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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>.
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==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]]

Latest revision as of 09:23, 28 December 2014


If youve already purchased a Trek bike or any other brand of bike and you prefer to bring your own bike on your tour, bring it along. Normal pedaling motion disengages the brake cane, allowing for normal pedaling. They both have some parts that are worse than another's, but it doesn't matter in general. its also important to look at the design of the bike some bikes are designed for specific porpuses that is to say , there bike s designed for sports , transport porpuses while other motobike are designed for prestige , for leisure porpuses before you go the market ldentify your porpuses why you want tobuy the mountai bike or any bike. When shopping for different bike accessories, parts, or whatever it is you are looking for, being a smart shopper is always in order.

Cross Country and all mountain style riders will lean towards a folding bead tire because they are much lighter and that is a crucial factor in those 2 styles of riding. You want to be pedaling as hard as you can while other riders right next to you are doing the exact same thing. An alternative to Clipless pedals are Mountain Bike shoes with Shimano Pedaling Dynamics (SPD). Cost shouldn't be a factor as a good helmet can be acquired for around $25-$45. Full suspensions are of course the most comfortable.

When you have any kind of issues regarding exactly where and the way to utilize Cannondale mountain bike sizing., it is possible to call us at the website. The steeper the angles, the more beneficial it would be for stability and high speed pedaling. If you do it as soon as you get home, then its done and you can concentrate on eating and relaxing. This Cannondale Mountain Bike is a beast, plain and simple. If you have a mountain bike or a road bike, it is important that you know how to care of it properly. The bike shop personnel should be able to advise you on how to choose the right frame size.

The inaugural Big Bear Gran Fondo takes place on July 26th in Big Bear, with distance options of 20K, 30K, 70K and 100K. This article discusses how to go about buying a mountain bike. In the purest sense of the word, a mountain bikes sole mission is to travel "off-road" only. The dedication and solid experience of the experts, it is not difficult to accomplish such goal. I've never been into cycling and wanted to give it a try for exercise.

And there are some people who rightfully bring up the issue of the fuel costs and environmental impact of producing the electricty used to power the bike in the first place. The tires are wider that are designed for diverse terrains. Something that sounds to me, to be nothing more than hard work. A twenty-inch wheeled bike are good for longer traveling time. It's about the journey and all that it entails, learning about yourself, making friends, being helped, and of course, helping others along the way".

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