Recurrence relation
Encyclopedia
In mathematics
, a recurrence relation is an equation
that recursively
defines a sequence
, once one or more initial terms are given: each further term of the sequence is defined as a function
of the preceding terms.
The term difference equation sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to any recurrence relation.
An example of a recurrence relation is the logistic map
:
with a given constant r; given the initial term x0 each subsequent term is determined by this relation.
Some simply defined recurrence relations can have very complex (chaotic
) behaviours, and they are a part of the field of mathematics known as nonlinear analysis
.
Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n.
s are the archetype of a linear, homogeneous recurrence relation with constant coefficients (see below). They are defined using the linear recurrence relation
with seed values:
Explicitly, recurrence yields the equations:
etc.
We obtain the sequence of Fibonacci numbers which begins:
It can be solved by methods described below yielding the closed form expression which involve powers of the two roots of the characteristic polynomial t2 = t + 1; the generating function
of the sequence is the rational function
where the d coefficients ci (for all i) are constants.
More precisely, this is an infinite list of simultaneous linear equations, one for each n>d−1. A sequence which satisfies a relation of this form is called a linear recurrence sequence or LRS. There are d degrees of freedom
for LRS, i.e., the initial values
can be taken to be any values but then the linear recurrence determines the sequence uniquely.
The same coefficients yield the characteristic polynomial
(also "auxiliary polynomial")
whose d roots play a crucial role in finding and understanding the sequences satisfying the recurrence. If the roots r1, r2, ... are all distinct, then the solution to the recurrence takes the form
where the coefficients ki are determined in order to fit the initial conditions of the recurrence.
When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of n. For instance, if the characteristic polynomial can be factored as (x − r)3, with the same root r occurring three times, then the solution would take the form
is a rational function
: the denominator is the polynomial obtained from the auxiliary polynomial by reversing the order of the coefficients, and the numerator is determined by the initial values of the sequence.
The simplest cases are periodic sequences,
, which have sequence 
and generating function a sum of geometric series:
More generally, given the recurrence relation:
with generating function
the series is annihilated at
and above by the polynomial:
That is, multiplying the generating function by the polynomial yields
as the coefficient on
, which vanishes (by the recurrence relation) for 
. Thus
so dividing yields
expressing the generating function as a rational function.
The denominator is
a transform of the auxiliary polynomial (equivalently, reversing the order of coefficients); one could also use any multiple of this, but this normalization is chosen both because of the simple relation to the auxiliary polynomial, and so that 
.
of real numbers: the first difference 
is defined as
.
The second difference
is defined as
,
which can be simplified to
.
More generally: the kth difference of the sequence
is written as 
is defined recursively as
.
The more restrictive definition of difference equation is an equation composed of an and its kth differences. (A widely used broader definition treats "difference equation" as synonymous with "recurrence relation". See for example rational difference equation and matrix difference equation
.)
Linear recurrence relations are difference equations, and conversely; since this is a simple and common form of recurrence, some authors use the two terms interchangeably. For example, the difference equation
is equivalent to the recurrence relation
Thus one can solve many recurrence relations by rephrasing them as difference equations, and then solving the difference equation, analogously to how one solves ordinary differential equations. However, the Ackermann numbers are an example of a recurrence relation that do not map to a difference equation, much less points on the solution to a differential equation.
See time scale calculus
for a unification of the theory of difference equations with that of differential equations.
Summation equation
s relate to difference equations as integral equation
s relate to differential equations.
has the obvious solution
with 
and the most general solution is 
with 
. The characteristic polynomial equated to zero (the characteristic equation
) is simply t − r = 0.
Solutions to such recurrence relations of higher order are found by systematic means, often using the fact that an = rn is a solution for the recurrence exactly when t = r is a root of the characteristic polynomial. This can be approached directly or using generating function
s (formal power series
) or matrices.
