15-metre class: Difference between revisions

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A '''rational difference equation''' is a nonlinear [[difference equation]] of the form<ref>[http://books.google.com/books?id=4Kb3lO31NcAC&printsec=frontcover&dq=on+third+order+rational+difference+equations&source=bl&ots=JSV5xuGLO3&sig=Y_oeukThSmjZhsLRbloxDPuHnSg&hl=en&ei=artgTOvYOcL-8Ab2lMTgCQ&sa=X&oi=book_result&ct=result&resnum=6&ved=0CCsQ6AEwBQ#v=onepage&q&f=false Dynamics of third-order rational difference equations with open problems and Conjectures]</ref><ref name="Ladas-Kulenovic">[http://books.google.com/books?id=zW7N4r64aZgC&printsec=frontcover&dq=on+second+order+rational+difference+equations&hl=en&ei=5b9gTPvTLoH78AaA6fyQCQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC8Q6AEwAA#v=onepage&q&f=false Dynamics of Second-order rational difference equations with open problems and Conjectures]</ref>
 
: <math>x_{n+1} = \frac{\alpha+\sum_{i=0}^k \beta_ix_{n-i}}{A+\sum_{i=0}^k B_ix_{n-i}},</math>
 
where the initial conditions <math>x_{0}, x_{-1},\dots, x_{-k}</math> are such that the denominator is never zero for any <math>n</math>.
 
==First-order rational difference equation==
A '''first-order rational difference equation''' is a nonlinear [[difference equation]] of the form
 
: <math>w_{t+1} = \frac{aw_t+b}{cw_t+d}.</math>
 
When <math>a,b,c,d</math> and the initial condition <math>w_{0}</math> are real numbers, this difference equation is called a '''Riccati difference equation'''.<ref name="Ladas-Kulenovic"/>
 
Such an equation can be solved by writing <math>w_t</math> as a nonlinear transformation of another variable <math>x_t</math> which itself evolves linearly.  Then standard methods can be used to solve the linear [[Recurrence relation#Solving|difference equation]] in <math>x_t</math>.
 
== Solving a first-order equation==
===First approach===
 
One approach <ref>Brand, Louis, "A sequence defined by a difference equation," ''[[American Mathematical Monthly]]'' 62, September 1955, 489&ndash;492.</ref> to developing the transformed variable <math>x_t</math>, when <math>ad-bc \neq 0</math>, is to write
 
: <math>y_{t+1}= \alpha - \frac{\beta}{y_t}</math>
 
where <math>\alpha = (a+d)/c</math> and <math>\beta = (ad-bc)/c^{2}</math> and where <math>w_t = y_t -d/c</math>. Further writing  <math>y_t = x_{t+1}/x_t</math> can be shown to yield
 
: <math>x_{t+2} - \alpha x_{t+1} + \beta x_t =0. \,</math>
 
===Second approach===
 
This approach <ref>Mitchell, Douglas W., "An analytic Riccati solution for two-target discrete-time control," ''[[Journal of Economic Dynamics and Control]]'' 24, 2000, 615&ndash;622.</ref>  gives a first-order difference equation for <math>x_t</math> instead of a second-order one, for the case in which <math>(d-a)^{2}+4bc</math> is non-negative.  Write  <math>x_t = 1/(\eta + w_t)</math> implying <math>w_t = (1- \eta x_t)/x_t</math>, where <math>\eta</math> is given by <math>\eta = (d-a+r)/2c</math> and where <math>r=\sqrt{(d-a)^{2}+4bc}</math>. Then it can be shown that <math>x_t</math> evolves according to
 
: <math>x_{t+1} = \frac{(d-\eta c)x_t}{\eta c+a} + \frac{c}{\eta c+a}.</math>
 
==Application==
 
It was shown in <ref>Balvers, Ronald J., and Mitchell, Douglas W., "Reducing the dimensionality of linear quadratic control problems," ''[[Journal of Economic Dynamics and Control]]'' 31, 2007, 141&ndash;159.</ref> that a dynamic [[matrix Riccati equation]] of the form
 
: <math> H_{t-1} = K +A'H_tA - A'H_tC(C'H_tC)^{-1}C'H_tA, \,</math>
 
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==
 
<references/>
 
==See also==
 
* Newth, Gerald, "World order from chaotic beginnings," ''[[Mathematical Gazette]]'' 88, March 2004, 39-45, for a [[trigometry|trigonometric]] approach.
 
* Simons, Stuart, "A non-linear difference equation," ''Mathematical Gazette'' 93, November 2009, 500-504.
 
{{DEFAULTSORT:Rational Difference Equation}}
[[Category:Algebra|Algebra]]
[[Category:Recurrence relations|Recurrence relations]]

Revision as of 20:59, 27 December 2013

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

xn+1=α+∑i=0kβixn−iA+∑i=0kBixn−i,

where the initial conditions x0,x−1,…,x−k are such that the denominator is never zero for any n.

First-order rational difference equation

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

wt+1=awt+bcwt+d.

When a,b,c,d and the initial condition w0 are real numbers, this difference equation is called a Riccati difference equation.[2]

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

Solving a first-order equation

First approach

One approach [3] to developing the transformed variable xt, when ad−bc≠0, is to write

yt+1=α−βyt

where α=(a+d)/c and β=(ad−bc)/c2 and where wt=yt−d/c. Further writing yt=xt+1/xt can be shown to yield

xt+2−αxt+1+βxt=0.

Second approach

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

xt+1=(d−ηc)xtηc+a+cηc+a.

Application

It was shown in [5] that a dynamic matrix Riccati equation of the form

Ht−1=K+A′HtA−A′HtC(C′HtC)−1C′HtA,

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

  1. ↑ Dynamics of third-order rational difference equations with open problems and Conjectures
  2. ↑ 2.0 2.1 Dynamics of Second-order rational difference equations with open problems and Conjectures
  3. ↑ Brand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly 62, September 1955, 489–492.
  4. ↑ Mitchell, Douglas W., "An analytic Riccati solution for two-target discrete-time control," Journal of Economic Dynamics and Control 24, 2000, 615–622.
  5. ↑ Balvers, Ronald J., and Mitchell, Douglas W., "Reducing the dimensionality of linear quadratic control problems," Journal of Economic Dynamics and Control 31, 2007, 141–159.

See also

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