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.

Notes

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 :None:None:`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.

Parameters

c_or_cr : array_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_finite : bool, 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.

workers : int, 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.

Returns

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

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

Efficient Toeplitz Matrix-Matrix Multiplication using FFT

See Also

solve_toeplitz

Solve a Toeplitz system using Levinson Recursion

toeplitz

Toeplitz matrix

Examples

[ 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.])

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.linalg._basic.matmul_toeplitz

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