Calculator Input
Example Data Table
| Example Polynomial | Coefficient Entry | Degree | Expected Root Pattern | Practical Note |
|---|---|---|---|---|
| z4 + 1 = 0 | 1, 0, 0, 0, 1 | 4 | Four unit-circle roots at quarter offsets | Useful for checking symmetry on the complex plane. |
| z3 - 1 = 0 | 1, 0, 0, -1 | 3 | One real root and a complex-conjugate pair | Good first test for argument sorting and residual checks. |
| z2 + (2+3i)z + (1-i) = 0 | 1, 2+3i, 1-i | 2 | Two fully complex roots | Confirms parsing for mixed real and imaginary coefficients. |
Formula Used
This calculator evaluates a polynomial of the form P(z) = a₀zⁿ + a₁zⁿ⁻¹ + ... + aₙ, where each coefficient may be real or complex.
Why this works: Durand–Kerner updates all root estimates together. It is effective for general polynomials and naturally supports complex roots. Residual magnitudes show how close each computed value is to a true root. Small residuals usually indicate a reliable solution.
How to Use This Calculator
- Enter coefficients from the highest power down to the constant term.
- Use complex notation such as 3+2i, -5i, or 4.
- Choose tolerance and iteration limits for the accuracy you want.
- Adjust the initial radius scale when convergence needs a stronger starting spread.
- Pick a root sorting mode that matches your review style.
- Submit the form to see roots, residuals, derivative magnitudes, and the complex-plane graph.
- Export the current result set as CSV or PDF when needed.
FAQs
1. Does this calculator accept complex coefficients?
Yes. You can enter values like 2+3i, -4i, 7, or 1.5-0.25i. The solver treats every coefficient as a complex number and computes roots accordingly.
2. Which polynomial degrees can it solve?
It can solve any degree you enter, provided the leading coefficient is not zero. Very high degrees may require more iterations and careful tolerance settings for stable convergence.
3. Why are residuals shown for every root?
Residuals measure how close P(z) is to zero after solving. Smaller values usually mean the computed root is numerically trustworthy and the iteration converged well.
4. What does the derivative magnitude tell me?
A very small derivative magnitude near a root may suggest a repeated or clustered root. It is a useful diagnostic when multiple roots are close together.
5. Why might repeated roots be harder to resolve?
Repeated roots reduce numerical separation between estimates. Iterations can slow down, and nearby roots may appear clustered. Increasing iterations often helps, but tiny residuals remain the main reliability check.
6. What does the initial radius scale change?
It changes the starting circle used by the Durand–Kerner method. A larger or smaller spread can improve convergence when a difficult polynomial resists the default starting pattern.
7. Why do root orders change between runs?
Root-finding methods do not guarantee a natural display order. This page lets you sort by real part, imaginary part, modulus, or argument for consistent review.
8. Can I keep a record of solved results?
Yes. Use the CSV button for spreadsheet analysis or the PDF button for a clean report. Both exports reflect the currently displayed solution set.