A rational difference equation is a nonlinear
difference equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
of the form
:
where the initial conditions
are such that the denominator never vanishes for any .
First-order rational difference equation
A first-order rational difference equation is a nonlinear difference equation of the form
:
When
and the initial condition
are
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, this difference equation is called a Riccati difference equation.
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
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
in
.
Equations of this form arise from the infinite resistor ladder problem.
Solving a first-order equation
First approach
One approach to developing the transformed variable
, when
, is to write
:
where
and
and where
.
Further writing
can be shown to yield
:
Second approach
This approach gives a first-order difference equation for
instead of a second-order one, for the case in which
is non-negative. Write
implying
, where
is given by
and where
. Then it can be shown that
evolves according to
:
Third approach
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
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
(in this case ''n'' = 1); the solution of this is
:
where
:
Application
It was shown in
[Balvers, Ronald J., and Mitchell, Douglas W., "Reducing the dimensionality of linear quadratic control problems," '']Journal of Economic Dynamics and Control
The ''Journal of Economic Dynamics and Control ''(JEDC) is a peer-reviewed scholarly journal devoted to computational economics, dynamic economic models, and macroeconomics. It is edited at the University of Amsterdam and published by Elsevier
...
'' 31, 2007, 141–159. that a dynamic
matrix Riccati equation of the form
:
which can arise in some
discrete-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...
problems, can be solved using the second approach above if the matrix ''C'' has only one more row than column.
References
Further reading
* Simons, Stuart, "A non-linear difference equation," ''Mathematical Gazette'' 93, November 2009, 500–504.
{{DEFAULTSORT:Rational Difference Equation
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
Recurrence relations
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
Dynamical systems