showing how numerical solution coming close to the real selution

Question

I want to plot the graph of the numerical solution of bisection method and show how it get close to the real solution . my code:

% f=input('please enter the function:');
%  xL=input('please enter the limits .from:');
%  XR=input('to:');
% eps=input('please enter epsilon:');
f=@(x)x.^2-1;
XR=2;
xL=-2;
XL=xL ;
eps=0.001;
subplot(2,1,1);
title('graph 1');
    ezplot(f);
    hold on ;
    % line([0 0],40);
    % line(-40,[0 0]);
    plot(XR,f(XR),'r*');
    plot(xL,f(xL),'r*');
    disp( 'the answers is : ');
    for df=xL:0.15:XR
        if f(xL)*f(df)<= 0
        xR=df;
        Roots = BisectionM(f,xR,xL,eps);
        plot(Roots,f(Roots),'gs');
        disp(Roots);
        xL=df;
        xR=XR;
        end
    end
subplot(2,1,2);
title('graph 2');
x0=fzero(f,xL);
sol = BisectionM(f,xR,xL,eps);
plot(1:1:100,ones(1,100)*x0);
hold on ;
plot(1:1:100,sol);

the function :

function[sol,Roots] = BisectionM(f,xR,xL,eps)
      while abs(xR - xL) > eps
             xM = (xR + xL) / 2;
             if (f(xL))*(f(xM)) > 0
                 xL = xM;
                 sol=xM;
                 plot(xL,f(xL),'.');
             else
                 xR = xM;
                 plot(xR,f(xR),'.');
                  sol=xM;
             end
             Roots = xM;

        end
    end

I don't know how to plot this so it will get closer to the solution (the blue line at the end). anyone?

Answer 1

I do not understand many things in your code, e.g. why BisectionM has two identical outputs under different variable names (sol, Roots), moreover only one output is used throughout the main function. Besides here is my guess what you might want:

The figure shows how the numerical solution converges with respect to iteration number. For this you have to store the iteration results in a vector (sol), please see below the modified code:

main.m

f=@(x)x.^2-1;
XR=2;
xL=-2;
XL=xL ;
eps=0.001;
subplot(2,1,1);
title('graph 1');
ezplot(f);
hold on ;
% line([0 0],40);
% line(-40,[0 0]);
plot(XR,f(XR),'r*');
plot(xL,f(xL),'r*');
disp( 'the answers is : ');
for df=xL:0.15:XR
    if f(xL)*f(df)<= 0
        xR=df;
        Roots = BisectionM(f,xR,xL,eps);
        plot(Roots(end),f(Roots(end)),'gs');
        disp(Roots(end));
        xL=df;
        xR=XR;
    end
end
subplot(2,1,2);
title('graph 2');
% give the wide interval again because it was changed previously
XR=2;
xL=-2;
x0=fzero(f,xL);
Roots = BisectionM(f,xR,xL,eps);
plot(1:length(Roots),ones(length(Roots),1)*x0, 'g');
hold on ;
plot(1:length(Roots),Roots,'Marker', '.');
xlabel('iteration number')

BisectionM.m

function Roots = BisectionM(f,xR,xL,eps)

% no preallocation because of while
ii = 1;
while abs(xR - xL) > eps
    xM = (xR + xL) / 2;
    if (f(xL))*(f(xM)) > 0
        xL = xM;
        sol=xM;
        plot(xL,f(xL),'.');
    else
        xR = xM;
        plot(xR,f(xR),'.');
        sol=xM;
    end
    Roots(ii) = xM;
    ii = ii + 1;
end
end