Newton Raphon's Method - Help

Hello all

I'm a student and recently acquainted with MatLab, and I have to program Newton's method on it for a work.

The thing is, I'm not sure how to code it in the program.

I have the following function and values: f(x)=e^x-4x^2, with a x0?2 and a max tolerance of 0.5*10^-3. However, I have no limit for max iterations (it says "Do the necessary number of iterations" to achieve that tolerance)

Knowing that Newton's expression is x(k+1)=xk-f(x)/f'(x), I've coded the derivative of that function symbolically (in a function file, not script).

Code:
    function [ y,f,f2 ] = derivf2(x)
    syms t
    f2=exp(t)-4.*t.^2;
    f=@(t)diff(f2);
    y=eval(subs(f,t,x));
  end

My question is. How do I code the actual method on Matlab itself?

I've tried several times, but I ran across some problems. My intention isn't to have the exercise completely solved, but to have a general idea on how to do it to help me solve this and other exercise.

Thanks in advance

EDIT: I'm a second year college student of Chemical Engineering, I've never faced programming in my entire life until now, so please be patient with me :)

Answer 1

[James Tursa]

I don't know why you would use all of the symbolic stuff inside your function (as opposed to just hard coding the derivative), but here is a way to get that part of it to work:

function [ y,f,f2 ] = derivf2(x)
syms t
f2 = exp(t) - 4.*t.^2;
f = diff(f2);
y = subs(f,t,x);
end

[António Aragão]

The derivative I had works, is the code you gave me an easier and simpler method of coding it?

[James Tursa]

Depends on how you are using this. If this is being called inside an iterative loop, why would you go through all of that symbolic stuff on each iteration? But if this is only called once at the start of your code just to get the symbolic expressions for the function and its derivative, which are then used downstream in your iterative code, then maybe this makes sense. Without seeing the rest of your code it is hard to advise.

[António Aragão]

What I am doing is probably easy but since I'm not familiar with programming it's hard for me to explain.

The exercise is to determine if the function converges for x0=2, doing the necessary iterations to satisfy the tolerance of 0.5e-3.

I've only defined the expression and the derivative, to call it on the script for the Newton's method itself.

I'm not sure if I'm making myself clear, and I apologize if not doing so

[James Tursa]

Do you have the iterative stuff coded yet? (i.e., the x(k+1)=xk-f(x)/f'(x) part). Or do you need help with that?

[António Aragão]

Sorry for the noob question, but is it just writing x(k+1)=xk-f(x)/f'(x) and the respective iterations (k)?

[James Tursa]

OK, just piecing parts of your code together (making use of your symbolic stuff), here is a start for you:

syms t
F = exp(t) - 4*t^2
FP = diff(F)
f = str2func(['@(t)' char(F)])
fp = str2func(['@(t)' char(FP)])
x = 2
tolerance = 0.5*10^-3
%
% You need to put your iterative code here for the x = x - f(x)/fp(x) part
%

So, for the code you need to insert above, you will need to create a loop of some kind with an appropriate test condition, and inside that loop will be the "Newton expression".

[António Aragão]

I've coded this so far, not sure if this is correct, but we have tried to get here logically according to the problem. We assumed 10 iterations, but we can't run it correctly yet, and we don't really know why we aren't understanding why

for
  k=0
  xk=2
  xa(k+1)=xk-(f(xk)./y(xk))
  Ea(k)=NaN
  for
      k=1:10
      xk=xa(k-1)
      xa(k+1)=xk-(f(xk)./y(xk))
      Ea(k)=abs(xa(k+1)-xa(k))
  end
  if
      Ea<=tolx
      break
  end
end

Thanks in advance