Recurrence relation
In mathematics, a recurrence relation is an equation that expresses the "n"th term of a sequence as a function of the "k" preceding terms, for some fixed "k" (independent from "n"), which is called the "order" of the relation. Once "k" initial terms of a sequence are given, the recurrence relation allows computing recursively all the remaining terms of the sequence. Most general results on recurrence relations are about "linear recurrences", which are recurrence relations such that the th term is linear with respect to its preceding terms. Among them, linear recurrences with constant coefficients, and linear recurrences  with polynomial coefficients are especially important. In the first case, this is because one can express the general term of the sequence as a closed-form expression of the index of the term. In the second case, this is because many common elementary and special functions have a Taylor series whose coefficients satisfy such a recurrence relation (see holonomic function). The concept can be extended to multidimensional arrays, that is, indexed families that are indexed by tuples of natural numbers. Definition. 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 where is a function, where is a set to which the elements of a sequence must belong. For any formula_3, this defines a unique sequence with formula_4 as its first element, called the "initial value". 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" has the form where formula_6 is a function that involves consecutive elements of the  sequence. In this case, initial values are needed for defining a sequence. Examples. Factorial. The factorial is defined by the recurrence relation and the initial condition Logistic map. An example of a recurrence relation is the logistic map: with a given constant formula_10; given the initial term formula_11 each subsequent term is determined by this relation. Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of formula_12. Fibonacci numbers. 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 with initial conditions Explicitly, the recurrence yields the equations etc. We obtain the sequence of Fibonacci numbers, which begins The recurrence can be solved by methods described below yielding Binet's formula, which involves powers of the two roots of the characteristic polynomial formula_19; the generating function of the sequence is the rational function Binomial coefficients. A simple example of a multidimensional recurrence relation is given by the binomial coefficients formula_21, which count the ways of selecting formula_22 elements out of a set of formula_12 elements. They can be computed by the recurrence  relation with the base cases formula_25. 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: The binomial coefficients can also be computed with a uni-dimensional recurrence: with the initial value formula_28 (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 involve very large integers as does the formula with factorials (if one uses formula_29 all involved integers are smaller than the final result). Difference operator and difference equations. The is an operator that maps sequences to sequences, and, more generally, functions to functions. It is commonly denoted formula_30 and is defined, in functional notation, as It is thus a special case of finite difference. When using the index notation for sequences, the definition becomes The parentheses around formula_33 and formula_34 are generally omitted, and formula_35  must be understood as the term of index in the sequence formula_36 and not formula_37 applied to the element formula_38 Given sequence formula_39 the of is formula_40 The is formula_41 A simple computation shows that More generally: the "th difference" is defined recursively as formula_43 and one has This relation can be inverted, giving A of order is an equation that involves the first differences of a sequence or a function, in the same way as a differential equation of order relates the first derivatives of a function. The two above relations allow transforming a recurrence relation of order into  a difference equation of order , and, conversely, a difference equation of order into recurrence relation of order . 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 is equivalent to the recurrence relation 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 recurrence relation, the two terms "recurrence relation" and "difference equation" are sometimes used interchangeably . See Rational difference equation and Matrix difference equation for example of uses of "difference equation" instead of "recurrence relation" Difference equations resemble to differential equations, and this resemblance is often used for mimic methods for solving differentiable equations for 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. 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 formula_12-dimensional grids. Functions defined on formula_12-grids can also be studied with partial difference equations. Solving. Solving first-order non-homogeneous recurrence relations with variable coefficients. Moreover, for the general first-order non-homogeneous linear recurrence relation with variable coefficients: there is also a nice method to solve it: Let Then If we apply the formula to formula_59 and take the limit formula_60, 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. 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 is given by the Bessel function, while is solved by 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 first-order rational difference equations. A first order rational difference equation has the  form formula_65. Such an equation can be solved by writing formula_66 as a nonlinear transformation of another variable formula_67 which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in formula_67. Stability. Stability of linear higher-order recurrences. The linear recurrence of order formula_69, has the characteristic equation 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 . In the first-order matrix difference equation with state vector formula_73 and transition matrix formula_74, formula_73 converges asymptotically to the steady state vector formula_76 if and only if all eigenvalues of the transition matrix formula_74 (whether real or complex) have an absolute value which is less than 1. Stability of nonlinear first-order recurrences. Consider the nonlinear first-order recurrence This recurrence is locally stable, meaning that it converges to a fixed point formula_76 from points sufficiently close to formula_76, if the slope of formula_81 in the neighborhood of formula_76 is smaller than unity in absolute value: that is, 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 formula_22 for formula_85. Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function with formula_81 appearing formula_22 times is locally stable according to the same criterion: where formula_76 is any point on the cycle. In a chaotic recurrence relation, the variable formula_73 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. When solving an ordinary differential equation numerically, one typically encounters a recurrence relation. For example, when solving the initial value problem with Euler's method and a step size formula_93, one calculates the values by the recurrence Systems of linear first order differential equations can be discretized exactly analytically using the methods shown in the discretization article. Applications. Biology. 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 with formula_98 representing the hosts, and formula_99 the parasites, at time formula_100. 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. Recurrence relations are also of fundamental importance in analysis of algorithms. 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 formula_12 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 formula_12. 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 the time complexity of which will be  formula_105. Digital signal processing. 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 formula_106 is: where formula_67 is the input at time formula_100, formula_110 is the output at time formula_100, and formula_112 controls how much of the delayed signal is fed back into the output. From this we can see that etc. Economics. Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical  and empirical economics. In particular, in macroeconomics one might develop a model of various broad sectors of the economy (the financial sector, the goods sector, the labor market, etc.) in which some agents' actions depend on lagged variables. The model would then be solved for current values of key variables (interest rate, real GDP, etc.) in terms of past and current values of other variables.