Consider, for example, a recurrence relation of the form
When does it have a solution of the same general form as an = rn? Substituting this guess (ansatz
) in the recurrence relation, we find that
must be true for all n > 1.
Dividing through by rn−2, we get that all these equations reduce to the same thing:
which is the characteristic equation of the recurrence relation. Solve for r to obtain the two roots λ1, λ2: these roots are known as the characteristic roots or eigenvalues of the characteristic equation. Different solutions are obtained depending on the nature of the roots: If these roots are distinct, we have the general solution
while if they are identical (when A2 + 4B = 0), we have
This is the most general solution; the two constants C and D can be chosen based on two given initial conditions a0 and a1 to produce a specific solution.
In the case of complex eigenvalues (which also gives rise to complex values for the solution parameters C and D), the use of complex numbers can be eliminated by rewriting the solution in trigonometric form. In this case we can write the eigenvalues as
Then it can be shown that 
can be rewritten as
where
Here E and F (or equivalently, G and
) are real constants which depend on the initial conditions.
In all cases—real distinct eigenvalues, real duplicated eigenvalues, and complex conjugate eigenvalues—the equation is stable
(that is, the variable a converges to a fixed value (specifically, zero)); if and only if both eigenvalues are smaller than one in absolute value
. In this second-order case, this condition on the eigenvalues can be shown to be equivalent to |A| < 1 − B < 2, which is equivalent to |B| < 1 and |A| < 1 − B.
The equation in the above example was homogeneous
, in that there was no constant term. If one starts with the non-homogeneous recurrence
with constant term K, this can be converted into homogeneous form as follows: The steady state
is found by setting bn = bn−1 = bn−2 = b* to obtain
Then the non-homogeneous recurrence can be rewritten in homogeneous form as
which can be solved as above.
The stability condition stated above in terms of eigenvalues for the second-order case remains valid for the general nth-order case: the equation is stable if and only if all eigenvalues of the characteristic equation are less than one in absolute value.
where
. Call this matrix C. Observe that
Determine an eigenbasis
corresponding to eigenvalues 
. Then express the seed (the initial conditions of the LRS) as a linear combination of the eigenbasis vectors:
Then it conveniently works out that:
This description is really no different from general method above, however it is more succinct. It also works nicely for situations like
Where there are several linked recurrences .
s. The z-transforms are a class of integral transforms that lead to more convenient algebraic manipulations and more straightforward solutions. There are cases in which obtaining a direct solution would be all but impossible, yet solving the problem via a thoughtfully chosen integral transform is straightforward.
(also "auxiliary polynomial")
such that each ci corresponds to each ci in the original recurrence relation (see the general form above). Suppose λ is a root of p(t) having multiplicity r. This is to say that (t − λ)r divides p(t). The following two properties hold:
As a result of this theorem a linear homogeneous recurrence relation with constant coefficients can be solved in the following manner:
The method for solving linear differential equations is similar to the method above—the "intelligent guess" (ansatz
) for linear differential equations with constant coefficients is
where λ is a complex number that is determined by substituting the guess into the differential equation.
This is not a coincidence. Considering the Taylor series
of the solution to a linear differential equation:
it can be seen that the coefficients of the series are given by the nth derivative of f(x) evaluated at the point a. The differential equation provides a linear difference equation relating these coefficients.
This equivalence can be used to quickly solve for the recurrence relationship for the coefficients in the power series solution of a linear differential equation.
The rule of thumb (for equations in which the polynomial multiplying the first term is non-zero at zero) is that:
and more generally
Example: The recurrence relationship for the Taylor series coefficients of the equation:
is given by
or
This example shows how problems generally solved using the power series solution method taught in normal differential equation classes can be solved in a much easier way.
Example: The differential equation
has solution
The conversion of the differential equation to a difference equation of the Taylor coefficients is
.
