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

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

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 $x_t$.

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

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

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

Third approach

The equation

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

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

$X_{t+1}=-(E+B{X}_t)(C+A{X}_t{)}^{-1},$

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

$\left(\begin{array}{c}{N}_t\\ D_t\end{array}\right)={\left(\begin{array}{cc}-B& -E\\ A& C\end{array}\right)}^t\left(\begin{array}{c}{X}_0\\ I\end{array}\right).$

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.

Further reading

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