MatLab using Fixed Point method to find a root - matlab

Question

I wanna find a root for the following function with an error less than 0.05%
f= 3*x*tan(x)=1
In the MatLab i've wrote that code to do so:
clc,close all
syms x;
x0 = 3.5
f= 3*x*tan(x)-1;
df = diff(f,x);
while (1)
x1 = 1 / 3*tan(x0)
%DIRV.. z= tan(x0)^2/3 + 1/3
er = (abs((x1 - x0)/x1))*100
if ( er <= 0.05)
break;
end
x0 = x1;
pause(1)
end
But It keeps running an infinite loop with error 200.00 I dunno why.

Answer 1

Don't use while true, as that's usually uncalled for and prone to getting stuck in infinite loops, like here. Simply set a limit on the while instead:
while er > 0.05
%//your code
end
Additionally, to prevent getting stuck in an infinite loop you can use an iteration counter and set a maximum number of iterations:
ItCount = 0;
MaxIt = 1e5; %// maximum 10,000 iterations
while er > 0.05 & ItCount<MaxIt
%//your code
ItCount=ItCount+1;
end

Answer 2

I see four points of discussion that I'll address separately:
Why does the error seemingly saturate at 200.0 and the loop continue infinitely?
The fixed-point iterator, as written in your code, is finding the root of f(x) = x - tan(x)/3; in other words, find a value of x at which the graphs of x and tan(x)/3 cross. The only point where this is true is 0. And, if you look at the value of the iterants, the value of x1 is approaching 0. Good.
The bad news is that you are also dividing by that value converging toward 0. While the value of x1 remains finite, in a floating point arithmetic sense, the division works but may become inaccurate, and er actually goes NaN after enough iterations because x1 underflowed below the smallest denormalized number in the IEEE-754 standard.
Why is er 200 before then? It is approximately 200 because the value of x1 is approximately 1/3 of the value of x0 since tan(x)/3 locally behaves as x/3 a la its Taylor Expansion about 0. And abs(1 - 3)*100 == 200.
Divisions-by-zero and relative orders-of-magnitude are why it is sometimes best to look at the absolute and relative error measures for both the values of the independent variable and function value. If need be, even putting an extremely (relatively) small finite, constant value in the denominator of the relative calculation isn't entirely a bad thing in my mind (I remember seeing it in some numerical recipe books), but that's just a band-aid for robustness's sake that typically hides a more serious error.
This convergence is far different compared to the Newton-Raphson iterations because it has absolutely no knowledge of slope and the fixed-point iteration will converge to wherever the fixed-point is (forgive the minor tautology), assuming it does converge. Unfortunately, if I remember correctly, fixed-point convergence is only guaranteed if the function is continuous in some measure, and tan(x) is not; therefore, convergence is not guaranteed since those pesky poles get in the way.
The function it appears you want to find the root of is f(x) = 3*x*tan(x)-1. A fixed-point iterator of that function would be x = 1/(3*tan(x)) or x = 1/3*cot(x), which is looking for the intersection of 3*tan(x) and 1/x. However, due to point number (2), those iterators still behave badly since they are discontinuous.
A slightly different iterator x = atan(1/(3*x)) should behave a lot better since small values of x will produce a finite value because atan(x) is continuous along the whole real line. The only drawback is that the domain of x is limited to the interval (-pi/2,pi/2), but if it converges, I think the restriction is worth it.
Lastly, for any similar future coding endeavors, I do highly recommend #Adriaan's advice. If would like a sort of compromise between the styles, most of my iterative functions are written with a semantic variable notDone like this:
iter = 0;
iterMax = 1E4;
tol = 0.05;
notDone = 0.05 < er & iter < iterMax;
while notDone
%//your code
iter = iter + 1;
notDone = 0.05 < er & iter < iterMax;
end
You can add flags and all that jazz, but that format is what I frequently use.

Answer 3

I believe that the code below achieves what you are after using Newton's method for the convergence. Please leave a comment if I have missed something.
% find x: 3*x*tan(x) = 1
f = #(x) 3*x*tan(x)-1;
dfdx = #(x) 3*tan(x)+3*x*sec(x)^2;
tolerance = 0.05; % your value?
perturbation = 1e-2;
converged = 1;
x = 3.5;
f_x = f(x);
% Use Newton s method to find the root
count = 0;
err = 10*tolerance; % something bigger than tolerance to start
while (err >= tolerance)
count = count + 1;
if (count > 1e3)
converged = 0;
disp('Did not converge.');
break;
end
x0 = x;
dfdx_x = dfdx(x);
if (dfdx_x ~= 0)
% Avoid division by zero
f_x = f(x);
x = x - f_x/dfdx_x;
else
% Perturb x and go back to top of while loop
x = x + perturbation;
continue;
end
err = (abs((x - x0)/x))*100;
end
if (converged)
disp(['Converged to ' num2str(x,'%10.8e') ' in ' num2str(count) ...
' iterations.']);
end