How to make a composite trapezoidal rule have more than 1 iteration?

Hi I have written the function for composite trapezoidal rule and I'm required to calculate the volume by integrating 1/(1+x^2) from 0 to 1. I've found the volume (0.7845) but how do i show the number of iterations and absolute error only when the accuracy of the estimation has improved by a decimal place. I'm supposed to fprintf like the as shown below but i don't understand how to make the trapezoidal rule have more than one iteration??

Thanks I'm new to matlab

Iter Pts    Vol_est    Error
2     3   0.41333333 0.05545687 (1 DP)
5     6   0.41799014 0.00888872 (2 DP)
15   16   0.41878025 0.00098765 (3 DP)
48   49   0.41886937 0.00009645 (4 DP)

Answer 1

[Walter Roberson]

The trapazoidal rule gets more accurate as you sample more finely. The "Iterations" is the number of pieces the range is subdivided in to. 2 pieces takes 3 points, one at the beginning, one in the middle, one in the end. 3 pieces takes 4 points, one at the beginning, one 1/3 of the way, one 2/3 of the way, one at the end. Notice in the table that Pts is always 1 greater than Iter: that is due to this relationship. "Iter" == "Subdivisions"

[Natalia Wong]

I don't understand how to do it. How do I increase the number of iterations? I can't use any user input

[Walter Roberson]

iter = 0;
error = inf;
while true
  iter = iter + 1;
  numpoint = iter + 1;
  sampling_points = linspace(0, 1, numpoint);
  do your approximation
  calculate error
  if there was enough of an improvement
    display something
  end
  if the approximation is close enough that you are done
    break
  end
end

[Natalia Wong]

Thanks Walter but I only need to display the no. of iterations only if the error has improved by a decimal points. How do I check if my error has improved by a decimal point? Is there a built in function?

[Walter Roberson]

There is no built in function to determine if an error has improved by a decimal point.

I would point out, though, that if abs(current_error) <= abs(recorded_error)/10 then the current error is at least one decimal place better.

If you want to know how many decimal places current_error is, think about log10