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Text of the page (random words):
his number k displaystyle k is called the order of the relation if the values of the first k displaystyle k numbers in the sequence have been given the rest of the sequence can be calculated by repeatedly applying the equation in linear recurrences the n th term is equated to a linear function of the k displaystyle k previous terms a famous example is the recurrence for the fibonacci numbers f n f n 1 f n 2 displaystyle f_ n f_ n 1 f_ n 2 where the order k displaystyle k is two and the linear function merely adds the two previous terms the sequences that satisfy a recurrence relation are exactly the same as the sequences that satisfy a difference equation more precisely a difference equation can be associated to every recurrence relation and conversely a recurrence relation can be associated to every difference equation such that the two processes are inverse one to the other and the sequences that satisfy one of the equations satisfies the other because of the similarity of difference equations with differential equations the methods of resolution of differential equations may often be applied to difference equations aand thus to recurrence relations the above example is a linear recurrence with constant coefficients for these recurrences one can express the general term of the sequence as a closed form expression of n displaystyle n as well linear recurrences with polynomial coefficients are also important since many common elementary functions and special functions have a taylor series whose coefficients satisfy such a recurrence relation see holonomic function solving a recurrence relation means obtaining a closed form solution a non recursive function of n displaystyle n the concept of a recurrence relation can be extended to multidimensional arrays that is indexed families that are indexed by tuples of natural numbers definition edit a recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones more precisely in the case where only the immediately preceding element is involved a recurrence relation has the form u n φ n u n 1 for n 0 displaystyle u_ n varphi n u_ n 1 quad text for quad n 0 where φ n x x displaystyle varphi mathbb n times x to x is a function where x is a set to which the elements of a sequence must belong for any u 0 x displaystyle u_ 0 in x this defines a unique sequence with u 0 displaystyle u_ 0 as its first element called the initial value 1 it is easy to modify the definition for getting sequences starting from the term of index 1 or higher this defines recurrence relation of first order a recurrence relation of order k has the form u n φ n u n 1 u n 2 u n k for n k displaystyle u_ n varphi n u_ n 1 u_ n 2 ldots u_ n k quad text for quad n geq k where φ n x k x displaystyle varphi mathbb n times x k to x is a function that involves k consecutive elements of the sequence in this case k initial values are needed for defining a sequence examples edit factorial edit the factorial is defined by the recurrence relation n n n 1 for n 0 displaystyle n n cdot n 1 quad text for quad n 0 and the initial condition 0 1 displaystyle 0 1 here u n displaystyle u_ n is written n displaystyle n this is an example of a linear recurrence with polynomial coefficients of order 1 where the simple polynomial in n coefficent is n displaystyle n as its only coefficient logistic map edit an example of a recurrence relation is the logistic map defined by x n 1 r x n 1 x n displaystyle x_ n 1 rx_ n 1 x_ n for a given constant r displaystyle r the behavior of the sequence depends dramatically on r displaystyle r but is stable when the initial condition x 0 displaystyle x_ 0 varies fibonacci numbers edit the recurrence of order two satisfied by the fibonacci numbers is the canonical example of a homogeneous linear recurrence relation with constant coefficients see below the fibonacci sequence is defined using the recurrence f n f n 1 f n 2 displaystyle f_ n f_ n 1 f_ n 2 with initial conditions f 0 0 displaystyle f_ 0 0 f 1 1 displaystyle f_ 1 1 explicitly the recurrence yields the equations f 2 f 1 f 0 displaystyle f_ 2 f_ 1 f_ 0 f 3 f 2 f 1 displaystyle f_ 3 f_ 2 f_ 1 f 4 f 3 f 2 displaystyle f_ 4 f_ 3 f_ 2 etc we obtain the sequence of fibonacci numbers which begins 0 1 1 2 3 5 8 13 21 34 55 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 di...