 z = [x y]' ;

 z = z - D*[f g]' ;

 x = z(1)
 y = z(2)
end

end

Answer 1

[Alan Stevens]

It depends on your initial guesses. Some work, some don't (not unusual for Newton's method!): Also, better practice to set D to be the Jacobian, rather than its inverse, then use backslash division in the iteration see below:

x = 2; y = 2;
[x,y] = newton3(x,y);

disp([x y])
disp([x^4+y^4-2*x*y^5  x^6+x^2+y^4-4 ])

function [x, y] = newton3(x,y)

  for N = 1:30

  D = [4*x^3-2*y^5 4*y^3-10*x*y^4; 6*x^5+2*x 4*y^3];  %%%%%%%

  f = x^4+y^4-2*x*y^5 ;
  g = x^6+x^2+y^4-4 ;
  
 z = [x y]' ;

 z = z - D\[f g]' ; %%%%%%%%

 x = z(1);
 y = z(2);
end

end

[John D'Errico]

If I could add, it is a really bad idea to hard code functions into your algorithms. So setting the matrix D here inside the loop, inside your function:

D = [4*x^3-2*y^5 4*y^3-10*x*y^4; 6*x^5+2*x 4*y^3];

Instead, learn to pass functions INTO your code. Think of the idea as a target, something you may want to learn for the future.

Yes, I know this is homework. But one day you might find it important. :)

[sanyer]

I tried to do the changes you proposed but the function continues to give the same values of x and y it receives as an input.

x0 = 2
y0 = 3
f = x^4+y^4-2*x*y^5 ;
g = x^6+x^2+y^4-4 ;

function [x1, y1] = newton3(f,g,x0,y0)

  for N = 1:30

  D = [diff(f,x) diff(f,y); diff(g,x) diff(g,y)];
  
 z = [x0 y0]' ;

 z = z - D\[f g]' ;

 x0 = z(1);
 y0 = z(2);
  end

  x1 = x0
  y1 = y0
  

end

[Alan Stevens]

The code doesn't actually call the function newton3!

[Alan Stevens]

Also, to define the functions you need

f = @(x,y) x^4 ...etc.

and you will need to define functions for dfdx, dfdy etc. if you are not using the Symbolic toolbox.

[sanyer]

Thank you! I feel so stupid now haha...