Calculator inputs
Choose a model, set initial values, define adaptive controls, and solve the initial value problem with Bulirsch–Stoer extrapolation.
Formula used
This page applies the Bulirsch–Stoer idea to a first-order initial value problem. A long segment
H is first solved using the modified midpoint scheme with increasing substep counts
n = 2, 4, 6, ....
The modified midpoint recursion is:
z₀ = y(x),
z₁ = z₀ + h f(x, z₀),
zₘ₊₁ = zₘ₋₁ + 2h f(x + mh, zₘ),
where h = H / n.
The segment estimate is then refined with Richardson extrapolation:
T(k,j) = T(k,j-1) + (T(k,j-1) - T(k-1,j-1)) / ((nₖ / nₖ₋ⱼ)^(2j) - 1).
A step is accepted when the extrapolated error estimate falls below the scaled tolerance. The next segment size is adjusted from the latest error and the chosen safety factor.
How to use this calculator
- Select a differential equation model that matches your problem.
- Enter the initial point x₀ and starting value y(x₀).
- Set the target endpoint where you want the numerical solution.
- Choose the initial segment size, tolerance, and step-size limits.
- Set extrapolation levels and maximum attempts for convergence control.
- Fill the model parameters used by the selected equation family.
- Press Compute solution to display results above the form.
- Review accepted and rejected steps, the graph, and export files if needed.
Example data table
| Model | Equation | x₀ | y₀ | Target x | Initial Segment | Tolerance | Approximate y(Target) |
|---|---|---|---|---|---|---|---|
| Linear | y' = 1.2y - 0.4 |
0.0 | 1.0 | 2.0 | 0.25 | 1e-8 | 7.682118 |
This sample illustrates a smooth linear test problem where adaptive Bulirsch–Stoer segments converge rapidly.
Frequently asked questions
1. What does the Bulirsch–Stoer method do?
It solves ordinary differential equations by combining modified midpoint steps with extrapolation. The method improves accuracy quickly on smooth problems and can outperform simpler fixed-step schemes.
2. When is this method a good choice?
It is best for smooth, nonstiff initial value problems where high accuracy matters. If the equation is stiff or has sharp discontinuities, implicit methods are usually safer.
3. Why are some steps rejected?
A rejection means the estimated extrapolation error exceeded the allowed tolerance. The solver then reduces the segment size and retries the same region with finer control.
4. What does extrapolation level mean?
Each level adds another modified midpoint estimate with more substeps. Higher levels can increase accuracy, but they also require more function evaluations and more runtime.
5. How should I choose the tolerance?
Start with a moderate target like 1e-6 or 1e-8. Tighten it only when the application truly needs more precision, because overly strict values may force very small segments.
6. What is the role of minimum and maximum segment size?
They limit how far the adaptive controller may shrink or grow each accepted segment. These bounds prevent unstable jumps and protect against excessive refinement.
7. Can this page solve any differential equation string directly?
This version uses several practical model families instead of a full symbolic parser. That keeps the page safer, faster, and easier to validate in a browser-ready workflow.
8. What do the CSV and PDF exports contain?
The exports include the step diagnostics table with status, interval bounds, step size, y estimate, error, and extrapolation levels. The PDF also includes a compact summary.