Advanced Complex Polynomial Roots Calculator

Calculator Inputs

Coefficients

Enter coefficients from highest degree to constant. Separate values with commas or new lines. Examples: 1, -3, 2 or 2+i, -4, 1-i.

Variable Symbol

Use a single letter such as z, x, or w.

Decimal Places

Controls displayed precision for roots and checks.

Maximum Iterations

Higher values may help difficult polynomials converge.

Tolerance

Typical values are 1e-8, 1e-10, or 1e-12.

Initial Radius

Leave blank for an automatic starting radius.

Plot X Minimum

Plot X Maximum

Plot Y Minimum

Plot Y Maximum

Normalize by leading coefficient

This often improves stability for numerical solving.

Example Data Table

Example	Input Coefficients	Polynomial	Expected Roots
Cubic unity roots	1, 0, 0, -1	z³ - 1	1, -0.5 + 0.866025i, -0.5 - 0.866025i
Quadratic with imaginary roots	1, 0, 4	z² + 4	2i, -2i
Mixed complex coefficients	1+i, -3, 2-i	(1+i)z² - 3z + (2-i)	Numerically estimated by the solver

Formula Used

This calculator solves P(z) = a_nzⁿ + a_n-1z^n-1 + ... + a₁z + a₀ = 0 using the Durand–Kerner iterative method.

z_k^(new) = z_k - P(z_k) / Π(z_k - z_j), for all j ≠ k

The method starts from equally spaced complex guesses on a circle, then repeatedly updates each estimate until the maximum correction is smaller than the chosen tolerance.

Residual checking is also applied: |P(z)| should be close to zero for each returned root. Smaller residuals usually mean better numerical accuracy.

How to Use This Calculator

Enter coefficients in descending power order.
Use commas or new lines between coefficients.
Complex entries may be written like 2+i, -3i, or 4-2i.
Pick display precision, tolerance, and iteration limit.
Leave the initial radius blank to use an automatic estimate.
Set plot ranges for the complex plane view.
Click Calculate Roots to display the results above the form.
Export the root table through CSV or PDF after solving.

Frequently Asked Questions

1) What does this calculator solve?

It finds numerical roots of polynomial equations, including real and imaginary solutions. It works for degree one and higher, provided you enter at least two coefficients.

2) Can I enter complex coefficients?

Yes. You can type values such as 2+i, -4i, 3-2i, or 5. Coefficients should still be listed from the highest power down to the constant term.

3) Why are some roots complex?

Many polynomials do not factor into real roots only. The complex plane includes imaginary components, so roots can appear with both real and imaginary parts.

4) What does the residual mean?

The residual is |P(z)| after substituting a computed root back into the polynomial. Smaller values mean the root satisfies the equation more closely.

5) Why normalize by the leading coefficient?

Normalization scales the polynomial so the leading coefficient becomes one. This often improves numerical stability and makes automatic starting values more reliable.

6) What if the solver does not converge?

Increase the iteration limit, tighten or relax tolerance carefully, or try a different initial radius. Repeated roots and ill-conditioned polynomials can converge more slowly.

7) Does the plot show the roots only?

Yes. The Plotly graph places each computed root on the Argand plane, where the horizontal axis is the real part and the vertical axis is the imaginary part.

8) Are repeated roots handled exactly?

Repeated or nearly repeated roots may still be found numerically, but they can be more sensitive. Check the residuals and convergence status to judge reliability.