Notes

For example, the 5x5 toeplitz matrix below should be represented as the linear array a on the right :

[ a0    a1   a2  a3  a4 ]
[ a-1   a0   a1  a2  a3 ]
[ a-2  a-1   a0  a1  a2 ] -> [a-4  a-3  a-2  a-1  a0  a1  a2  a3  a4]
[ a-3  a-2  a-1  a0  a1 ]
[ a-4  a-3  a-2  a-1 a0 ]

Parameters

a : array, dtype=double or complex128, shape=(2n-1,)

The first column of the matrix in reverse order (without the diagonal) followed by the first (see below)

b : array, dtype=double or complex128, shape=(n,)

The right hand side vector. Both a and b must have the same type (double or complex128).

Returns

x : arrray, shape=(n,)

The solution vector

reflection_coeff : array, shape=(n+1,)

Toeplitz reflection coefficients. When a is symmetric Toeplitz and b is a[n:] , as in the solution of autoregressive systems, then reflection_coeff also correspond to the partial autocorrelation function.

Solve a linear Toeplitz system using Levinson recursion.

Examples

See :

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