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5 89 the recurrence can be solved by methods described below yielding binet s formula which involves powers of the two roots of the characteristic polynomial t 2 t 1 displaystyle t 2 t 1 the generating function of the sequence is the rational function t 1 t t 2 displaystyle frac t 1 t t 2 binomial coefficients edit a simple example of a multidimensional recurrence relation is given by the binomial coefficients n k displaystyle tbinom n k which count the ways of selecting k displaystyle k elements out of a set of n displaystyle n elements they can be computed by the recurrence relation n k n 1 k 1 n 1 k displaystyle binom n k binom n 1 k 1 binom n 1 k with the base cases n 0 n n 1 displaystyle tbinom n 0 tbinom n n 1 using this formula to compute the values of all binomial coefficients generates an infinite array called pascal s triangle the same values can also be computed directly by a different formula that is not a recurrence but uses factorials multiplication and division not just additions n k n k n k displaystyle binom n k frac n k n k the binomial coefficients can also be computed with a uni dimensional recurrence n k n k 1 n k 1 k displaystyle binom n k binom n k 1 n k 1 k with the initial value n 0 1 textstyle binom n 0 1 the division is not displayed as a fraction for emphasizing that it must be computed after the multiplication for not introducing fractional numbers this recurrence is widely used in computers because it does not require to build a table as does the bi dimensional recurrence and does not involve very large integers as does the formula with factorials if one uses n k n n k textstyle binom n k binom n n k all involved integers are smaller than the final result difference operator and difference equations edit the difference operator is an operator that maps sequences to sequences and more generally functions to functions it is commonly denoted δ displaystyle delta and is defined in functional notation as δ f x f x 1 f x displaystyle delta f x f x 1 f x it is thus a special case of finite difference when using the index notation for sequences the definition becomes δ a n a n 1 a n displaystyle delta a _ n a_ n 1 a_ n the parentheses around δ f displaystyle delta f and δ a displaystyle delta a are generally omitted and δ a n displaystyle delta a_ n must be understood as the term of index n in the sequence δ a displaystyle delta a and not δ displaystyle delta applied to the element a n displaystyle a_ n given sequence a a n n n displaystyle a a_ n _ n in mathbb n the first difference of a is δ a displaystyle delta a the second difference is δ 2 a δ δ a δ δ a displaystyle delta 2 a delta circ delta a delta delta a a simple computation shows that δ 2 a n a n 2 2 a n 1 a n displaystyle delta 2 a_ n a_ n 2 2a_ n 1 a_ n more generally the k th difference is defined recursively as δ k δ δ k 1 displaystyle delta k delta circ delta k 1 and one has δ k a n t 0 k 1 t k t a n k t displaystyle delta k a_ n sum _ t 0 k 1 t binom k t a_ n k t this relation can be inverted giving a n k a n k 1 δ a n k k δ k a n displaystyle a_ n k a_ n k choose 1 delta a_ n cdots k choose k delta k a_ n a difference equation of order k is an equation that involves the k first differences of a sequence or a function in the same way as a differential equation of order k relates the k first derivatives of a function the two above relations allow transforming a recurrence relation of order k into a difference equation of order k and conversely a difference equation of order k into recurrence relation of order k each transformation is the inverse of the other and the sequences that are solution of the difference equation are exactly those that satisfies the recurrence relation for example the difference equation 3 δ 2 a n 2 δ a n 7 a n 0 displaystyle 3 delta 2 a_ n 2 delta a_ n 7a_ n 0 is equivalent to the recurrence relation 3 a n 2 4 a n 1 8 a n displaystyle 3a_ n 2 4a_ n 1 8a_ n in the sense that the two equations are satisfied by the same sequences as it is equivalent for a sequence to satisfy a recurrence relation or to be the solution of a difference equation the use of the term difference equation is not limited to equations using a difference operator 2 3 and the two terms recurrence relation and difference equation can be used interchangeably 4 see rational difference equation linear constant coefficient difference equation and matrix difference equation for examples of using difference equation instead of recurrence relation difference equations resemble differential equations and this resemblance is often used to mimic methods for solving differentiable equations to apply to solving difference equations and therefore recurrence relations summation equations relate to difference equations as integral equations relate to differential equations see time scale calculus for a unification of the theory of difference equations with that of differential