15-metre class: Difference between revisions

Revision as of 21:59, 27 December 2013

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

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 $x_{0},x_{-1},\dots ,x_{-k}$ 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

w_{t+1}={\frac {aw_{t}+b}{cw_{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.^[2]

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 $x_{t}$ .

Solving a first-order equation

First approach

One approach ^[3] to developing the transformed variable $x_{t}$ , when $ad-bc\neq 0$ , is to write

y_{t+1}=\alpha -{\frac {\beta }{y_{t}}}

where $\alpha =(a+d)/c$ and $\beta =(ad-bc)/c^{2}$ 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}-\alpha x_{t+1}+\beta x_{t}=0.\,

Second approach

This approach ^[4] gives a first-order difference equation for $x_{t}$ instead of a second-order one, for the case in which $(d-a)^{2}+4bc$ is non-negative. Write $x_{t}=1/(\eta +w_{t})$ implying $w_{t}=(1-\eta x_{t})/x_{t}$ , where $\eta$ is given by $\eta =(d-a+r)/2c$ and where $r={\sqrt {(d-a)^{2}+4bc}}$ . Then it can be shown that $x_{t}$ evolves according to

x_{t+1}={\frac {(d-\eta c)x_{t}}{\eta c+a}}+{\frac {c}{\eta c+a}}.

Application

It was shown in ^[5] 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

↑ Dynamics of third-order rational difference equations with open problems and Conjectures
↑ ^2.0 ^2.1 Dynamics of Second-order rational difference equations with open problems and Conjectures
↑ Brand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly 62, September 1955, 489–492.
↑ Mitchell, Douglas W., "An analytic Riccati solution for two-target discrete-time control," Journal of Economic Dynamics and Control 24, 2000, 615–622.
↑ 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.

@@ Line 1: / Line 1: @@
-Some say that high quality kitchen knives are as crucial as the chef. In his book, An Edge in the Kitchen: The Ultimate Guide to Kitchen Knives — How to Buy Them, Hold Them Razor Sharp, and Use Them Like a Pro , Chad Ward discusses this new class of stamped knives, which are cut and precision ground from high-alloy steel and have an edge finish on par with forged knives.  Get your knives sharpened professionally about as soon as a year (this is what Alton Brown suggests).<br><br>Simply because America's Test Kitchen and Consumer Reports both rated the Wüsthof Classic set extremely, I decided I required to test this one particular additional.  Wüsthof sent me the 8-piece set and I used the knives to make a number of meals more than the course of a few days.  Out of the box, the knives have beensharp and created quick work of cutting up herbs, nuts, meat and other ingredients for a variety of meals.  They make knives for just about each and every probable want.<br><br>Use the heel to open can, break the seal on jars, or even open beer bottles! Use the back to tenderize meat, or crush tougher components like lemongrass to release flavor, but be cautious not to reduce oneself! Use the back and/or heel of the knife to crack open cooked crab and lobster shells, then you can fish out that great meat much more easily. You use the honing rod each time you use the knife, this keeps the blade straight and will permit for much easier cutting. I use them a number of occasions a day.<br><br>Carbon steel advocates claim that their knives take a keener edge, hold it longer and are simpler to resharpen than stainless steel knives.  Stainless steel customers claim that carbon steel knives are unsanitary, leave an off taste in foods and that stainless knives hold an edge longer than their carbon counterparts.  For those who have any questions regarding wherever as well as the best way to work with Best Damascus Chef Knife ([http://www.thebestkitchenknivesreviews.com/best-chef-knives-reviews/ Http://Www.Thebestkitchenknivesreviews.Com/Best-Chef-Knives-Reviews]), you possibly can call us from the web-site. Chef Knives to Go offers Japanese, Euro and American, but the Japanese selection is excellent.<br><br>Use your knives on the right cutting surfaces and adhere to the knife manufacturer's care and upkeep ideas.  I recommend possessing your knives sharpened by certified experts anytime possible. The new cooks basically don't have an understanding of the significance of the right knives.  Peeling garlic requires a lot of time, that is unless you use a garlic peeler.  Use it to get your dishes performed faster, whether it's a marinade or lemonade.
+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]]

15-metre class: Difference between revisions

Revision as of 21:59, 27 December 2013

Contents

First-order rational difference equation

Solving a first-order equation

First approach

Second approach

Application

References

See also

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15-metre class: Difference between revisions

Revision as of 21:59, 27 December 2013

First-order rational difference equation

Solving a first-order equation

First approach

Second approach

Application

References

See also

Navigation menu

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