Polynomial division

Kolmogoroff cross-spectral factorization, therefore, provides a tool to factor the helical 1-D filter of length 2N_x + 1 into minimum-phase causal and anti-causal filters of length N_x +1. Fortunately, filter coefficients drop away rapidly from either end. In practice, small-valued coefficients can be safely discarded, without violating the minimum-phase requirement; so for a given grid-size, the cost of the matrix inversion scales linearly with the size of the grid.

The unitary form of equation (3) can be maintained by factoring the right-hand-side matrix, ${\bf A}_2$ in equation (6), with Kolmogoroff before applying it to ${\bf q}_z$.

$$\begin{eqnarray} {\bf L}_1 {\bf U}_1 \; {\bf q}_{z+1} & = & {\bf L}_2 {\bf U}_2 ... ...&=& \frac{{\bf L}_2 {\bf U}_2}{{\bf L}_1 {\bf U}_1} \; {\bf q}_{z}\end{eqnarray}$$ (11) (12)