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 n.

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 .

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 , 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 more general matrix equation

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