Rational difference equation


A rational difference equation is a nonlinear difference equation of the form
where the initial conditions 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
When and the initial condition are real numbers, this difference equation is called a Riccati difference equation.
Such an equation can be solved by writing as a nonlinear transformation of another variable which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in.

Solving a first-order equation

First approach

One approach to developing the transformed variable, when, is to write
where and and where.
Further writing can be shown to yield

Second approach

This approach gives a first-order difference equation for instead of a second-order one, for the case in which is non-negative. Write implying, where is given by and where. Then it can be shown that evolves according to

Third approach

The equation
can also be solved by treating it as a special case of the
where all of A, B, C, E, and X are n×n matrices ; the solution of this is
where

Application

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