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Rational difference equation
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<h2 id="first-order-rational-difference-equation">First-order rational difference equation</h2> <p>A first-order rational difference equation is a nonlinear difference equation of the form </p> w t + 1 = a w t + b c w t + d . {\displaystyle w_{t+1}={\frac {aw_{t}+b}{cw_{t}+d}}.} <p>When a , b , c , d {\displaystyle a,b,c,d} and the initial condition w 0 {\displaystyle w_{0}} are <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, this difference equation is called a Riccati difference equation.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>Such an equation can be solved by writing w t {\displaystyle w_{t}} as a nonlinear transformation of another variable x t {\displaystyle x_{t}} which itself evolves linearly. Then standard methods can be used to solve the <a href="/facts/Linear_difference_equation/gs9PR0D7">linear difference equation</a> in x t {\displaystyle x_{t}} . </p><p>Equations of this form arise from the infinite resistor ladder problem.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h2 id="solving-a-first-order-equation">Solving a first-order equation</h2> <h3>First approach</h3> <p>One approach<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> to developing the transformed variable x t {\displaystyle x_{t}} , when a d − b c ≠ 0 {\displaystyle ad-bc\neq 0} , is to write </p> y t + 1 = α − β y t {\displaystyle y_{t+1}=\alpha -{\frac {\beta }{y_{t}}}} <p>where α = ( a + d ) / c {\displaystyle \alpha =(a+d)/c} and β = ( a d − b c ) / c 2 {\displaystyle \beta =(ad-bc)/c^{2}} and where w t = y t − d / c {\displaystyle w_{t}=y_{t}-d/c} . </p><p>Further writing y t = x t + 1 / x t {\displaystyle y_{t}=x_{t+1}/x_{t}} can be shown to yield </p> x t + 2 − α x t + 1 + β x t = 0. {\displaystyle x_{t+2}-\alpha x_{t+1}+\beta x_{t}=0.} <h3>Second approach</h3> <p>This approach<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> gives a first-order difference equation for x t {\displaystyle x_{t}} instead of a second-order one, for the case in which ( d − a ) 2 + 4 b c {\displaystyle (d-a)^{2}+4bc} is non-negative. Write x t = 1 / ( η + w t ) {\displaystyle x_{t}=1/(\eta +w_{t})} implying w t = ( 1 − η x t ) / x t {\displaystyle w_{t}=(1-\eta x_{t})/x_{t}} , where η {\displaystyle \eta } is given by η = ( d − a + r ) / 2 c {\displaystyle \eta =(d-a+r)/2c} and where r = ( d − a ) 2 + 4 b c {\displaystyle r={\sqrt {(d-a)^{2}+4bc}}} . Then it can be shown that x t {\displaystyle x_{t}} evolves according to </p> x t + 1 = ( d − η c η c + a ) x t + c η c + a . {\displaystyle x_{t+1}=\left({\frac {d-\eta c}{\eta c+a}}\right)\!x_{t}+{\frac {c}{\eta c+a}}.} <h3>Third approach</h3> <p>The equation </p> w t + 1 = a w t + b c w t + d {\displaystyle w_{t+1}={\frac {aw_{t}+b}{cw_{t}+d}}} <p>can also be solved by treating it as a special case of the <a href="/facts/Matrix_difference_equation/YoDHzvIj">more general matrix equation</a> </p> X t + 1 = − ( E + B X t ) ( C + A X t ) − 1 , {\displaystyle X_{t+1}=-(E+BX_{t})(C+AX_{t})^{-1},} <p>where all of <i>A, B, C, E,</i> and <i>X</i> are <i>n</i> × <i>n</i> <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> (in this case <i>n</i> = 1); the solution of this is<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> X t = N t D t − 1 {\displaystyle X_{t}=N_{t}D_{t}^{-1}} <p>where </p> ( N t D t ) = ( − B − E A C ) t ( X 0 I ) . {\displaystyle {\begin{pmatrix}N_{t}\\D_{t}\end{pmatrix}}={\begin{pmatrix}-B&-E\\A&C\end{pmatrix}}^{t}{\begin{pmatrix}X_{0}\\I\end{pmatrix}}.} <h2 id="application">Application</h2> <p>It was shown in <a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> that a dynamic <a href="/facts/Matrix_Riccati_equation/lVWtEIQ9">matrix Riccati equation</a> of the form </p> H t − 1 = K + A ′ H t A − A ′ H t C ( C ′ H t C ) − 1 C ′ H t A , {\displaystyle H_{t-1}=K+A'H_{t}A-A'H_{t}C(C'H_{t}C)^{-1}C'H_{t}A,} <p>which can arise in some <a href="/facts/Discrete-time/RR8wpS9e">discrete-time</a> <a href="/facts/Optimal_control/9SmcQHjJ">optimal control</a> problems, can be solved using the second approach above if the matrix <i>C</i> has only one more row than column. </p> <h2 id="further-reading">Further reading</h2> <ul><li>Simons, Stuart, "A non-linear difference equation," <i>Mathematical Gazette</i> 93, November 2009, 500–504.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Skellam, J.G. (1951). “Random dispersal in theoretical populations”, Biometrika 38 196−218, eqns (41,42) <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Camouzis, Elias; Ladas, G. (November 16, 2007). Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures. CRC Press. ISBN 9781584887669 – via Google Books. <a href="9781584887669" target="_blank">9781584887669</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Kulenovic, Mustafa R. S.; Ladas, G. (July 30, 2001). Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures. CRC Press. ISBN 9781420035384 – via Google Books. <a href="9781420035384" target="_blank">9781420035384</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Newth, Gerald, "World order from chaotic beginnings", Mathematical Gazette 88, March 2004, 39-45 gives a trigonometric approach. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Kulenovic, Mustafa R. S.; Ladas, G. (July 30, 2001). Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures. CRC Press. ISBN 9781420035384 – via Google Books. <a href="9781420035384" target="_blank">9781420035384</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Equivalent resistance in ladder circuit". Stack Exchange. Retrieved 21 February 2022. <a href="https://physics.stackexchange.com/q/121297" target="_blank">https://physics.stackexchange.com/q/121297</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>"Thinking Recursively: How to Crack the Infinite Resistor Ladder Puzzle!". Youtube. Retrieved 21 February 2022. <a href="https://www.youtube.com/watch?v=rqckorUt2ck" target="_blank">https://www.youtube.com/watch?v=rqckorUt2ck</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Brand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly 62, September 1955, 489–492. online <a href="/wiki/American_Mathematical_Monthly" target="_blank">/wiki/American_Mathematical_Monthly</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Mitchell, Douglas W., "An analytic Riccati solution for two-target discrete-time control," Journal of Economic Dynamics and Control 24, 2000, 615–622. <a href="/wiki/Journal_of_Economic_Dynamics_and_Control" target="_blank">/wiki/Journal_of_Economic_Dynamics_and_Control</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Martin, C. F., and Ammar, G., "The geometry of the matrix Riccati equation and associated eigenvalue method," in Bittani, Laub, and Willems (eds.), The Riccati Equation, Springer-Verlag, 1991. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>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. <a href="/wiki/Journal_of_Economic_Dynamics_and_Control" target="_blank">/wiki/Journal_of_Economic_Dynamics_and_Control</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> </ol>