15-metre class

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 is never zero 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

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

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.

See also

• Newth, Gerald, "World order from chaotic beginnings," Mathematical Gazette 88, March 2004, 39-45, for a trigonometric approach.

• Simons, Stuart, "A non-linear difference equation," Mathematical Gazette 93, November 2009, 500-504.