It is easy to see that the nth derivative of eax evaluated at 0 is an
and the solution is the sum of the solution of the homogeneous and the particular solutions. Another method to solve an inhomogeneous recurrence is the method of symbolic differentiation. For example, consider the following recurrence:
This is an inhomogeneous recurrence. If we substitute
, we obtain the recurrence
Subtracting the original recurrence from this equation yields
or equivalently
This is a homogeneous recurrence which can be solved by the methods explained above. In general, if a linear recurrence has the form
where
are constant coefficients and p(n) is the inhomogeneity, then if 
is a polynomial with degree r, then this inhomogeneous recurrence can be reduced to a homogeneous recurrence by applying the method of symbolic differencing r times.
Moreover, for the general first-order linear inhomogeneous recurrence relation with variable coefficient(s)
, 
, there is also a nice method to solve it:
, and many special functions. For example, the solution to
is given by
the Bessel function
, while
is solved by
the confluent hypergeometric series.
A first order rational difference equation has the form
. 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 
.
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.
In the first-order matrix difference equation
with state vector x and transition matrix A, x converges asymptotically to the steady state vector x* if and only if all eigenvalues of the transition matrix A (whether real or complex) have an absolute value
which is less than 1.
This recurrence is locally stable
, meaning that it converges
to a fixed point x* from points sufficiently close to x*, if and only if the slope of f in the neighborhood of x* 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 k for k > 1. Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function
with f appearing k times is locally stable according to the same criterion:
where x* is any point on the cycle.
In a chaotic
recurrence relation, the variable 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.
numerically
, one typically encounters a recurrence relation. For example, when solving the initial value problem
with Euler's method and a step size h, 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.
dynamics. For example, the Fibonacci number
s 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. 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 Nt representing the hosts, and Pt the parasites, at time 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.
, 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 filter
s.
For example, the equation for a "feedforward" IIR comb filter
of delay T is:
Where
is the input at time t, 
is the output at time t, and 
controls how much of the delayed signal is fed back into the output. From this we can see that
etc.
variables. See also time series analysis.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a recurrence relation is an equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
that recursively
Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
defines a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
, once one or more initial terms are given: each further term of the sequence is defined as a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of the preceding terms.
The term difference equation sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to any recurrence relation.
An example of a recurrence relation is the logistic map
Logistic map
The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
:
with a given constant r; given the initial term x0 each subsequent term is determined by this relation.
Some simply defined recurrence relations can have very complex (chaotic
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
) behaviours, and they are a part of the field of mathematics known as nonlinear analysis
Nonlinearity
In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable to be solved for cannot be...
.
Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n.
Fibonacci numbers
The Fibonacci numberFibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
s are the archetype of a linear, homogeneous recurrence relation with constant coefficients (see below). They are defined using the linear recurrence relation
with seed values:
Explicitly, recurrence yields the equations:
etc.
We obtain the sequence of Fibonacci numbers which begins:
- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
It can be solved by methods described below yielding the closed form expression which involve powers of the two roots of the characteristic polynomial t2 = t + 1; the generating function
Generating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
of the sequence is the rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
Linear homogeneous recurrence relations with constant coefficients
An order d linear homogeneous recurrence relation with constant coefficients is an equation of the form:where the d coefficients ci (for all i) are constants.
More precisely, this is an infinite list of simultaneous linear equations, one for each n>d−1. A sequence which satisfies a relation of this form is called a linear recurrence sequence or LRS. There are d degrees of freedom
Degrees of freedom
Degrees of freedom can mean:* Degrees of freedom , independent displacements and/or rotations that specify the orientation of the body or system...
for LRS, i.e., the initial values
can be taken to be any values but then the linear recurrence determines the sequence uniquely.The same coefficients yield the characteristic polynomial
Characteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
(also "auxiliary polynomial")
whose d roots play a crucial role in finding and understanding the sequences satisfying the recurrence. If the roots r1, r2, ... are all distinct, then the solution to the recurrence takes the form
where the coefficients ki are determined in order to fit the initial conditions of the recurrence.
