# Rational difference equation

A rational difference equation is a nonlinear difference equation of the form[1][2][3][4]

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

where the initial conditions ${\displaystyle x_{0},x_{-1},\dots ,x_{-k}}$ 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

${\displaystyle w_{t+1}={\frac {aw_{t}+b}{cw_{t}+d}}.}$

When ${\displaystyle a,b,c,d}$ and the initial condition ${\displaystyle w_{0}}$ are real numbers, this difference equation is called a Riccati difference equation.[3]

Such an equation can be solved by writing ${\displaystyle w_{t}}$ as a nonlinear transformation of another variable ${\displaystyle x_{t}}$ which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in ${\displaystyle x_{t}}$.

## Solving a first-order equation

### First approach

One approach [5] to developing the transformed variable ${\displaystyle x_{t}}$, when ${\displaystyle ad-bc\neq 0}$, is to write

${\displaystyle y_{t+1}=\alpha -{\frac {\beta }{y_{t}}}}$

where ${\displaystyle \alpha =(a+d)/c}$ and ${\displaystyle \beta =(ad-bc)/c^{2}}$ and where ${\displaystyle w_{t}=y_{t}-d/c}$.

Further writing ${\displaystyle y_{t}=x_{t+1}/x_{t}}$ can be shown to yield

${\displaystyle x_{t+2}-\alpha x_{t+1}+\beta x_{t}=0.\,}$

### Second approach

This approach [6] gives a first-order difference equation for ${\displaystyle x_{t}}$ instead of a second-order one, for the case in which ${\displaystyle (d-a)^{2}+4bc}$ is non-negative. Write ${\displaystyle x_{t}=1/(\eta +w_{t})}$ implying ${\displaystyle w_{t}=(1-\eta x_{t})/x_{t}}$, where ${\displaystyle \eta }$ is given by ${\displaystyle \eta =(d-a+r)/2c}$ and where ${\displaystyle r={\sqrt {(d-a)^{2}+4bc}}}$. Then it can be shown that ${\displaystyle x_{t}}$ evolves according to

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

### Third approach

The equation

${\displaystyle w_{t+1}={\frac {aw_{t}+b}{cw_{t}+d}}}$

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

${\displaystyle X_{t+1}=-(E+BX_{t})(C+AX_{t})^{-1},}$

where all of A, B, C, E, and X are n×n matrices (in this case n=1); the solution of this is[7]

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

where

${\displaystyle {\begin{pmatrix}N_{t}\\D_{t}\end{pmatrix}}={\begin{pmatrix}-B&-E\\A&C\end{pmatrix}}^{t}{\begin{pmatrix}X_{0}\\I\end{pmatrix}}.}$

## Application

It was shown in [8] that a dynamic matrix Riccati equation of the form

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

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.

## References

1. ^ Skellam, J.G. (1951). “Random dispersal in theoretical populations”, Biometrika 38 196−218, eqns (41,42)
2. ^ Dynamics of third-order rational difference equations with open problems and Conjectures
3. ^ a b Dynamics of Second-order rational difference equations with open problems and Conjectures
4. ^ Newth, Gerald, "World order from chaotic beginnings", Mathematical Gazette 88, March 2004, 39-45 gives a trigonometric approach.
5. ^ Brand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly 62, September 1955, 489–492. online
6. ^ Mitchell, Douglas W., "An analytic Riccati solution for two-target discrete-time control," Journal of Economic Dynamics and Control 24, 2000, 615–622.
7. ^ Martin, C. F., and Ammar, G., "The geometry of the matrix Riccati equation and associated eigenvalue method," in Bittani, Laub, and Willems (eds.), The Riccati Equation, Springer-Verlag, 1991.
8. ^ Balvers, Ronald J., and Mitchell, Douglas W., "Reducing the dimensionality of linear quadratic control problems," Journal of Economic Dynamics and Control 31, 2007, 141–159.