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solve_toeplitz(c_or_cr, b, check_finite=True)

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

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.
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.

Solve a Toeplitz system using Levinson Recursion

Examples

[ 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.

>>> 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.])

See :

Back References

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

scipy.linalg._basic.matmul_toeplitz scipy.linalg._basic.solve_toeplitz scipy.linalg._special_matrices.toeplitz

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