Rational difference equation – Wikipedia

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[edit]

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[edit]

First approach[edit]

One approach^[7] to developing the transformed variable

${displaystyle x_{t}}$

, when

${displaystyle ad-bcneq 0}$

, 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

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

Application[edit]

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.

References[edit]

Further reading[edit]

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