How to loop intermidpoint statement until the absolute error falls within the tolerance of?

This is my code. I am trying to loop, increasing "nargin" until the error falls within the tolerance of 0.001. But my code is broken.

<!-- start of code-->

function [integral] = intmidpoint2(N)
%INTMIDPOINT numerical integration using the composite midpoint rule
 n=1
 %this is the aim of absolute error required
aim = 1e-4;
%define the integration limits
    a=0;
    b=2;
%define the number of sub-intervals to be used
    if nargin == 1
        n=N;
    else
        n=45;
    end
    h=(b-a)/n;
%approximate integral using the composite mid-point rule
      integral=0.0;
%add the contribution of each sub-interval in turn

      for i=1:n
          x_left = a+(i-1)*h;
          x_right = a+i*h;
          x_half = (x_left+x_right)/2.0;
          f_half = f(x_half);
          integral = integral+h*f_half;
      end
       abs_err=abs(4-integral);
       while aim < abs_err
           goto 2
  if abs_err>aim
  fprintf('Midpoint approximation to integral using %g subintervals is %g.  \n Data outside of tolerance.  Abosolute error is %g, which is greater than 0.001. Increase the number of steps.  \n',n,integral,abs_err);
  else if abs_err<aim
        fprintf('Midpoint approximation to integral using %g subintervals is %g.  \n Data within tolerance.  Absolute error is %g, which is less than 0.001. \n',n,integral,abs_err);
  n=n+1;        
  break
      end
     end
  end
  end

%subfunction defines the integrand
function [f_value] = f(x)
    f_value = (x)^3;

end

<!--End of code-->

Answer 1

[Geoff Hayes]

Tyw7 - I notice that there is the following couple of lines of code

 while aim < abs_err
     goto 2

What is the intent behind this line? There is no goto statement in MATLAB so if you are trying to repeat the block of code that precedes this, you will have to find an alternative.

If you want to change the number of subintervals n until the error falls within the desired tolerance, then you could do something like the following. First, set n as you have done already

%this is the aim of absolute error required
aim = 1e-4;
%define the integration limits
a=0;
b=2;
%define the number of sub-intervals to be used
if nargin == 1
    n=N;
else
    n=45;
end

Now set up the while loop that will continue approximating the definite integral until the condition has been satisfied

abs_error = realmax;

while abs_error > aim 
    h=(b-a)/n;
    integral=0.0;
    for k=1:n
        % etc.
    end
    abs_err=abs(4-integral);
    n = n + 1;
end

So on the first iteration of the while loop, we estimate the integral given the chosen n. We calculate the abs_err and increment n so that on subsequent iterations of the loop (if necessary) we can increase the number of subintervals in our interval.

Note that a problem with the above code is that what happens if we never converge to aim. Then we get stuck in the loop and never exit. When using while loops in this manner, it is good practice to include a maximum allowed iterations count so that eventually the loop will terminate when this count has been exceeded.

[Tyw7]

While I haven't test out this line of the code yet (I'm now in bed), would it print the integral of the chosen step before continuing to find the "ideal" one?

Edit: And abs_error is a valid command in MatLab?

Also, how do you set the max allowed iteration count?

[Tyw7]

Edit:

reverting back to old code. It doesn't seem to print the final statement that says that data is within tolerance.

function [integral] = intmidpoint(N)
clear
clc
%INTMIDPOINT numerical integration using the composite midpoint rule
    clc
%define the integration limits
    a=0;
    b=2;
    %this is the aim of absolute error required
aim = 0.001;

%define the number of sub-intervals to be used
    if nargin == 1
        n=N;
    else
        n=1;
    end
    h=(b-a)/n;
%approximate integral using the composite mid-point rule
      integral=0.0;
%add the contribution of each sub-interval in turn

abs_err = realmax;
while abs_err>aim;
        h=(b-a)/n;
    integral=0.0;
     for i=1:n;
        x_left = a+(i-1)*h;
        x_right = a+i*h;
        x_half = (x_left+x_right)/2.0;
        f_half = f(x_half);
        integral = integral+h*f_half;
        abs_err=abs(4-integral);

       end;
       fprintf('2 Midpoint approximation to integral using %g subintervals is %g.  \n Data outside of tolerance.  Abosolute error is %g, which is greater than 0.001. Increase the number of steps.  \n',n,integral,abs_err);
       n = n + 1; 
      if abs_err>aim;
          continue;
     fprintf('test Midpoint approximation to integral using %g subintervals is %g.  \n Data within tolerance.  Absolute error is %g, which is less than 0.001. \n',n,integral,abs_err);        
      end;

end;
end

%subfunction defines the integrand
function [f_value] = f(x)
    f_value = (x)^3;

end

[Geoff Hayes]

To put a maximum number of iterations for your while loop to execute, you could do something like

 maxIterations = 100;
 atIter        = 0;

 while abs_err>aim && atIter <= maxIterations
     atIter = atIter + 1;
     % etc.
 end

I'm not clear on what your question as to whether abs_error is a valid MATLAB command. In the above code, abs_error is a local variable.