Rational difference equation

A rational difference equation is a nonlinear difference equation of the form
where the initial conditions formula_2 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 formula_4 and the initial condition formula_5 are real numbers, this difference equation is called a Riccati difference equation.
Such an equation can be solved by writing formula_6 as a nonlinear transformation of another variable formula_7 which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in formula_7.
Solving a first-order equation.
First approach.
One approach to developing the transformed variable formula_7, when formula_10, is to write
where formula_12 and formula_13 and where formula_14.
Further writing formula_15 can be shown to yield
Second approach.
This approach gives a first-order difference equation for formula_7 instead of a second-order one, for the case in which formula_18 is non-negative. Write formula_19 implying formula_20, where formula_21 is given by formula_22 and where formula_23. Then it can be shown that formula_7 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 (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.