The problem that I am dealing with is to compute the determinant of a $2\times 2$ Toeplitz matrix[1] (in general I would like to generalize to a more general case, but let's consider the easiest case to begin). I know that in the scalar case, the determinant has an analytical asymptotic expression [2] \begin{equation} \mbox{det}\;T_n[\varphi]=G[\varphi]^{n+1}E[\varphi] \end{equation} where \begin{eqnarray} G[\varphi]=\exp\left[\frac{1}{2\pi}\int_0^{2\pi}d\theta \log \varphi(e^{i\theta})\right]\\ E[\varphi]\sim\exp\sum_{k=1}^{\infty}k(log\;\varphi)_k(log\;\varphi)_{-k}\;, \end{eqnarray} where the subindex $k$ denotes the discrete Fourier transform of the symbol $\varphi$.
In the matritial case things become more complex [2]: there is no a general solution for a general symbol. Only under some restricting conditions analytic expressions can be obtain. My question then is about approximating the expression of the determinant. Is there a cheaper method to compute the $2\times 2$ determinant, than just computing it? Thanks for the help
Note: The $2\times 2$ symbol is continuous and without any singularity. Also its entries are in the Wiener class.
[1]http://ee.stanford.edu/~gray/toeplitz.pdf
[2]http://www.sciencedirect.com/science/article/pii/0001870876901134