equations from sequences to grids edit single variable or one dimensional recurrence relations are about sequences i e functions defined on one dimensional grids multi variable or n dimensional recurrence relations are about n displaystyle n dimensional grids functions defined on n displaystyle n grids can also be studied with partial difference equations 5 solving edit solving linear recurrence relations with constant coefficients edit main article linear recurrence with constant coefficients solving first order non homogeneous recurrence relations with variable coefficients edit moreover for the general first order non homogeneous linear recurrence relation with variable coefficients a n 1 f n a n g n f n 0 displaystyle a_ n 1 f_ n a_ n g_ n qquad f_ n neq 0 there is also a nice method to solve it 6 a n 1 f n a n g n displaystyle a_ n 1 f_ n a_ n g_ n a n 1 k 0 n f k f n a n k 0 n f k g n k 0 n f k displaystyle frac a_ n 1 prod _ k 0 n f_ k frac f_ n a_ n prod _ k 0 n f_ k frac g_ n prod _ k 0 n f_ k a n 1 k 0 n f k a n k 0 n 1 f k g n k 0 n f k displaystyle frac a_ n 1 prod _ k 0 n f_ k frac a_ n prod _ k 0 n 1 f_ k frac g_ n prod _ k 0 n f_ k let a n a n k 0 n 1 f k displaystyle a_ n frac a_ n prod _ k 0 n 1 f_ k then a n 1 a n g n k 0 n f k displaystyle a_ n 1 a_ n frac g_ n prod _ k 0 n f_ k m 0 n 1 a m 1 a m a n a 0 m 0 n 1 g m k 0 m f k displaystyle sum _ m 0 n 1 a_ m 1 a_ m a_ n a_ 0 sum _ m 0 n 1 frac g_ m prod _ k 0 m f_ k a n k 0 n 1 f k a 0 m 0 n 1 g m k 0 m f k displaystyle frac a_ n prod _ k 0 n 1 f_ k a_ 0 sum _ m 0 n 1 frac g_ m prod _ k 0 m f_ k a n k 0 n 1 f k a 0 m 0 n 1 g m k 0 m f k displaystyle a_ n left prod _ k 0 n 1 f_ k right left a_ 0 sum _ m 0 n 1 frac g_ m prod _ k 0 m f_ k right if we apply the formula to a n 1 1 h f n h a n h g n h displaystyle a_ n 1 1 hf_ nh a_ n hg_ nh and take the limit h 0 displaystyle h to 0 we get the formula for first order linear differential equations with variable coefficients the sum becomes an integral and the product becomes the exponential function of an integral solving general homogeneous linear recurrence relations edit many homogeneous linear recurrence relations may be solved by means of the generalized hypergeometric series special cases of these lead to recurrence relations for the orthogonal polynomials and many special functions for example the solution to j n 1 2 n z j n j n 1 displaystyle j_ n 1 frac 2n z j_ n j_ n 1 is given by j n j n z displaystyle j_ n j_ n z the bessel function while b n m n 1 2 n b z m n n m n 1 0 displaystyle b n m_ n 1 2n b z m_ n nm_ n 1 0 is solved by m n m n b z displaystyle m_ n m n b z the confluent hypergeometric series sequences which are the solutions of linear difference equations with polynomial coefficients are called p recursive for these specific recurrence equations algorithms are known which find polynomial rational or hypergeometric solutions solving general non homogeneous linear recurrence relations with constant coefficients edit furthermore for the general non homogeneous linear recurrence relation with constant coefficients one can solve it based on variation of parameter 7 solving first order rational difference equations edit main article rational difference equation a first order rational difference equation has the form w t 1 a w t b c w t d displaystyle w_ t 1 tfrac aw_ t b cw_ t d 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 linear difference equation in x t displaystyle x_ t stability edit stability of linear higher order recurrences edit the linear recurrence of order d displaystyle d a n c 1 a n 1 c 2 a n 2 c d a n d displaystyle a_ n c_ 1 a_ n 1 c_ 2 a_ n 2 cdots c_ d a_ n d has the characteristic equation λ d c 1 λ d 1 c 2 λ d 2 c d λ 0 0 displaystyle lambda d c_ 1 lambda d 1 c_ 2 lambda d 2 cdots c_ d lambda 0 0 the recurrence is stable meaning that the iterates converge asymptotically to a fixed value if and only if the eigenvalues i e the roots of the characteristic equation whether real or complex are all less than unity in absolute value stability of linear first order matrix recurrences edit main article matrix difference equation in the first order matrix difference equation x t x a x t 1 x displaystyle x_ t x a x_ t 1 x with state vector x displaystyle x and transition matrix a displaystyle a x displaystyle x converges asymptotically to the steady state vector x displaystyle x if and only if all eigenvalues of the transition matrix a displaystyle a whether real or complex have an absolute value which is less than 1 stability of nonlinear first order recurrences edit consider the nonlinear first order recurrence x n f x n 1 displaystyle x_ n f x_ n 1 this recurrence is locally stable meaning that it converges to a fixed point x displaystyle x from points sufficiently close to x displaystyle x if the slope of f displaystyle f in the neighborhood of x displaystyle x is smaller than unity in absolute value that is f x 1 displaystyle f x 1 a