maths.numerical_analysis.weierstrass_method

Functions

weierstrass_method(→ numpy.ndarray)

Approximates all complex roots of a polynomial using the

Module Contents

maths.numerical_analysis.weierstrass_method.weierstrass_method(polynomial: collections.abc.Callable[[numpy.ndarray], numpy.ndarray], degree: int, roots: numpy.ndarray | None = None, max_iter: int = 100) → numpy.ndarray

Approximates all complex roots of a polynomial using the Weierstrass (Durand-Kerner) method. Args:

polynomial: A function that takes a NumPy array of complex numbers and returns

the polynomial values at those points.

degree: Degree of the polynomial (number of roots to find). Must be ≥ 1. roots: Optional initial guess as a NumPy array of complex numbers.

Must have length equal to ‘degree’. If None, perturbed complex roots of unity are used.

max_iter: Number of iterations to perform (default: 100).

Returns:

np.ndarray: Array of approximated complex roots.

Raises:

ValueError: If degree < 1, or if initial roots length doesn’t match the degree.

Note:

• Root updates are clipped to prevent numerical overflow.

Example:

>>> import numpy as np
>>> def check(poly, degree, expected):
...     roots = weierstrass_method(poly, degree)
...     return np.allclose(np.sort(roots), np.sort(expected))

>>> check(
...     lambda x: x**2 - 1,
...     2,
...     np.array([-1, 1]))
True

>>> check(
...     lambda x: x**3 - 4.5*x**2 + 5.75*x - 1.875,
...     3,
...     np.array([1.5, 0.5, 2.5])
... )
True

See Also:

https://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method