The forward difference formula can be expressed as :
f'(x0) = (1/h)[f(x0+h)-f(x0)]-(h/2)f''(x0)-(h^2/6)f'''(x0)+O(h^3).
Use extrapolation to dervive an O(h^3) formula for f'(x0)
To derive an O(h^3) formula for f'(x0) using extrapolation, we can use Richardson's extrapolation method. This method allows us to estimate the value of a derivative with improved accuracy by combining multiple approximations at different step sizes.
Step 1: Start with the forward difference formula:
f'(x0) = (1/h)[f(x0+h) - f(x0)] - (h/2)f''(x0) - (h^2/6)f'''(x0) + O(h^3)
Step 2: Replace h with a smaller step size, such as h/2:
f'(x0) = (1/(h/2))[f(x0+(h/2)) - f(x0)] - ((h/2)/2)f''(x0) - ((h/2)^2/6)f'''(x0) + O((h/2)^3)
Step 3: Apply the forward difference formula again with the step size h/2:
f'(x0) = (1/(h/2))[f(x0 + (h/2) + (h/2)) - f(x0 + (h/2))] - ((h/2)/2)f''(x0 + (h/2)) - ((h/2)^2/6)f'''(x0 + (h/2)) + O((h/2)^3)
Step 4: Simplify the expression by expanding the terms:
f'(x0) = (2/h)[f(x0 + h) - f(x0 + (h/2))] - (h/4)f''(x0 + (h/2)) - (h^2/24)f'''(x0 + (h/2)) + O(h^3)
Step 5: Combine the common terms on the right-hand side:
f'(x0) = (4/h)[f(x0 + h) - f(x0 + (h/2))] - (3h/4)f''(x0 + (h/2)) - (h^2/8)f'''(x0 + (h/2)) + O(h^3)
This is the O(h^3) formula for f'(x0) obtained using extrapolation. It provides a more accurate estimation of the derivative compared to the original forward difference formula.