nonlinear recurrence could have multiple fixed points in which case some fixed points may be locally stable and others locally unstable for continuous f two adjacent fixed points cannot both be locally stable a nonlinear recurrence relation could also have a cycle of period k displaystyle k for k 1 displaystyle k 1 such a cycle is stable meaning that it attracts a set of initial conditions of positive measure if the composite function g x f f f x displaystyle g x f circ f circ cdots circ f x with f displaystyle f appearing k displaystyle k times is locally stable according to the same criterion g x 1 displaystyle g x 1 where x displaystyle x is any point on the cycle in a chaotic recurrence relation the variable x displaystyle x stays in a bounded region but never converges to a fixed point or an attracting cycle any fixed points or cycles of the equation are unstable see also logistic map dyadic transformation and tent map relationship to differential equations edit when solving an ordinary differential equation numerically one typically encounters a recurrence relation for example when solving the initial value problem y t f t y t y t 0 y 0 displaystyle y t f t y t y t_ 0 y_ 0 with euler s method and a step size h displaystyle h one calculates the values y 0 y t 0 y 1 y t 0 h y 2 y t 0 2 h displaystyle y_ 0 y t_ 0 y_ 1 y t_ 0 h y_ 2 y t_ 0 2h dots by the recurrence y n 1 y n h f t n y n t n t 0 n h displaystyle y_ n 1 y_ n hf t_ n y_ n t_ n t_ 0 nh systems of linear first order differential equations can be discretized exactly analytically using the methods shown in the discretization article applications edit mathematical biology edit some of the best known difference equations have their origins in the attempt to model population dynamics for example the fibonacci numbers were once used as a model for the growth of a rabbit population the logistic map is used either directly to model population growth or as a starting point for more detailed models of population dynamics in this context coupled difference equations are often used to model the interaction of two or more populations for example the nicholson bailey model for a host parasite interaction is given by n t 1 λ n t e a p t displaystyle n_ t 1 lambda n_ t e ap_ t p t 1 n t 1 e a p t displaystyle p_ t 1 n_ t 1 e ap_ t with n t displaystyle n_ t representing the hosts and p t displaystyle p_ t the parasites at time t displaystyle t integrodifference equations are a form of recurrence relation important to spatial ecology these and other difference equations are particularly suited to modeling univoltine populations computer science edit recurrence relations are also of fundamental importance in analysis of algorithms 8 9 if an algorithm is designed so that it will break a problem into smaller subproblems divide and conquer its running time is described by a recurrence relation a simple example is the time an algorithm takes to find an element in an ordered vector with n displaystyle n elements in the worst case a naive algorithm will search from left to right one element at a time the worst possible scenario is when the required element is the last so the number of comparisons is n displaystyle n a better algorithm is called binary search however it requires a sorted vector it will first check if the element is at the middle of the vector if not then it will check if the middle element is greater or lesser than the sought element at this point half of the vector can be discarded and the algorithm can be run again on the other half the number of comparisons will be given by c 1 1 displaystyle c_ 1 1 c n 1 c n 2 displaystyle c_ n 1 c_ n 2 the time complexity of which will be o log 2 n displaystyle o log _ 2 n digital signal processing edit in digital signal processing recurrence relations can model feedback in a system where outputs at one time become inputs for future time they thus arise in infinite impulse response iir digital filters for example the equation for a feedforward iir comb filter of delay t displaystyle t is y t 1 α x t α y t t displaystyle y_ t 1 alpha x_ t alpha y_ t t where x t displaystyle x_ t is the input at time t displaystyle t y t displaystyle y_ t is the output at time t displaystyle t and α displaystyle alpha controls how much of the delayed signal is fed back into the output from this we can see that y t 1 α x t α 1 α x t t α y t 2 t displaystyle y_ t 1 alpha x_ t alpha 1 alpha x_ t t alpha y_ t 2t y t 1 α x t α α 2 x t t α 2 y t 2 t displaystyle y_ t 1 alpha x_ t alpha alpha 2 x_ t t alpha 2 y_ t 2t etc economics edit see also time series analysis and simultaneous equations model recurrence relations especially linear recurrence relations are used extensively in both theoretical and empirical economics 10 11 in particular in macroeconomics one might develop a model of various broad sectors of the economy the financial sector the goods sector the labor m...
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