This MATLAB function, `ministep`, computes the minimum step size required...

July 4, 2025 at 01:17 AM

function minimum_step = ministep(f,a,b,h0,ea,maxit) % % inputs: % f = function to be evaluated % % h0 = initial step size % ea = relative error threshhold (default: 0.1%) % maxit = maximum iterations (default: 50) if nargin < 5 || isempty(ea), ea=0.01; end if nargin < 6 || isempty(maxit), maxit=50; end iter = 0; n=(b-a)/h0; v = trap(f,a,b,n); while (1) n_old=n; n=n*2; iter = iter + 1; test = ((trap(f,a,b,n_old)-trap(f,a,b,n))/trap(f,a,b,n_old))*100; if test < ea, break, end end if iter > maxit, end minimum_step = (b-a)/n; end

This MATLAB function, ministep, computes the minimum step size required for numerical integration using the trapezoidal rule (trap appears to refer to a function performing the trapezoidal integration). It starts with an initial step size h0, then iteratively refines the integration by halving the step size until a specified error threshold ea is met or a maximum number of iterations maxit is reached.

Input parameters:

  1. f: The function handle to be integrated.
  2. a, b: The integration bounds (a as the lower limit, b as the upper limit).
  3. h0: The initial step size for the integration.
  4. ea (optional): The relative error threshold for stopping the refinement. Defaults to 0.01% if not provided.
  5. maxit (optional): The maximum number of iterations allowed. Defaults to 50 if not provided.

Process flow:

  1. The code first checks if the optional inputs ea and maxit are missing or empty. If so, default values are assigned.
  2. The number of sub-intervals (n) for the trapezoidal rule is calculated as (b - a) / h0.
  3. The initial value of the integration is computed using the trap function.
  4. A while loop is entered to iteratively refine the step size:
    • In each iteration, the number of sub-intervals n is doubled.
    • The relative error is calculated as the percentage difference between the integration results of the current and previous steps.
    • If the relative error is less than the threshold ea, the loop terminates.
  5. If the iteration count exceeds maxit, the loop ends.
  6. The minimum step size, which corresponds to the final value of (b - a) / n, is returned as the output.

Output:

  • The function returns the minimum_step, which is the smallest step size that satisfies the relative error threshold.

Important Notes:

  • The function assumes the existence of another function trap, which performs trapezoidal integration. The specifics of trap are not provided.
  • If the maximum number of iterations is exceeded, the code simply exits (though this behavior might need refinement since it doesn't raise an explicit error or handle the condition robustly).
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