scipy.linalg.

fiedler_companion#

scipy.linalg.fiedler_companion(a)[source]#

Returns a Fiedler companion matrix

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

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.

See also

companion

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.

Added in version 1.3.0.

References

[1]

M. Fiedler, “ A note on companion matrices”, Linear Algebra and its Applications, 2003, DOI:10.1016/S0024-3795(03)00548-2

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
array([[  16.,  -86.,    1.,    0.],
       [   1.,    0.,    0.,    0.],
       [   0.,  176.,    0., -105.],
       [   0.,    1.,    0.,    0.]])
>>> eigvals(fc)
array([7.+0.j, 5.+0.j, 3.+0.j, 1.+0.j])