Step 1 - Data and Initialization:

You have a numerical method that provides approximations of a value, which may not be very accurate.

Step 2 - Apply the Method with Different Step Sizes:

Use the chosen numerical method to compute the value of interest with multiple step sizes (e.g., h, h/2, h/4).

Step 3 - Extrapolation Formula:

Apply the Richardson extrapolation formula, which combines the results from different step sizes to provide a more accurate estimate. The formula is typically expressed as:

R(h) = 2^n * F(h/2) - F(h) / (2^n - 1)

Where R(h) is the extrapolated result, F(h) is the result obtained with the original step size h, and n is the order of accuracy of the method.

Step 4 - Convergence Check:

You can assess the accuracy and convergence of the Richardson extrapolation by comparing the results obtained with different step sizes.

Step 5 - Output:

The result of the Richardson extrapolation, R(h), is a more accurate approximation of the desired value compared to the original numerical method.

Richardson extrapolation is a powerful technique for improving the accuracy of numerical approximations, particularly when dealing with methods of known order of accuracy.