# Rational difference equation

A rational difference equation is a nonlinear difference equation of the form

$x_{n+1}={\frac {\alpha +\sum _{i=0}^{k}\beta _{i}x_{n-i}}{A+\sum _{i=0}^{k}B_{i}x_{n-i}}}~,$ 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 $x_{t}$ .

## Solving a first-order equation

### 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×n matrices (in this case n=1); the solution of this is

$X_{t}=N_{t}D_{t}^{-1}$

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

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

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.