scipy.linalg.

solve_toeplitz#

scipy.linalg.solve_toeplitz(c_or_cr, b, check_finite=True)[source]#

Solve the equation T @ x = b for x, where T is a Toeplitz matrix defined by c_or_cr.

The Toeplitz matrix has constant diagonals, with c as its first column and r as its first row. If r is not given, r == conjugate(c) is assumed.

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 Batched Linear Operations for details.

Parameters:

c_or_crarray_like or tuple of (array_like, array_like): The vector c, or a tuple of arrays (c, r). If not supplied, r = conjugate(c) is assumed; in this case, if c[0] is real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row of the Toeplitz matrix is [c[0], r[1:]].
b(M,) or (M, K) array_like: Right-hand side in T x = b.
check_finitebool, optional: Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (result entirely NaNs) if the inputs do contain infinities or NaNs.

Returns:

x(M,) or (M, K) ndarray: The solution to the system T @ x = b. Shape of return matches shape of b.

See also

toeplitz: Toeplitz matrix

Notes

The solution is computed using Levinson-Durbin recursion, which is faster than generic least-squares methods, but can be less numerically stable.

Examples

Solve the Toeplitz system T @ x = b, where:

    [ 1 -1 -2 -3]       [1]
T = [ 3  1 -1 -2]   b = [2]
    [ 6  3  1 -1]       [2]
    [10  6  3  1]       [5]

To specify the Toeplitz matrix, only the first column and the first row are needed.

>>> import numpy as np
>>> c = np.array([1, 3, 6, 10])    # First column of T
>>> r = np.array([1, -1, -2, -3])  # First row of T
>>> b = np.array([1, 2, 2, 5])

>>> from scipy.linalg import solve_toeplitz, toeplitz
>>> x = solve_toeplitz((c, r), b)
>>> x
array([ 1.66666667, -1.        , -2.66666667,  2.33333333])

Check the result by creating the full Toeplitz matrix and multiplying it by x. We should get b.

>>> T = toeplitz(c, r)
>>> T.dot(x)
array([ 1.,  2.,  2.,  5.])