Enter two numeric approximations to the same quantity, computed externally using the same second-order method: A(h) at the original step size and A(h/2) after halving that step size. The calculator returns an extrapolated estimate and the signed correction to the finer approximation. Inputs are dimensionless numbers. The method coefficients are fixed at 4 and 3.
If A(h) = L + c₂h² + O(h⁴), halving h divides the leading error term by four. Combining 4A(h/2) − A(h) cancels that term, and dividing by three gives L + O(h⁴). Fourth-order improvement requires this stronger error expansion; second-order accuracy alone does not guarantee it. See Richardson extrapolation.
For the illustrative pair A(h) = 1.04 and A(h/2) = 1.01, the extrapolated estimate is (4 × 1.01 − 1.04) / 3 = 1. The signed correction is (1.01 − 1.04) / 3 = −0.01, so the extrapolated estimate is 0.01 below the finer approximation. This arithmetic does not establish that the true value is 1.
Use the same method, target quantity and settings for both approximations, changing only the step size from h to h/2. For quadrature, keep the integration interval unchanged; for a derivative, keep the evaluation point unchanged. The assumed error expansion must apply at the supplied step sizes.
The signed correction is not a certified error bound, and the extrapolated estimate is not guaranteed to be more accurate. Roundoff, cancellation, noisy inputs and premature rounding can undermine improvement. Preserve available digits when entering the approximations. Equal inputs produce zero correction but do not prove accuracy.
This calculator does not parse expressions, sample functions, determine the method’s order or certify convergence. Results display up to eight decimal places; a displayed zero correction can reflect rounding. Arithmetic outside the supported numeric range produces an unavailable result, not a valid zero.