When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of n. For instance, if the characteristic polynomial can be factored as (x − r)3, with the same root r occurring three times, then the solution would take the form
Rational generating function
Linear recursive sequences are precisely the sequences whose generating functionGenerating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
is a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
: the denominator is the polynomial obtained from the auxiliary polynomial by reversing the order of the coefficients, and the numerator is determined by the initial values of the sequence.
The simplest cases are periodic sequences,
, which have sequence and generating function a sum of geometric series:More generally, given the recurrence relation:
with generating function
the series is annihilated at
and above by the polynomial:That is, multiplying the generating function by the polynomial yields
as the coefficient on
, which vanishes (by the recurrence relation) for . Thusso dividing yields
expressing the generating function as a rational function.
The denominator is
a transform of the auxiliary polynomial (equivalently, reversing the order of coefficients); one could also use any multiple of this, but this normalization is chosen both because of the simple relation to the auxiliary polynomial, and so that .Relationship to difference equations narrowly defined
Given an ordered sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of real numbers: the first difference is defined as.The second difference
is defined as,which can be simplified to
.More generally: the kth difference of the sequence
is written as is defined recursively as.The more restrictive definition of difference equation is an equation composed of an and its kth differences. (A widely used broader definition treats "difference equation" as synonymous with "recurrence relation". See for example rational difference equation and matrix difference equation
Matrix difference equation
A matrix difference equation is a difference equation in which the value of a vector of variables at one point in time is related to its own value at one or more previous points in time, using matrices. Occasionally, the time-varying entity may itself be a matrix instead of a vector...
.)
Linear recurrence relations are difference equations, and conversely; since this is a simple and common form of recurrence, some authors use the two terms interchangeably. For example, the difference equation
is equivalent to the recurrence relation
Thus one can solve many recurrence relations by rephrasing them as difference equations, and then solving the difference equation, analogously to how one solves ordinary differential equations. However, the Ackermann numbers are an example of a recurrence relation that do not map to a difference equation, much less points on the solution to a differential equation.
See time scale calculus
Time scale calculus
In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems...
for a unification of the theory of difference equations with that of differential equations.
Summation equation
Summation equation
In mathematics, a summation equation or discrete integral equation is an equation in which an unknown function appears under a summation sign. The theories of summation equations and integral equations can be unified as integral equations on time scales using time scale calculus...
s relate to difference equations as integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
s relate to 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 n-dimensional grids. Functions defined on n-grids can also be studied with partial difference equations.General methods
For order 1 no theory is needed; the recurrencehas the obvious solution
with and the most general solution is with . The characteristic polynomial equated to zero (the characteristic equationCharacteristic equation
Characteristic equation may refer to:* Characteristic equation , used to solve linear differential equations* Characteristic equation, a characteristic polynomial equation in linear algebra used to find eigenvalues...
) is simply t − r = 0.
Solutions to such recurrence relations of higher order are found by systematic means, often using the fact that an = rn is a solution for the recurrence exactly when t = r is a root of the characteristic polynomial. This can be approached directly or using generating function
Generating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
s (formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
) or matrices.
Consider, for example, a recurrence relation of the form
When does it have a solution of the same general form as an = rn? Substituting this guess (ansatz
Ansatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
) in the recurrence relation, we find that
must be true for all n > 1.Dividing through by rn−2, we get that all these equations reduce to the same thing:
which is the characteristic equation of the recurrence relation. Solve for r to obtain the two roots λ1, λ2: these roots are known as the characteristic roots or eigenvalues of the characteristic equation. Different solutions are obtained depending on the nature of the roots: If these roots are distinct, we have the general solution
while if they are identical (when A2 + 4B = 0), we have
This is the most general solution; the two constants C and D can be chosen based on two given initial conditions a0 and a1 to produce a specific solution.
