Computed Numerical Solution
The best converged estimate appears below the header and above the form, followed by method comparisons, iteration tables, and graphs.
Best Method
Best Root Estimate
Best Residual |f(x)|
Method Comparison Summary
| Method | Status | Root Estimate | f(x) | Iterations | Approx Error | Notes |
|---|
Function Plot and Estimated Roots
The curve shows the input function over the chosen graph range. Marker points indicate converged root estimates from successful methods.
Convergence by Iteration
This graph tracks the estimate sequence for each applicable method, helping you compare speed, stability, and direction of convergence.
Iteration Details
Numerical Solution Finder Inputs
Use explicit multiplication like 2*x. Supported functions include sin, cos, tan, exp, sqrt, log, ln, abs, sec, csc, and cot.
Example Data Table
This example uses a standard nonlinear equation to compare different root-finding methods under the same tolerance and iteration settings.
| Example Equation | Bracket [a, b] | x₀ | x₁ | g(x) | λ | Tolerance | Expected Root |
|---|---|---|---|---|---|---|---|
x^3 - x - 2 = 0 |
[1, 2] | 1.5 | 2.0 | blank | 0.1 | 0.000001 | ≈ 1.52138 |
cos(x) - x = 0 |
[0, 1] | 0.5 | 1.0 | cos(x) |
blank | 0.000001 | ≈ 0.73909 |
Formula Used
Bisection Method
For a valid bracket where f(a)f(b) < 0, compute the midpoint:
x_n = (a + b) / 2
Choose the next interval by checking the sign of f(a)f(x_n).
False Position Method
Use a secant line on bracket endpoints:
x_n = (a f(b) - b f(a)) / (f(b) - f(a))
The sign test updates the bracket while preserving enclosure.
Newton-Raphson Method
Iterate from one guess using the derivative:
x_(n+1) = x_n - f(x_n) / f'(x_n)
This calculator estimates f'(x) numerically with a centered finite difference.
Secant Method
Use two prior guesses without an explicit derivative:
x_(n+1) = x_n - f(x_n)(x_n - x_(n-1)) / (f(x_n) - f(x_(n-1)))
It often converges faster than bracketing when guesses are good.
Fixed-Point Iteration
Rewrite the equation as x = g(x) and iterate:
x_(n+1) = g(x_n)
If g(x) is blank, this calculator uses g(x) = x - λf(x).
Stopping Conditions
The calculator stops when either condition becomes true:
|f(x_n)| ≤ tolerance
|x_n - x_(n-1)| ≤ tolerance
For bracketing methods, interval width is also checked.
How to Use This Calculator
- Enter the equation as
f(x), not the full equality. For example, typex^3 - x - 2. - Choose one method or run all applicable methods to compare speed and stability.
- Provide a bracket
[a, b]for bracketing methods, and guessesx₀orx₁for open methods. - Set tolerance, maximum iterations, decimal places, and graph range.
- Click Find Numerical Solution to display the result above the form, then review summaries, tables, and graphs.
- Use the export buttons to save comparison results in CSV or PDF format.
FAQs
1) What does this calculator solve?
It solves nonlinear equations of one variable by finding numerical approximations to roots where f(x) = 0. It compares several iterative methods using the same inputs.
2) When should I use bisection?
Use bisection when you know a valid interval containing a sign change. It is slower than open methods, but it is usually very stable and predictable.
3) Why can Newton-Raphson fail?
Newton-Raphson may fail when the derivative is near zero, the starting guess is poor, or the function changes sharply. In those cases, bracketing methods may be safer.
4) What is a good tolerance value?
A tolerance of 1e-6 is a practical default for many engineering and math tasks. Use a smaller value when you need more precision and a larger value for faster estimates.
5) Why is a bracket required for some methods?
Bisection and false position require an interval where the function changes sign. This confirms that at least one root lies inside the interval for continuous functions.
6) What should I enter for fixed-point iteration?
You can supply a custom g(x) expression if you already rearranged the equation. Otherwise, leave it blank and use the relaxation value λ to generate x - λf(x).
7) What do the graphs show?
The first graph plots the function and marks converged roots. The second graph shows how each method’s estimate changes by iteration, helping you compare convergence behavior.
8) Can this tool solve every equation?
No numerical method works for every equation and every starting value. Success depends on function behavior, valid inputs, and choosing a method suitable for the problem.