Nonlinear Solver Visualizer
Visualise not only the answer, but also Newton iterations and Homotopy path tracking for nonlinear systems with up to ten variables.
Problem and method
Use a built-in problem or enter a custom system of 1–10 equations. All calculations run locally in the browser.
Enter equations
Enter each equation as Fᵢ(x) = 0. A right-hand side after "=" is also accepted. Use x1–x10 for variables and write multiplication explicitly with *.
+ - * / ^ ( ) sin cos tan exp log sqrt abs pi e
Initial value / start point
Solutions at λ = 1
| Solution | Path Step | Residual ||F(x)|| | x |
|---|
One-variable Newton method
The horizontal axis is x and the vertical axis is f(x). Cyan is the function, the yellow dashed line is the tangent, pink is the current point, and green is the next candidate obtained from the tangent.
Newton Iteration Path
Variable History
The horizontal axis is the iteration step and the vertical axis is each variable value. Line colours correspond to x1–x10 shown in the chart.
Residual History
The horizontal axis is the iteration step and the vertical axis is the logarithmic residual norm ||F(x)||. Lower values mean the iterate is closer to a solution.
Iteration History
| Step | λ | Arc length s | Residual | Step / Corrector | x |
|---|
What this demo shows
Newton's method solves for a correction using the Jacobian. Homotopy treats z=(x,λ) as one unknown vector and advances arclength s along the solution path rather than forcing λ to increase. A tangent predictor is corrected by solving H(x,λ)=0 together with a perpendicular hyperplane condition, so the path can continue through folds where λ decreases.
J(xₖ) Δx = −F(xₖ)H(x,λ) = (1−λ)G(x) + λF(x)[Hₓ Hλ] t = 0tᵀ(z − z predictor) = 0