In the case of complex eigenvalues (which also gives rise to complex values for the solution parameters C and D), the use of complex numbers can be eliminated by rewriting the solution in trigonometric form. In this case we can write the eigenvalues as
Then it can be shown that can be rewritten aswhere
Here E and F (or equivalently, G and
) are real constants which depend on the initial conditions.In all cases—real distinct eigenvalues, real duplicated eigenvalues, and complex conjugate eigenvalues—the equation is stable
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...
(that is, the variable a converges to a fixed value (specifically, zero)); if and only if both eigenvalues are smaller than one in absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
. In this second-order case, this condition on the eigenvalues can be shown to be equivalent to |A| < 1 − B < 2, which is equivalent to |B| < 1 and |A| < 1 − B.
The equation in the above example was homogeneous
Homogeneous differential equation
The term homogeneous differential equation has several distinct meanings.One meaning is that a first-order ordinary differential equation is homogeneous if it has the formwhere F is a homogeneous function of degree zero; that is to say, that F = F.In a related, but distinct, usage, the term linear...
, in that there was no constant term. If one starts with the non-homogeneous recurrence
with constant term K, this can be converted into homogeneous form as follows: The steady state
Steady state
A system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...
is found by setting bn = bn−1 = bn−2 = b* to obtain
Then the non-homogeneous recurrence can be rewritten in homogeneous form as
which can be solved as above.
The stability condition stated above in terms of eigenvalues for the second-order case remains valid for the general nth-order case: the equation is stable if and only if all eigenvalues of the characteristic equation are less than one in absolute value.
Solving via linear algebra
Given a linearly recursive sequence, let C be the transpose of the companion matrix of its characteristic polynomial, that iswhere
. Call this matrix C. Observe thatDetermine an eigenbasis
corresponding to eigenvalues . Then express the seed (the initial conditions of the LRS) as a linear combination of the eigenbasis vectors:Then it conveniently works out that:
This description is really no different from general method above, however it is more succinct. It also works nicely for situations like
Where there are several linked recurrences .
Solving with z-transforms
Certain difference equations, in particular Linear constant coefficient difference equations, can be solved using z-transformZ-transform
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....
s. The z-transforms are a class of integral transforms that lead to more convenient algebraic manipulations and more straightforward solutions. There are cases in which obtaining a direct solution would be all but impossible, yet solving the problem via a thoughtfully chosen integral transform is straightforward.
Theorem
Given a linear homogeneous recurrence relation with constant coefficients of order d, let p(t) be the characteristic polynomialCharacteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
(also "auxiliary polynomial")
such that each ci corresponds to each ci in the original recurrence relation (see the general form above). Suppose λ is a root of p(t) having multiplicity r. This is to say that (t − λ)r divides p(t). The following two properties hold:
- Each of the r sequences
satisfies the recurrence relation. - Any sequence satisfying the recurrence relation can be written uniquely as a linear combination of solutions constructed in part 1 as λ varies over all distinct roots of p(t).
As a result of this theorem a linear homogeneous recurrence relation with constant coefficients can be solved in the following manner:
- Find the characteristic polynomial p(t).
- Find the roots of p(t) counting multiplicity.
- Write an as a linear combination of all the roots (counting multiplicity as shown in the theorem above) with unknown coefficients bi.
-
- This is the general solution to the original recurrence relation.
-
- (q is the multiplicity of λ*)
- 4. Equate each
from part 3 (plugging in 
into the general solution of the recurrence relation) with the known values 
from the original recurrence relation. However, the values an from the original recurrence relation used do not have to be contiguous, just d of them are needed (i.e., for an original linear homogeneous recurrence relation of order 3 one could use the values a0, a1, a4). This process will produce a linear system of d equations with d unknowns. Solving these equations for the unknown coefficients 
of the general solution and plugging these values back into the general solution will produce the particular solution to the original recurrence relation that fits the original recurrence relation's initial conditions (as well as all subsequent values 
of the original recurrence relation).
