Rational Difference Equation

A rational difference equation is a nonlinear difference equation of the form
: $x_ = \frac~,$
where the initial conditions $x_, x_,\dots, x_$ are such that the denominator never vanishes for any .

First-order rational difference equation

A first-order rational difference equation is a nonlinear difference equation of the form

: $w_ = \frac.$

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 $ad-bc \neq 0$, is to write
: $y_= \alpha - \frac$
where $\alpha = (a+d)/c$ and $\beta = (ad-bc)/c^$ and where $w_t = y_t -d/c$.

Further writing $y_t = x_/x_t$ can be shown to yield
: $x_ - \alpha x_ + \beta x_t = 0.$

Second approach

This approach gives a first-order difference equation for $x_t$ instead of a second-order one, for the case in which $(d-a)^+4bc$ 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 $r=\sqrt$. Then it can be shown that $x_t$ evolves according to

: $x_ = \left(\frac\right)\!x_t + \frac.$

Third approach

The equation

: $w_ = \frac$

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

: $X_ = -(E+BX_t)(C+AX_t)^,$

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

: $X_t = N_tD_t^$

where

: $\begin N_ \\ D_\end = \begin -B & -E \\ A & C \end^t\begin X_0\\ I \end.$

Application

It was shown in 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. that a dynamic matrix Riccati equation of the form

: $H_ = K +A'H_tA - A'H_tC(C'H_tC)^C'H_tA,$

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

Further reading

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

{{DEFAULTSORT:Rational Difference Equation
Algebra
Recurrence relations
Dynamical systems