matmul_toeplitz(c_or_cr, x, check_finite=False, workers=None)
This function returns the matrix multiplication between a Toeplitz matrix and a dense matrix.
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 Toeplitz matrix is embedded in a circulant matrix and the FFT is used to efficiently calculate the matrix-matrix product.
Because the computation is based on the FFT, integer inputs will result in floating point outputs. This is unlike NumPy's matmul, which preserves the data type of the input.
This is partly based on the implementation that can be found in , licensed under the MIT license. More information about the method can be found in reference . References and have more reference implementations in Python.
The vector c, or a tuple of arrays (c, r). Whatever the actual shape of c, it will be converted to a 1-D array. 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:]]. Whatever the actual shape of r, it will be converted to a 1-D array.
Matrix with which to multiply.
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.
To pass to scipy.fft.fft and ifft. Maximum number of workers to use for parallel computation. If negative, the value wraps around from os.cpu_count(). See scipy.fft.fft for more details.
The result of the matrix multiplication T @ x. Shape of return matches shape of x.
Efficient Toeplitz Matrix-Matrix Multiplication using FFT
solve_toeplitz
toeplitz
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
x = np.array([[1, 10], [2, 11], [2, 11], [5, 19]])
from scipy.linalg import toeplitz, matmul_toeplitz
matmul_toeplitz((c, r), x)
toeplitz(c, r) @ x
n = 1000000
matmul_toeplitz([1] + [0]*(n-1), np.ones(n))
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