The method for solving linear differential equations is similar to the method above—the "intelligent guess" (ansatz
Ansatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
) for linear differential equations with constant coefficients is
where λ is a complex number that is determined by substituting the guess into the differential equation.This is not a coincidence. Considering the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
of the solution to a linear differential equation:
it can be seen that the coefficients of the series are given by the nth derivative of f(x) evaluated at the point a. The differential equation provides a linear difference equation relating these coefficients.
This equivalence can be used to quickly solve for the recurrence relationship for the coefficients in the power series solution of a linear differential equation.
The rule of thumb (for equations in which the polynomial multiplying the first term is non-zero at zero) is that:
and more generally
Example: The recurrence relationship for the Taylor series coefficients of the equation:
is given by
or
This example shows how problems generally solved using the power series solution method taught in normal differential equation classes can be solved in a much easier way.
Example: The differential equation
has solution
The conversion of the differential equation to a difference equation of the Taylor coefficients is
.It is easy to see that the nth derivative of eax evaluated at 0 is an
Solving non-homogeneous recurrence relations
If the recurrence is inhomogeneous, a particular solution can be found by the method of undetermined coefficientsMethod of undetermined coefficients
In mathematics, the method of undetermined coefficients, also known as the lucky guess method, is an approach to finding a particular solution to certain inhomogeneous ordinary differential equations and recurrence relations...
and the solution is the sum of the solution of the homogeneous and the particular solutions. Another method to solve an inhomogeneous recurrence is the method of symbolic differentiation. For example, consider the following recurrence:
This is an inhomogeneous recurrence. If we substitute
, we obtain the recurrenceSubtracting the original recurrence from this equation yields
or equivalently
This is a homogeneous recurrence which can be solved by the methods explained above. In general, if a linear recurrence has the form
where
are constant coefficients and p(n) is the inhomogeneity, then if is a polynomial with degree r, then this inhomogeneous recurrence can be reduced to a homogeneous recurrence by applying the method of symbolic differencing r times.Moreover, for the general first-order linear inhomogeneous recurrence relation with variable coefficient(s)
, , there is also a nice method to solve it:
- Let
, - Then
General linear homogeneous recurrence relations
Many linear homogeneous recurrence relations may be solved by means of the generalized hypergeometric series. Special cases of these lead to recurrence relations for the orthogonal polynomialsOrthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
, and many special functions. For example, the solution to
is given by
the Bessel function
Bessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
, while
is solved by
the confluent hypergeometric series.
Solving a first order rational difference equation
Main article: Rational difference equationA first order rational difference equation has the form
. 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 .Stability of linear higher-order recurrences
The linear recurrence of order d,has the characteristic equation
Characteristic equation
Characteristic equation may refer to:* Characteristic equation , used to solve linear differential equations* Characteristic equation, a characteristic polynomial equation in linear algebra used to find eigenvalues...
The recurrence is stable
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...
, 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
Main article: Matrix difference equationMatrix difference equation
A matrix difference equation is a difference equation in which the value of a vector of variables at one point in time is related to its own value at one or more previous points in time, using matrices. Occasionally, the time-varying entity may itself be a matrix instead of a vector...
In the first-order matrix difference equation
with state vector x and transition matrix A, x converges asymptotically to the steady state vector x* if and only if all eigenvalues of the transition matrix A (whether real or complex) have an absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
which is less than 1.
Stability of nonlinear first-order recurrences
Consider the nonlinear first-order recurrenceThis recurrence is locally stable
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...
, meaning that it converges
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
to a fixed point x* from points sufficiently close to x*, if and only if the slope of f in the neighborhood of x* 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 k for k > 1. Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function
with f appearing k times is locally stable according to the same criterion:
where x* is any point on the cycle.
In a chaotic
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
recurrence relation, the variable 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
Logistic map
The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
, dyadic transformation
Dyadic transformation
The dyadic transformation is the mapping produced by the rule...
