Rational difference equation

A rational difference equation is a nonlinear difference equation of the form^[1][2][3][4]

${\displaystyle x_{n+1}={\frac {\alpha +\sum _{i=0}^{k}\beta _{i}x_{n-i}}{A+\sum _{i=0}^{k}B_{i}x_{n-i}}}~,}$

where the initial conditions ${\displaystyle x_{0},x_{-1},\dots ,x_{-k}}$ 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

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

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]

Solving a first-order equation

First approach

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

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

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

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

Application

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.