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Module « scipy.linalg »

Fonction matmul_toeplitz - module scipy.linalg

Signature de la fonction matmul_toeplitz

def matmul_toeplitz(c_or_cr, x, check_finite=False, workers=None) 

Description

help(scipy.linalg.matmul_toeplitz)

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.

.. warning::

    Beginning in SciPy 1.17, multidimensional input will be treated as a batch,
    not ``ravel``\ ed. To preserve the existing behavior, ``ravel`` arguments
    before passing them to `matmul_toeplitz`.

Parameters
----------
c_or_cr : array_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:]]``.
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`.

See Also
--------
toeplitz : Toeplitz matrix
solve_toeplitz : Solve a Toeplitz system using Levinson Recursion

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 `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.

.. versionadded:: 1.6.0

References
----------
.. [1] 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:
   https://github.com/cornellius-gp/gpytorch

.. [2] 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:
   http://www.netlib.org/utk/people/JackDongarra/etemplates/node384.html

.. [3] R. Scheibler, E. Bezzam, I. Dokmanic, Pyroomacoustics: A Python
   package for audio room simulations and array processing algorithms,
   Proc. IEEE ICASSP, Calgary, CA, 2018.
   https://github.com/LCAV/pyroomacoustics/blob/pypi-release/
   pyroomacoustics/adaptive/util.py

.. [4] 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.

Examples
--------
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.

>>> 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)
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.], shape=(1000000,))

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