, and tent map.
Relationship to differential equations
When solving an ordinary differential equationOrdinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
numerically
Numerical ordinary differential equations
Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...
, one typically encounters a recurrence relation. For example, when solving the initial value problem
Initial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
with Euler's method and a step size h, 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
Discretization
In mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers...
article.
Biology
Some of the best-known difference equations have their origins in the attempt to model populationPopulation
A population is all the organisms that both belong to the same group or species and live in the same geographical area. The area that is used to define a sexual population is such that inter-breeding is possible between any pair within the area and more probable than cross-breeding with individuals...
dynamics. For example, the Fibonacci number
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
s were once used as a model for the growth of a rabbit population.
The logistic map
Logistic map
The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
is used either directly to model population growth, or as a starting point for more detailed models. 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 Nt representing the hosts, and Pt the parasites, at time t.
Integrodifference equations are a form of recurrence relation important to spatial ecology
Ecology
Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...
. These and other difference equations are particularly suited to modeling univoltine
Voltinism
Voltinism is a term used in biology to indicate the number of broods or generations of an organism in a year. The term is particularly in use in sericulture, where silkworm varieties vary in their voltinism....
populations.
Digital signal processing
In digital signal processingDigital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of 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
Infinite impulse response
Infinite impulse response is a property of signal processing systems. Systems with this property are known as IIR systems or, when dealing with filter systems, as IIR filters. IIR systems have an impulse response function that is non-zero over an infinite length of time...
(IIR) digital filter
Digital filter
In electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is...
s.
For example, the equation for a "feedforward" IIR comb filter
Comb filter
In signal processing, a comb filter adds a delayed version of a signal to itself, causing constructive and destructive interference. The frequency response of a comb filter consists of a series of regularly spaced spikes, giving the appearance of a comb....
of delay T is:
Where
is the input at time t, is the output at time t, and controls how much of the delayed signal is fed back into the output. From this we can see thatetc.
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 exogenous variables and lagged endogenousEndogeneity (economics)
In an econometric model, a parameter or variable is said to be endogenous when there is a correlation between the parameter or variable and the error term. Endogeneity can arise as a result of measurement error, autoregression with autocorrelated errors, simultaneity, omitted variables, and sample...
variables. See also time series analysis.
See also
- Iterated functionIterated functionIn mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...
- Matrix difference equationMatrix difference equationA matrix difference equation is a difference equation in which the value of a vector of variables at one point in time is related to its own value at one or more previous points in time, using matrices. Occasionally, the time-varying entity may itself be a matrix instead of a vector...
- Orthogonal polynomialsOrthogonal polynomialsIn mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
- RecursionRecursionRecursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
- Recursion (computer science)Recursion (computer science)Recursion in computer science is a method where the solution to a problem depends on solutions to smaller instances of the same problem. The approach can be applied to many types of problems, and is one of the central ideas of computer science....
- Lagged Fibonacci generatorLagged Fibonacci generatorA Lagged Fibonacci generator is an example of a pseudorandom number generator. This class of random number generator is aimed at being an improvement on the 'standard' linear congruential generator...
- Master theoremMaster theoremIn the analysis of algorithms, the master theorem provides a cookbook solution in asymptotic terms for recurrence relations of types that occur in the analysis of many divide and conquer algorithms...
- Circle points segments proof
- Continued fractionContinued fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
- Time scale calculusTime scale calculusIn mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems...
- Integrodifference equation
- Combinatorial principlesCombinatorial principlesIn proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used.The rule of sum, rule of product, and inclusion-exclusion principle are often used for enumerative purposes. Bijective proofs are utilized to demonstrate that two sets...
- Infinite impulse responseInfinite impulse responseInfinite impulse response is a property of signal processing systems. Systems with this property are known as IIR systems or, when dealing with filter systems, as IIR filters. IIR systems have an impulse response function that is non-zero over an infinite length of time...