Calculator Inputs
Plotly Graph
The graph shows the function curve, the horizontal axis, and the computed root marker when a solution is available.
Formula Used
Bisection: The midpoint is x = (a + b) / 2. The interval is repeatedly halved where the sign change remains.
False Position: The estimate is x = (a*f(b) - b*f(a)) / (f(b) - f(a)). It uses a secant line inside a valid bracket.
Newton-Raphson: The update rule is x(n+1) = x(n) - f(x(n))/f'(x(n)). It is usually fast near a good starting value.
Secant: The update rule is x(n+1) = x(n) - f(x(n)) * (x(n) - x(n-1)) / (f(x(n)) - f(x(n-1))). It avoids the explicit derivative.
How to Use This Calculator
- Enter the nonlinear function using
xas the variable. - Select a numerical method that matches your available information.
- For interval methods, provide lower and upper bounds with opposite function signs.
- For Newton or Secant, provide one or two starting guesses.
- Set tolerance, iteration limit, decimal places, and graph range.
- Press Solve Equation to display the result above the form.
- Review the iteration table, residual, convergence status, and plotted root.
- Use the CSV and PDF buttons to export the generated report.
Example Data Table
| Example | Function | Method | Starting Data | Approximate Root |
|---|---|---|---|---|
| Polynomial Root | x^3 - x - 2 |
Newton-Raphson | x0 = 1.5 |
1.52137971 |
| Trigonometric Equation | cos(x) - x |
Secant | x0 = 0, x1 = 1 |
0.73908513 |
| Bracketed Root | x^2 - 3 |
Bisection | a = 1, b = 2 |
1.73205081 |
| Exponential Balance | exp(-x) - x |
False Position | a = 0, b = 1 |
0.56714329 |
Frequently Asked Questions
1. What does this nonlinear equation solver calculate?
It estimates a real root of a nonlinear equation by applying one of several numerical methods. It also reports residuals, convergence progress, iteration history, and a plotted function curve.
2. Which method should I choose first?
Choose Bisection or False Position when you know a valid bracket. Choose Newton when you have a strong starting guess and derivative. Choose Secant when you have two good guesses but no derivative.
3. Why do interval methods need opposite signs?
A sign change across the interval suggests at least one real root lies inside when the function is continuous. Without that sign change, Bisection and False Position may not work correctly.
4. Can I leave the derivative field blank?
Yes. When using Newton-Raphson, the calculator can estimate the derivative numerically. A correct symbolic derivative is still preferable because it often improves stability and speed.
5. What functions and syntax are supported?
You can use operators like +, -, *, /, and ^, plus functions such as sin, cos, tan, sqrt, log, log10, exp, and abs.
6. Why might convergence fail?
Convergence may fail because of poor initial guesses, missing sign changes, flat derivatives, discontinuities, or too few iterations. Tight tolerances can also slow or prevent practical convergence.
7. What does the residual tell me?
The residual is the function value at the reported root estimate. A smaller residual usually means the estimate is closer to an actual solution of the equation.
8. What do the CSV and PDF downloads include?
The CSV export contains the iteration table. The PDF export summarizes the method, root, residual, convergence status, and includes the full iteration history for reporting or study.