scipy.linalg.

fiedler#

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

Returns a symmetric Fiedler matrix

Given an sequence of numbers a, Fiedler matrices have the structure F[i, j] = np.abs(a[i] - a[j]), and hence zero diagonals and nonnegative entries. A Fiedler matrix has a dominant positive eigenvalue and other eigenvalues are negative. Although not valid generally, for certain inputs, the inverse and the determinant can be derived explicitly as given in [1].

Parameters:

a(…, n,) array_like: Coefficient array. N-dimensional arrays are treated as a batch: each slice along the last axis is a 1-D coefficient array.

Returns:

F(…, n, n) ndarray: Fiedler matrix. For batch input, each slice of shape (n, n) along the last two dimensions of the output corresponds with a slice of shape (n,) along the last dimension of the input.

See also

circulant, toeplitz

Notes

Added in version 1.3.0.

Array API Standard Support

fiedler has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	✅	✅
Dask	✅	n/a

See Support for the array API standard for more information.

References

[1]

J. Todd, “Basic Numerical Mathematics: Vol.2 : Numerical Algebra”, 1977, Birkhauser, DOI:10.1007/978-3-0348-7286-7

Examples

>>> import numpy as np
>>> from scipy.linalg import det, inv, fiedler
>>> a = [1, 4, 12, 45, 77]
>>> n = len(a)
>>> A = fiedler(a)
>>> A
array([[ 0,  3, 11, 44, 76],
       [ 3,  0,  8, 41, 73],
       [11,  8,  0, 33, 65],
       [44, 41, 33,  0, 32],
       [76, 73, 65, 32,  0]])

The explicit formulas for determinant and inverse seem to hold only for monotonically increasing/decreasing arrays. Note the tridiagonal structure and the corners.

>>> Ai = inv(A)
>>> Ai[np.abs(Ai) < 1e-12] = 0.  # cleanup the numerical noise for display
>>> Ai
array([[-0.16008772,  0.16666667,  0.        ,  0.        ,  0.00657895],
       [ 0.16666667, -0.22916667,  0.0625    ,  0.        ,  0.        ],
       [ 0.        ,  0.0625    , -0.07765152,  0.01515152,  0.        ],
       [ 0.        ,  0.        ,  0.01515152, -0.03077652,  0.015625  ],
       [ 0.00657895,  0.        ,  0.        ,  0.015625  , -0.00904605]])
>>> det(A)
15409151.999999998
>>> (-1)**(n-1) * 2**(n-2) * np.diff(a).prod() * (a[-1] - a[0])
15409152