Calculator Input
Define two quadratic-style nonlinear equations in x and y, then choose a starting guess.
Example Data Table
This example solves a circle and line intersection using Newton iterations.
| Example | Equation 1 | Equation 2 | x0 | y0 | Tolerance | Max Iterations | Approximate Root |
|---|---|---|---|---|---|---|---|
| Circle and line | x² + y² - 4 = 0 | x - y - 1 = 0 | 1.5 | 0.5 | 0.000001 | 25 | (1.822876, 0.822876) |
| Shifted intersection | x² + xy + y² - 3 = 0 | x + y - 1 = 0 | 1 | 0 | 0.000001 | 30 | Depends on starting guess |
Formula Used
This solver handles a 2×2 nonlinear system:
f₁(x, y) = 0 and
f₂(x, y) = 0.
Newton’s method builds a linear correction at each step by using the Jacobian matrix:
[ ∂f₂/∂x ∂f₂/∂y ]
The update solves:
J(xₖ, yₖ) [Δx, Δy]ᵀ = -F(xₖ, yₖ)
Then the next estimate becomes:
xₖ₊₁ = xₖ + Δx and
yₖ₊₁ = yₖ + Δy.
For the quadratic-style equations used here, the partial derivatives are:
∂f/∂x = 2ax + by + d
∂f/∂y = bx + 2cy + e
The solver stops when both the correction size and residual norm are within the selected tolerance.
How to Use This Calculator
- Enter coefficients for the first nonlinear equation.
- Enter coefficients for the second nonlinear equation.
- Provide starting guesses for x and y.
- Set a tolerance and maximum iteration count.
- Click Solve System to run Newton iterations.
- Review the approximate root, residual norm, and table.
- Use the graph to inspect convergence behavior.
- Export the detailed report using CSV or PDF buttons.
Frequently Asked Questions
1) What does this solver compute?
It estimates a solution pair for two nonlinear equations in two variables. The page iteratively updates x and y until the corrections and residuals become very small.
2) Why do starting values matter?
Newton’s method is local. Different starting guesses can lead to different roots, slow convergence, or divergence. Good initial values usually improve stability and speed.
3) What happens if the Jacobian determinant is near zero?
A near-zero determinant means the linear correction step becomes unstable or undefined. The solver stops and asks for a different starting guess or a different equation setup.
4) Why does the result include a residual norm?
The residual norm measures how closely the computed point satisfies both equations. Smaller values mean the estimated root fits the system better.
5) Can this page solve any nonlinear system?
This version is designed for quadratic-style two-variable systems built from x², xy, y², x, y, and a constant. It still covers many practical educational examples.
6) What tolerance should I use?
A tolerance like 1e-6 works well for many classroom problems. Tighter tolerances can improve accuracy, but they may require better initial guesses or more iterations.
7) Why might the solver fail to converge?
Convergence can fail because of poor starting values, a singular Jacobian, extreme coefficients, or a system without a nearby real root. Adjusting guesses often helps.
8) What do the CSV and PDF exports include?
They include the equation summary, approximate solution, residual information, and the iteration table. This makes the result easier to archive, share, or review later.