Question

develop a matlab code to plot: fn(n)=x-x^(n+1) for different n's till solution convergence then the matlab code should contain a least square approach to prove the convergence of solution.

Answer 1

%
% Solve f(x) = 0 using bisection method.
%
% The function is defined in func(x);  
% Input
tol = 1.e-10;
a = 1.0;  
b = 2.0;
nmax = 100;
% Initialization  
itcount = 0;
error = 1.0;
% Graph of the function
xval = linspace(a,b,100);
for i=1:100
fval(i) = func(xval(i));
end plot(xval,fval);
grid on;  
hold on;
% iteration begins here
while (itcount <= nmax && error >= tol)
itcount = itcount + 1;
% Generate and save iteratres
x = a + (b-a)/2;
z(itcount) = x;
fa = func(a);  
fb = func(b);
fx = func(x);  
error = abs(fx);
% error = abs(x - xold);
if (error < tol)
x_final = x;
else
if (fa*fx < 0)
% root is between a and x
b = x;
else
% root is between x and b
a = x;
end
end
plot(z(1:itcount),zeros(itcount,1),'r+');
pause(5)  
end
if (itcount < nmax);
val = func(x);
fprintf(1,'Converged solution after %5d iterations',itcount);
fprintf(1,' is %15.7e, %e \n',x_final, val);
else
fprintf(1,'Not converged after %5d iterations',nmax);
end
function val = func(x)
%val = x^3 + 4 * x^2 - 10;
val = x^3 - x - 3;
%val = sin(x);  
end