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Module « scipy.linalg »

Fonction solve_toeplitz - module scipy.linalg

Signature de la fonction solve_toeplitz

def solve_toeplitz(c_or_cr, b, check_finite=True) 

Description

help(scipy.linalg.solve_toeplitz)

Solve a Toeplitz system using Levinson Recursion

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 `solve_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:]]``.
b : (M,) or (M, K) array_like
    Right-hand side in ``T x = b``.
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.

Returns
-------
x : (M,) or (M, K) ndarray
    The solution to the system ``T x = b``. Shape of return matches shape
    of `b`.

See Also
--------
toeplitz : Toeplitz matrix

Notes
-----
The solution is computed using Levinson-Durbin recursion, which is faster
than generic least-squares methods, but can be less numerically stable.

Examples
--------
Solve the Toeplitz system T x = b, where::

        [ 1 -1 -2 -3]       [1]
    T = [ 3  1 -1 -2]   b = [2]
        [ 6  3  1 -1]       [2]
        [10  6  3  1]       [5]

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
>>> b = np.array([1, 2, 2, 5])

>>> from scipy.linalg import solve_toeplitz, toeplitz
>>> x = solve_toeplitz((c, r), b)
>>> x
array([ 1.66666667, -1.        , -2.66666667,  2.33333333])

Check the result by creating the full Toeplitz matrix and
multiplying it by `x`.  We should get `b`.

>>> T = toeplitz(c, r)
>>> T.dot(x)
array([ 1.,  2.,  2.,  5.])

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