Question

II. Using Newton’s method, write a MATLAB program to find the fixed point of the following
function:
?(?) = √? + ??
accurate to at least 8 decimal places. (HINT: finding the fixed point of f(x) is the same as finding the zero of g(x) = f(x) − x. ) The output of this program should display in a single table (i) the solution for the fixed point, (ii) the initial guess, (iii) the number of iterations it took to reach an
answer, and (iv) the stopping rule (i.e. the tolerance). Remember that your initial guess should be
contained in the interval [−πe, ∞), as f is defined only on this interval. Finally, you should also
store each successive guess of the final solution in a vector and plot them, give your figure a title
and label the horizontal axis “iterations” and the vertical axis “guesses”.

Answer 1

MATLAB code for the problem is provided below, please comment if any doubts:

MATLAB code:

%set the function here
g=@(x) sqrt(x)+pi()*exp(1);

%set the inital guess
init=1;
x0 = init;
x1 = g(x0);

%set the alue of n
n=8;
%compute the epsilon
epsi= 5*10^-(n+1);

%declare the iteration vector
itr = 0;

%iteration array
iteraArr=[];

%guess array
guessArr=[];

%the while loop
while (abs(x1-x0) > epsi && itr<100)
     %update the itr
    itr = itr + 1;
    x0 = x1;
  
    %find the next guess
    x1 = g(x0);
  
    %set the iteration and guess array values
    iterArr(itr)=itr;
    guessArr(itr)=x1;
end

%display the results as table
h={'solution' 'inital_guess' 'number of iterations' 'Tolerance'};
data=[x1 init itr epsi];
f=figure;
t=uitable(f,'data',data,'columnname',h);

%plot
figure
plot(iterArr, guessArr)
title('Iterations Vs Guess')
xlabel('iterations')
ylabel('guesses')

Output: