Calculator Input
Enter the function in terms of x. Use explicit multiplication such as 2*x and powers such as x^3. Supported functions include sin, cos, tan, sqrt, log, exp, abs, floor, and ceil.
Example Data Table
| Function | x₀ | x₁ | x₂ | Tolerance | Max Iterations | Approximate Root | Converged |
|---|---|---|---|---|---|---|---|
| x^3 - x - 1 | 0 | 1 | 2 | 0.000001 | 25 | 1.324718 | Yes |
| x^2 - 2 | 0 | 1 | 2 | 0.000001 | 25 | 1.414214 | Yes |
| cos(x) - x | 0 | 0.5 | 1 | 0.000001 | 25 | 0.739085 | Yes |
Formula Used
Muller’s method fits a quadratic curve through three recent points and then uses a root of that quadratic as the next estimate. The larger-magnitude denominator is chosen to reduce loss of significance.
This calculator can stop by absolute error, relative error, or the magnitude of f(x₃). In real-root mode, it stops if the discriminant becomes negative.
How to Use This Calculator
- Enter a function in terms of x, such as x^3 - x - 1.
- Provide three distinct starting values x₀, x₁, and x₂ near the expected root.
- Set tolerance, function threshold, maximum iterations, and the preferred stopping criterion.
- Keep real-root mode checked if you only want real-valued progress.
- Click Calculate to view the result summary above the form and inspect the full iteration table below.
- Use the CSV or PDF buttons to export the computed summary and iteration history.
Frequently Asked Questions
1. What does Muller’s method do?
It estimates a root of a nonlinear equation by fitting a quadratic through three nearby points and projecting the next approximation from that curve.
2. Why are three starting values required?
Three values define the quadratic interpolation step. That extra curvature information can help Muller’s method move faster than simpler two-point methods.
3. Can this version return complex roots?
This implementation focuses on real-root workflows. When real-root mode is enabled, it stops if the discriminant becomes negative during an iteration.
4. What function formats are accepted?
Use x, numbers, parentheses, and standard functions like sin, cos, sqrt, log, exp, and abs. Write multiplication explicitly as 2*x.
5. Which stopping criterion should I choose?
Absolute error works well when root scale is known. Relative error suits varying magnitudes. Function value is useful when residual size matters most.
6. Why is the larger denominator selected?
Choosing the denominator with larger magnitude reduces cancellation and usually gives a more stable numerical update for the next root estimate.
7. Why might the method fail to converge?
Poor starting values, repeated guesses, undefined function evaluations, flat local behavior, or complex intermediate steps can prevent reliable convergence.
8. Is Muller’s method only for polynomials?
No. It is often used for general nonlinear functions, provided the expression can be evaluated and the chosen starting values are reasonable.