Girish Mahajan (Editor)

Rational difference equation

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A rational difference equation is a nonlinear difference equation of the form

Contents

x n + 1 = α + ∑ i = 0 k β i x n − i A + ∑ i = 0 k B i x n − i   ,

where the initial conditions x 0 , x − 1 , … , 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

w t + 1 = a w t + b c w t + d .

When a , b , c , d and the initial condition w 0 are real numbers, this difference equation is called a Riccati difference equation.

Such an equation can be solved by writing w t as a nonlinear transformation of another variable x t which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in x t .

First approach

One approach to developing the transformed variable x t , when a d − b c ≠ 0 , is to write

y t + 1 = α − β y t

where α = ( a + d ) / c and β = ( a d − b c ) / c 2 and where w t = y t − d / c .

Further writing y t = x t + 1 / x t can be shown to yield

x t + 2 − α x t + 1 + β x t = 0.

Second approach

This approach gives a first-order difference equation for x t instead of a second-order one, for the case in which ( d − a ) 2 + 4 b c is non-negative. Write x t = 1 / ( η + w t ) implying w t = ( 1 − η x t ) / x t , where η is given by η = ( d − a + r ) / 2 c and where r = ( d − a ) 2 + 4 b c . Then it can be shown that x t evolves according to

x t + 1 = ( d − η c η c + a ) x t + c η c + a .

Third approach

The equation

w t + 1 = a w t + b c w t + d

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

X t + 1 = − ( E + B X t ) ( C + A X 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

X t = N t D t − 1

where

( N t D t ) = ( − B − E A C ) t ( X 0 I ) .

Application

It was shown in that a dynamic matrix Riccati equation of the form

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

Rational difference equation Wikipedia
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