scipy.linalg.matmul_toeplitz(c_or_cr, x, check_finite=False, workers=None)[source]

Efficient Toeplitz Matrix-Matrix Multiplication using FFT

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.

c_or_crarray_like or tuple of (array_like, array_like)

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.

x(M,) or (M, K) array_like

Matrix with which to multiply.

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.

workersint, optional

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.

T @ x(M,) or (M, K) ndarray

The result of the matrix multiplication T @ x. Shape of return matches shape of x.

See also


Toeplitz matrix


Solve a Toeplitz system using Levinson Recursion


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 [1], licensed under the MIT license. More information about the method can be found in reference [2]. References [3] and [4] have more reference implementations in Python.

New in version 1.6.0.



Jacob R Gardner, Geoff Pleiss, David Bindel, Kilian Q Weinberger, Andrew Gordon Wilson, “GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration” with contributions from Max Balandat and Ruihan Wu. Available online:


J. Demmel, P. Koev, and X. Li, “A Brief Survey of Direct Linear Solvers”. In Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, editors. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, 2000. Available at:


R. Scheibler, E. Bezzam, I. Dokmanic, Pyroomacoustics: A Python package for audio room simulations and array processing algorithms, Proc. IEEE ICASSP, Calgary, CA, 2018. pyroomacoustics/adaptive/


Marano S, Edwards B, Ferrari G and Fah D (2017), “Fitting Earthquake Spectra: Colored Noise and Incomplete Data”, Bulletin of the Seismological Society of America., January, 2017. Vol. 107(1), pp. 276-291.


Multiply the Toeplitz matrix T with matrix x:

    [ 1 -1 -2 -3]       [1 10]
T = [ 3  1 -1 -2]   x = [2 11]
    [ 6  3  1 -1]       [2 11]
    [10  6  3  1]       [5 19]

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

>>> 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)
array([[-20., -80.],
       [ -7.,  -8.],
       [  9.,  85.],
       [ 33., 218.]])

Check the result by creating the full Toeplitz matrix and multiplying it by x.

>>> toeplitz(c, r) @ x
array([[-20, -80],
       [ -7,  -8],
       [  9,  85],
       [ 33, 218]])

The full matrix is never formed explicitly, so this routine is suitable for very large Toeplitz matrices.

>>> n = 1000000
>>> matmul_toeplitz([1] + [0]*(n-1), np.ones(n))
array([1., 1., 1., ..., 1., 1., 1.])

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