Rational difference equation

A rational difference equation is a nonlinear difference equation of the form
 * $$x_{n+1} = \frac{\alpha+\sum_{i=0}^k \beta_ix_{n-i}}{A+\sum_{i=0}^k B_ix_{n-i}}~,$$

where the initial conditions $$x_{0}, x_{-1},\dots, x_{-k}$$ are such that the denominator never vanishes for any $n$.

First-order rational difference equation
A first-order rational difference equation is a nonlinear difference equation of the form


 * $$w_{t+1} = \frac{aw_t+b}{cw_t+d}.$$

When $$a,b,c,d$$ and the initial condition $$w_0$$ are real numbers, this difference equation is called a Riccati difference equation.

Such an equation can be solved by writing $$w_t$$ as a nonlinear transformation of another variable $$x_t$$ which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in $$x_t$$.

Equations of this form arise from the infinite resistor ladder problem.

First approach
One approach to developing the transformed variable $$x_t$$, when $$ad-bc \neq 0$$, is to write
 * $$y_{t+1}= \alpha - \frac{\beta}{y_t}$$

where $$\alpha = (a+d)/c$$ and $$\beta = (ad-bc)/c^{2}$$ and where $$w_t = y_t -d/c$$.

Further writing $$y_t = x_{t+1}/x_t$$ can be shown to yield
 * $$x_{t+2} - \alpha x_{t+1} + \beta x_t = 0.$$

Second approach
This approach gives a first-order difference equation for $$x_t$$ instead of a second-order one, for the case in which $$(d-a)^{2}+4bc$$ is non-negative. Write $$x_t = 1/(\eta + w_t)$$  implying $$w_t = (1- \eta x_t)/x_t$$, where $$\eta$$ is given by $$\eta = (d-a+r)/2c$$ and where $$r=\sqrt{(d-a)^{2}+4bc}$$. Then it can be shown that $$x_t$$ evolves according to


 * $$x_{t+1} = \left(\frac{d-\eta c}{\eta c+a}\right)\!x_t + \frac{c}{\eta c+a}.$$

Third approach
The equation


 * $$w_{t+1} = \frac{aw_t+b}{cw_t+d}$$

can also be solved by treating it as a special case of the more general matrix equation


 * $$X_{t+1} = -(E+BX_t)(C+AX_t)^{-1},$$

where all of A, B, C, E, and X are n&thinsp;×&thinsp;n matrices (in this case n = 1); the solution of this is


 * $$X_t = N_tD_t^{-1}$$

where


 * $$\begin{pmatrix} N_{t} \\ D_{t}\end{pmatrix} = \begin{pmatrix} -B & -E \\ A & C \end{pmatrix}^t\begin{pmatrix} X_0\\ I \end{pmatrix}.$$

Application
It was shown in that a dynamic matrix Riccati equation of the form


 * $$H_{t-1} = K +A'H_tA - A'H_tC(C'H_tC)^{-1}C'H_tA,$$

which can arise in some discrete-time optimal control problems, can be solved using the second approach above if the matrix C has only one more row than column.