Calculator inputs
Select a preset or enter custom equations in the form f(x, y) = 0. Use explicit multiplication such as 2*x.
Example data table
| Example | Equation 1 | Equation 2 | Initial guess | Approximate root |
|---|---|---|---|---|
| Circle and line | x² + y² − 5 = 0 | x − y − 1 = 0 | (1.6, 0.6) | (2, 1) |
| Exponential and line | exp(x) + y − 3 = 0 | x + y − 2 = 0 | (0.3, 1.5) | (0, 2) |
| Sine and parabola | sin(x) + y − 1 = 0 | x² + y − 1 = 0 | (0.2, 0.9) | (0, 1) |
Formula used
Vector form: F(X) = [f₁(x, y), f₂(x, y)]ᵀ and X = [x, y]ᵀ.
Newton update: Xₖ₊₁ = Xₖ − λ J(Xₖ)⁻¹ F(Xₖ), where λ is the damping factor.
Jacobian matrix: J(X) = [[∂f₁/∂x, ∂f₁/∂y], [∂f₂/∂x, ∂f₂/∂y]].
Finite-difference approximation: ∂f/∂x ≈ (f(x + h, y) − f(x − h, y)) / (2h), and similarly for y.
Residual norm: ||F|| = √(f₁² + f₂²). Convergence is accepted when the residual or step size becomes sufficiently small.
How to use this calculator
- Choose a preset or type two custom equations written as expressions equal to zero.
- Enter starting guesses for x and y near the expected solution.
- Set tolerance, maximum iterations, finite-difference step, and damping factor.
- Click Solve system to place the results directly above the form.
- Review convergence status, residual norm, the iteration table, and the Plotly graph.
- Use the export buttons to save the iteration log as CSV or PDF.
Frequently asked questions
1. What kind of systems can this page solve?
It solves two nonlinear equations with two unknowns. Equations may include powers, trigonometric functions, exponentials, logarithms, and mixed terms, provided they return finite values near the chosen guess.
2. Which numerical method does it use?
The page uses a damped Newton-Raphson method for systems. It builds a numerical Jacobian with central finite differences, solves the 2×2 linear update, and repeats until convergence or termination.
3. Why do initial guesses matter so much?
Newton-style solvers are local methods. A poor starting point may converge to another root, stall near a singular Jacobian, or diverge entirely. Start near a realistic solution whenever possible.
4. What does the damping factor change?
The damping factor scales the Newton step. A value of 1 applies the full update. Smaller values often improve stability when the raw step is too aggressive or the system is highly nonlinear.
5. What does a singular Jacobian mean?
A singular or nearly singular Jacobian means the local linear model cannot be inverted reliably. This often happens near flat regions, overlapping constraints, or unsuitable guesses.
6. How should I choose the finite-difference step?
Use a small positive value such as 0.0001. Too large reduces derivative accuracy, while too small may amplify rounding errors. Moderate values usually balance both effects well.
7. Can I enter equations with implicit multiplication?
Use explicit multiplication for reliable evaluation. Enter 2*x instead of 2x, and write x*y instead of juxtaposed variables.
8. What should I do if the solver does not converge?
Try better starting guesses, reduce the damping factor, increase the iteration limit modestly, or reformulate the equations. Also inspect the residual graph and Jacobian determinant trend for trouble spots.