Editing Levinson recursion (section)

=== Background ===
Matrix equations follow the form

: <math>\mathbf M \, \vec x = \vec y.</math>

The Levinson–Durbin algorithm may be used for any such equation, as long as '''M''' is a known [[Toeplitz matrix]] with a nonzero main diagonal. Here <math>\vec y</math> is a known [[vector space|vector]], and <math>\vec x</math> is an unknown vector of numbers ''x''<sub>''i''</sub> yet to be determined.

For the sake of this article, ''ê''<sub>''i''</sub> is a vector made up entirely of zeroes, except for its ''i''th place, which holds the value one. Its length will be implicitly determined by the surrounding context. The term ''N'' refers to the width of the matrix above – '''M''' is an ''N''×''N'' matrix. Finally, in this article, superscripts refer to an ''inductive index'', whereas subscripts denote indices. For example (and definition), in this article, the matrix '''T'''<sup>''n''</sup> is an ''n''×''n'' matrix that copies the upper left ''n''×''n'' block from '''M''' – that is, ''T''<sup>''n''</sup><sub>''ij''</sub> = ''M''<sub>''ij''</sub>.

'''T'''<sup>''n''</sup> is also a Toeplitz matrix, meaning that it can be written as

: <math>\mathbf T^n = \begin{bmatrix}
    t_0    & t_{-1}  & t_{-2}  & \dots  & t_{-n+1}   \\
    t_1    & t_0     & t_{-1}  & \dots  & t_{-n+2} \\
    t_2    & t_1     & t_0     & \dots  & t_{-n+3} \\
    \vdots & \vdots  & \vdots  & \ddots & \vdots   \\
    t_{n-1}& t_{n-2} & t_{n-3} & \dots  & t_0
  \end{bmatrix}. </math>