`scipy.linalg._special_matrices:fiedler_companion`

source: build-install/usr/lib/python3.14/site-packages/scipy/linalg/_special_matrices.py:1024

Signature

def fiedler_companion(a)

Summary

Returns a Fiedler companion matrix.

Extended Summary

Given a polynomial coefficient array a, this function forms a pentadiagonal matrix with a special structure whose eigenvalues coincides with the roots of a.

Array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see linalg_batch for details.

The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see linalg_batch for details. The NEP 5 signature of this function is (i)->(i-1,i-1).

Parameters

a : (..., N) array_like: 1-D array of polynomial coefficients in descending order with a nonzero leading coefficient. For N < 2, an empty array is returned. N-dimensional arrays are treated as a batch: each slice along the last axis is a 1-D array of polynomial coefficients.

Returns

c : (..., N-1, N-1) ndarray: Resulting companion matrix. For batch input, each slice of shape (N-1, N-1) along the last two dimensions of the output corresponds with a slice of shape (N,) along the last dimension of the input.

Notes

Similar to companion, each leading coefficient along the last axis of the input should be nonzero. If the leading coefficient is not 1, other coefficients are rescaled before the array generation. To avoid numerical issues, it is best to provide a monic polynomial.

Examples

import numpy as np
from scipy.linalg import fiedler_companion, eigvals
p = np.poly(np.arange(1, 9, 2))  # [1., -16., 86., -176., 105.]
fc = fiedler_companion(p)
fc
eigvals(fc)

✓

array([7.+0.j, 5.+0.j, 3.+0.j, 1.+0.j])

Aliases

scipy.linalg.fiedler_companion