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

${\displaystyle w_{t+1}={\frac {aw_{t}+b}{cw_{t}+d}}.}$ 

When ${\displaystyle a,b,c,d}$  and the initial condition ${\displaystyle w_{0}}$  are real numbers, this difference equation is called a Riccati difference equation.^[3]

Such an equation can be solved by writing ${\displaystyle w_{t}}$  as a nonlinear transformation of another variable ${\displaystyle x_{t}}$  which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in ${\displaystyle x_{t}}$ .

Equations of this form arise from the infinite resistor ladder problem.^[5][6]

First approach edit

One approach^[7] to developing the transformed variable ${\displaystyle x_{t}}$ , when ${\displaystyle ad-bc\neq 0}$ , is to write

${\displaystyle y_{t+1}=\alpha -{\frac {\beta }{y_{t}}}}$ 

where ${\displaystyle \alpha =(a+d)/c}$  and ${\displaystyle \beta =(ad-bc)/c^{2}}$  and where ${\displaystyle w_{t}=y_{t}-d/c}$ .

Further writing ${\displaystyle y_{t}=x_{t+1}/x_{t}}$  can be shown to yield

${\displaystyle x_{t+2}-\alpha x_{t+1}+\beta x_{t}=0.}$ 

Second approach edit

This approach^[8] gives a first-order difference equation for ${\displaystyle x_{t}}$  instead of a second-order one, for the case in which ${\displaystyle (d-a)^{2}+4bc}$  is non-negative. Write ${\displaystyle x_{t}=1/(\eta +w_{t})}$  implying ${\displaystyle w_{t}=(1-\eta x_{t})/x_{t}}$ , where ${\displaystyle \eta }$  is given by ${\displaystyle \eta =(d-a+r)/2c}$  and where ${\displaystyle r={\sqrt {(d-a)^{2}+4bc}}}$ . Then it can be shown that ${\displaystyle x_{t}}$  evolves according to

${\displaystyle x_{t+1}=\left({\frac {d-\eta c}{\eta c+a}}\right)\!x_{t}+{\frac {c}{\eta c+a}}.}$ 

Third approach edit

The equation

${\displaystyle w_{t+1}={\frac {aw_{t}+b}{cw_{t}+d}}}$ 

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

${\displaystyle X_{t+1}=-(E+BX_{t})(C+AX_{t})^{-1},}$ 

where all of A, B, C, E, and X are n × n matrices (in this case n = 1); the solution of this is^[9]

${\displaystyle X_{t}=N_{t}D_{t}^{-1}}$ 

where

${\displaystyle {\begin{pmatrix}N_{t}\\D_{t}\end{pmatrix}}={\begin{pmatrix}-B&-E\\A&C\end{pmatrix}}^{t}{\begin{pmatrix}X_{0}\\I\end{pmatrix}}.}$ 

It was shown in ^[10] that a dynamic matrix Riccati equation of the form

${\displaystyle H_{t-1}=K+A'H_{t}A-A'H_{t}C(C'H_{t}C)^{-1}C'H_{t}A,}$ 

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.

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