A first-order rational difference equation is a nonlinear difference equation of the form
When and the initial condition are real numbers, this difference equation is called a Riccati difference equation.[3]
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 in .
Equations of this form arise from the infinite resistor ladder problem.[5][6]
Solving a first-order equationedit
First approachedit
One approach[7] to developing the transformed variable , when , is to write
where and and where .
Further writing can be shown to yield
Second approachedit
This approach[8] 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
which can arise in some discrete-timeoptimal control problems, can be solved using the second approach above if the matrix C has only one more row than column.
^Camouzis, Elias; Ladas, G. (November 16, 2007). Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures. CRC Press. ISBN9781584887669 – via Google Books.
^ abKulenovic, Mustafa R. S.; Ladas, G. (July 30, 2001). Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures. CRC Press. ISBN9781420035384 – via Google Books.
^Newth, Gerald, "World order from chaotic beginnings", Mathematical Gazette 88, March 2004, 39-45 gives a trigonometric approach.
^"Equivalent resistance in ladder circuit". Stack Exchange. Retrieved 21 February 2022.
^"Thinking Recursively: How to Crack the Infinite Resistor Ladder Puzzle!". Youtube. Retrieved 21 February 2022.
^Brand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly62, September 1955, 489–492. online
^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.
^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.
Further readingedit
Simons, Stuart, "A non-linear difference equation," Mathematical Gazette 93, November 2009, 500–504.
April 12, 2024
rational, difference, equation, rational, difference, equation, nonlinear, difference, equation, form, 0kβixn, 0kbixn, displaystyle, frac, alpha, beta, where, initial, conditions, displaystyle, dots, such, that, denominator, never, vanishes, contents, first, o. A rational difference equation is a nonlinear difference equation of the form 1 2 3 4 xn 1 a i 0kbixn iA i 0kBixn i displaystyle 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 x0 x 1 x k displaystyle x 0 x 1 dots x k are such that the denominator never vanishes for any n Contents 1 First order rational difference equation 2 Solving a first order equation 2 1 First approach 2 2 Second approach 2 3 Third approach 3 Application 4 References 5 Further readingFirst order rational difference equation editA first order rational difference equation is a nonlinear difference equation of the form wt 1 awt bcwt d displaystyle w t 1 frac aw t b cw t d nbsp When a b c d displaystyle a b c d nbsp and the initial condition w0 displaystyle w 0 nbsp are real numbers this difference equation is called a Riccati difference equation 3 Such an equation can be solved by writing wt displaystyle w t nbsp as a nonlinear transformation of another variable xt displaystyle x t nbsp which itself evolves linearly Then standard methods can be used to solve the linear difference equation in xt displaystyle x t nbsp Equations of this form arise from the infinite resistor ladder problem 5 6 Solving a first order equation editFirst approach edit One approach 7 to developing the transformed variable xt displaystyle x t nbsp when ad bc 0 displaystyle ad bc neq 0 nbsp is to write yt 1 a byt displaystyle y t 1 alpha frac beta y t nbsp where a a d c displaystyle alpha a d c nbsp and b ad bc c2 displaystyle beta ad bc c 2 nbsp and where wt yt d c displaystyle w t y t d c nbsp Further writing yt xt 1 xt displaystyle y t x t 1 x t nbsp can be shown to yield xt 2 axt 1 bxt 0 displaystyle x t 2 alpha x t 1 beta x t 0 nbsp Second approach edit This approach 8 gives a first order difference equation for xt displaystyle x t nbsp instead of a second order one for the case in which d a 2 4bc displaystyle d a 2 4bc nbsp is non negative Write xt 1 h wt displaystyle x t 1 eta w t nbsp implying wt 1 hxt xt displaystyle w t 1 eta x t x t nbsp where h displaystyle eta nbsp is given by h d a r 2c displaystyle eta d a r 2c nbsp and where r d a 2 4bc displaystyle r sqrt d a 2 4bc nbsp Then it can be shown that xt displaystyle x t nbsp evolves according to xt 1 d hchc a xt chc a displaystyle x t 1 left frac d eta c eta c a right x t frac c eta c a nbsp Third approach edit The equation wt 1 awt bcwt d displaystyle w t 1 frac aw t b cw t d nbsp can also be solved by treating it as a special case of the more general matrix equation Xt 1 E BXt C AXt 1 displaystyle X t 1 E BX t C AX t 1 nbsp where all of A B C E and X are n n matrices in this case n 1 the solution of this is 9 Xt NtDt 1 displaystyle X t N t D t 1 nbsp where NtDt B EAC t X0I displaystyle begin pmatrix N t D t end pmatrix begin pmatrix B amp E A amp C end pmatrix t begin pmatrix X 0 I end pmatrix nbsp Application editIt was shown in 10 that a dynamic matrix Riccati equation of the form Ht 1 K A HtA A HtC C HtC 1C HtA displaystyle H t 1 K A H t A A H t C C H t C 1 C H t A nbsp 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 edit Skellam J G 1951 Random dispersal in theoretical populations Biometrika 38 196 218 eqns 41 42 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 b 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 Newth Gerald World order from chaotic beginnings Mathematical Gazette 88 March 2004 39 45 gives a trigonometric approach Equivalent resistance in ladder circuit Stack Exchange Retrieved 21 February 2022 Thinking Recursively How to Crack the Infinite Resistor Ladder Puzzle Youtube Retrieved 21 February 2022 Brand Louis A sequence defined by a difference equation American Mathematical Monthly 62 September 1955 489 492 online Mitchell Douglas W An analytic Riccati solution for two target discrete time control Journal of Economic Dynamics and Control 24 2000 615 622 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 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 Further reading editSimons Stuart A non linear difference equation Mathematical Gazette 93 November 2009 500 504 Retrieved from https en wikipedia org w index php title Rational difference equation amp oldid 1134074515, wikipedia, wiki, book, books, library,