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bundles / scipy 1.17.1 / scipy / linalg / _basic / matmul_toeplitz

function

scipy.linalg._basic:matmul_toeplitz

source: /scipy/linalg/_basic.py :2259

Signature

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

Summary

Efficient Toeplitz Matrix-Matrix Multiplication using FFT

Extended Summary

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 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 linalg_batch for details.

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.

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.

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)
✓
Check the result by creating the full Toeplitz matrix and multiplying it by ``x``. 
toeplitz(c, r) @ x
✓
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))
✓

See also

solve_toeplitz

Solve a Toeplitz system using Levinson Recursion

toeplitz

Toeplitz matrix

Aliases

  • scipy.linalg.matmul_toeplitz

Referenced by

This package