Newton Raphson Solver Inputs
Use explicit multiplication like 2*x. Supported functions include
sin, cos, tan,
exp, log, sqrt,
abs, and constants pi, e.
Example Data Table
| Example Item | Value |
|---|---|
| Function | x^3 - x - 2 |
| Derivative | 3*x^2 - 1 |
| Initial Guess | 1.5 |
| Tolerance | 0.000001 |
| Maximum Iterations | 10 |
| Iteration | xn | f(xn) | f′(xn) | xn+1 | Approx. Error |
|---|---|---|---|---|---|
| 1 | 1.500000 | 0.875000 | 5.750000 | 1.347826 | 0.152174 |
| 2 | 1.347826 | 0.100682 | 4.449905 | 1.325200 | 0.022626 |
| 3 | 1.325200 | 0.002058 | 4.268468 | 1.324718 | 0.000482 |
| 4 | 1.324718 | 0.000001 | 4.264633 | 1.324718 | 0.000000 |
Formula Used
Newton Raphson update:
xn+1 = xn - f(xn) / f′(xn)
Step error: |xn+1 - xn|
Residual: |f(xn+1)|
Numerical derivative option: f′(x) ≈ [f(x+h) - f(x-h)] / (2h)
The solver keeps iterating until the step error or residual falls below the selected tolerance, or until the iteration limit is reached.
How to Use This Calculator
- Enter a nonlinear function using
xas the variable. - Choose an analytical derivative or switch to the numerical derivative option.
- Provide an initial guess close to the expected root.
- Set step tolerance, function tolerance, and the iteration limit.
- Optionally enter chart bounds for a custom plotting range.
- Press Solve Equation to view the result summary, iteration table, exports, and convergence charts.
Frequently Asked Questions
1. What does this solver calculate?
It estimates a root of a nonlinear equation where f(x) equals zero. The solver also shows each iteration, error size, residual, and convergence pattern.
2. Why is the initial guess important?
Newton Raphson is sensitive to the starting value. A guess near the true root usually improves speed and stability, while a poor guess can slow convergence or cause divergence.
3. When should I use the numerical derivative option?
Use it when finding or typing the exact derivative is inconvenient. It is helpful for experimentation, though an analytical derivative is usually faster and more precise.
4. What happens if the derivative becomes zero?
The update formula divides by the derivative. If the derivative is zero or extremely small, the next step becomes unstable, so the solver stops and reports that condition.
5. Why did the solver stop without converging?
It may hit the iteration limit, encounter a tiny derivative, or move into values where the function becomes unstable. A different starting guess often helps.
6. What is the difference between step tolerance and function tolerance?
Step tolerance checks how much the estimate changes between iterations. Function tolerance checks how close the computed point is to making the function value zero.
7. Which math functions are supported?
You can use functions such as sin, cos, tan, exp, log, log10, sqrt, abs, pow, floor, ceil, and round. Use explicit multiplication symbols.
8. Can I export the iteration results?
Yes. After solving, use the CSV and PDF buttons to download the iteration table and summary for records, reports, or later review.