A rational difference equation is a nonlinear difference equation of the form[1][2][3][4] where the initial conditions
are such that the denominator never vanishes for any n. 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.
[3] 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.
[5][6] One approach[7] to developing the transformed variable
, is to write where
α = ( a + d )
β = ( a d − b c )
{\displaystyle \beta =(ad-bc)/c^{2}}
Further writing
can be shown to yield This approach[8] gives a first-order difference equation for
instead of a second-order one, for the case in which
{\displaystyle (d-a)^{2}+4bc}
is non-negative.
Write
( η +
implying
{\displaystyle r={\sqrt {(d-a)^{2}+4bc}}}
Then it can be shown that
evolves according to 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[9] where It was shown in [10] 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.