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

$w_{t+1}=\frac{a{w}_t+b}{c{w}_t+d}.$

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 $ad-bc\ne 0$, 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

$H_{t-1}=K+A^{\prime }{H}_{t}A-A^{\prime }{H}_{t}C(C^{\prime }{H}_{t}C{)}^{-1}{C}^{\prime }{H}_{t}A,$

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.