I have a matrix-valued continued fraction defined in the following way: $\alpha_n$ and $\beta_n$ are matrices, and I am interested in the quantity $A_1$, where all the $A_n$, $n = 1, 2, \dots$ are given by $A_n = \left[ 1 - \beta_n A_{n+1} \right]^{-1} \alpha_n$
I know that $\lim\limits_{n\to \infty} A_n = 0$, so a way to calculate $A_1$ is to start at some cutoff $N$ and evaluate $A_1$ up to that value, and then to re-calculate it for a larger cutoff $N$ until the change in the result is sufficiently small. What I do not like about this approach is that I cannot reuse intermediate results from smaller cutoffs when I calculate the result for the larger cutoff.
I know that for ordinary (i.e. scalar) continued fractions, there is the modified Lentz's version as described in Numerical Recipes and I wondered if I there is a similar method for my matrix-valued example.
In case it helps, maybe some information regarding the matrices $\alpha_n$ and $\beta_n$.
- They are sparse
- They are not square matrices, with $\alpha_n$ having more columns than rows and $\beta_n$ having more rows than columns.
Motivation: I work on an implementation/refinement on a neat method to compute lattice Green's functions as given in the following reference.
Berciu, M., & Cook, A. M. (2010). Efficient computation of lattice Greenʼs functions for models with nearest-neighbour hopping. EPL (Europhysics Letters), 92(4), 40003
Without much detail: We have quantities $G(n_x,n_y)$ and for these we have coupled recurrence relations that link quantities where one of the two arguments gets reduced or increased by 1. Noting that the relations couple quantities that differ in $M := n_x + n_y$ by $\pm 1$, we can define vectors $V_M$ that contain all quantities $G(n_x, n_z)$ with $n_x + n_y = M = const$, and then the recurrence relations can be written in matrix form as $V_M = \alpha_M V_{M-1} + \beta_M V_{M+1}$ Some physicla arguments demand that $V_M = 0$ for very large $M$, and using this as a hard cut-off, we can write $V_M = A_M V_{M-1}$ where $A_M$ is given by the recurrence relation I mention above.
If I choose the cut-off by hand, I can compute the result, in the same way that I could compute a normal continued fraction approximatively by just computing the $n$-th convergent for a large $n$. But I am interested in having a cut-off that automatically ensures that the calculation actually converged.