Calculator Inputs
Enter three approximations from progressively smaller step sizes. Add an exact value if known to compare actual error reduction.
Example Data Table
Use this sample when checking the calculator workflow. It approximates a quantity using three step sizes and order two convergence.
| Level | Step Size | Approximation | Notes |
|---|---|---|---|
| A(h1) | 0.4000 | 1.4833333333 | Coarse estimate |
| A(h2) | 0.2000 | 1.4916666667 | Refined estimate |
| A(h3) | 0.1000 | 1.4958333333 | Finer estimate |
| Reference | 0.0000 | 1.5000000000 | Optional exact value |
Formula Used
Richardson extrapolation removes the dominant truncation error term when a numerical method follows a predictable order of convergence.
Single extrapolation:
R = A(h/r) + (A(h/r) - A(h)) / (rp - 1)
Here, A(h) is the coarse approximation, A(h/r) is the refined approximation, r is the refinement ratio, and p is the method order.
Repeated extrapolation:
Apply the same idea to previously extrapolated values, producing a second-level estimate that often improves accuracy further.
Observed order:
p_obs = ln((A(h1)-A(h2)) / (A(h2)-A(h3))) / ln(r)
This estimate is valid when the refinement ratio is consistent and the error behaves asymptotically.
How to Use This Calculator
- Enter the numerical quantity you are approximating, such as a derivative, integral, or limit.
- Provide three approximations computed with decreasing step sizes.
- Enter each corresponding step size. Smaller values should represent finer grids or smaller increments.
- Set the expected convergence order p of your method.
- Optionally add an exact or benchmark value to compare actual error.
- Press Calculate to display extrapolated estimates above the form.
- Review the graph, estimated changes, and improvement percentages.
- Export your result summary using the CSV or PDF buttons.
Frequently Asked Questions
1. What does Richardson extrapolation do?
It combines approximations from different step sizes to cancel the leading truncation error term. This usually gives a more accurate estimate than the finest raw approximation alone.
2. When should I use this calculator?
Use it when your numerical method has a known error order, and you can compute the same quantity at multiple step sizes. It is common in differentiation, integration, and finite difference work.
3. Why do I need three approximations?
Three approximations allow two first-level extrapolations and one second-level extrapolation. They also help estimate the observed convergence order when the refinement ratio is consistent.
4. What is the meaning of p?
The value p is the expected convergence order. For example, many centered finite difference schemes are second order, so their leading error decreases proportionally to h squared.
5. Do the step sizes need a constant ratio?
A constant ratio is strongly preferred, especially for estimating the observed order. The calculator still computes first-level extrapolations with unequal ratios, but interpretation becomes less clean.
6. What if my extrapolated result looks worse?
That can happen when round-off error dominates, the order p is wrong, or the solution is not yet in the asymptotic convergence range. Check the method assumptions and step-size quality.
7. Is the exact value required?
No. The exact value is optional. Without it, the calculator still estimates changes between raw and extrapolated values, but it cannot report true errors or improvement percentages.
8. Can I use this for integrals and derivatives?
Yes. The method is general. It works for many numerical approximations, including derivatives, definite integrals, limits, and grid-based solutions, provided the error model follows a predictable order.