Pattern defining an infinite sequence of numbers
In
mathematics
, a
recurrence relation
is an
equation
according to which the
th term of a
sequence
of numbers is equal to some combination of the previous terms. Often, only
previous terms of the sequence appear in the equation, for a parameter
that is independent of
; this number
is called the
order
of the relation. If the values of the first
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
previous terms. A famous example is the recurrence for the
Fibonacci numbers
,
where the order
is two and the linear function merely adds the two previous terms. This example is a
linear recurrence with constant coefficients
, because the coefficients of the linear function (1 and 1) are constants that do not depend on
. For these recurrences, one can express the general term of the sequence as a
closed-form expression
of
. As well,
linear recurrences with polynomial coefficients
depending on
are also important, because many common elementary 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
.
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
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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
X
is a set to which the elements of a sequence must belong. For any
, this defines a unique sequence with
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
where
is a function that involves
k
consecutive elements of the sequence.
In this case,
k
initial values are needed for defining a sequence.
Examples
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Factorial
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The
factorial
is defined by the recurrence relation
and the initial condition
This is an example of a
linear recurrence with polynomial coefficients
of order 1, with the simple polynomial
as its only coefficient.
Logistic map
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An example of a recurrence relation is the
logistic map
:
with a given constant
; given the initial term
, each subsequent term is determined by this relation.
Fibonacci numbers
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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
- 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
; the
generating function
of the sequence is the
rational function
Binomial coefficients
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A simple example of a multidimensional recurrence relation is given by the
binomial coefficients
, which count the ways of selecting
elements out of a set of
elements.
They can be computed by the recurrence relation
with the base cases
. 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
(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
all involved integers are smaller than the final result).
Difference operator and difference equations
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The
difference operator
is an
operator
that maps
sequences
to sequences, and, more generally,
functions
to functions. It is commonly denoted
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
and
are generally omitted, and
must be understood as the term of index
n
in the sequence
and not
applied to the element
Given
sequence
the
first difference
of
a
is
The
second difference
is
A simple computation shows that
More generally: the
k
th difference
is defined recursively as
and one has
This relation can be inverted, giving
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
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 difference equation, 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 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
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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
-dimensional grids. Functions defined on
-grids can also be studied with
partial difference equations
.
[2]
Solving
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Solving linear recurrence relations with constant coefficients
[
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Solving first-order non-homogeneous recurrence relations with variable coefficients
[
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Moreover, for the general first-order non-homogeneous linear recurrence relation with variable coefficients:
there is also a nice method to solve it:
[3]
Let
Then
If we apply the formula to
and take the limit
, 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
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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
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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
.
Stability
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Stability of linear higher-order recurrences
[
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The linear recurrence of order
,
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
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In the first-order matrix difference equation
with state vector
and transition matrix
,
converges asymptotically to the steady state vector
if and only if all eigenvalues of the transition matrix
(whether real or complex) have an
absolute value
which is less than 1.
Stability of nonlinear first-order recurrences
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Consider the nonlinear first-order recurrence
This recurrence is
locally stable
, meaning that it
converges
to a fixed point
from points sufficiently close to
, if the slope of
in the neighborhood of
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
for
. Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function
with
appearing
times is locally stable according to the same criterion:
where
is any point on the cycle.
In a
chaotic
recurrence relation, the variable
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
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]
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
, 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
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Mathematical biology
[
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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
representing the hosts, and
the parasites, at time
.
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
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Recurrence relations are also of fundamental importance in
analysis of algorithms
.
[4]
[5]
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
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
.
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
.
Digital signal processing
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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
is:
where
is the input at time
,
is the output at time
, and
controls how much of the delayed signal is fed back into the output. From this we can see that
etc.
Economics
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Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics.
[6]
[7]
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.
See also
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References
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- ^
Jacobson, Nathan, Basic Algebra 2 (2nd ed.), § 0.4. pg 16.
- ^
Partial difference equations
, Sui Sun Cheng, CRC Press, 2003,
ISBN
978-0-415-29884-1
- ^
"Archived copy"
(PDF)
.
Archived
(PDF)
from the original on 2010-07-05
. Retrieved
2010-10-19
.
{{
cite web
}}
: CS1 maint: archived copy as title (
link
)
- ^
Cormen, T. et al,
Introduction to Algorithms
, MIT Press, 2009
- ^
R. Sedgewick, F. Flajolet,
An Introduction to the Analysis of Algorithms
, Addison-Wesley, 2013
- ^
Stokey, Nancy L.
;
Lucas, Robert E. Jr.
;
Prescott, Edward C.
(1989).
Recursive Methods in Economic Dynamics
. Cambridge: Harvard University Press.
ISBN
0-674-75096-9
.
- ^
Ljungqvist, Lars
;
Sargent, Thomas J.
(2004).
Recursive Macroeconomic Theory
(Second ed.). Cambridge: MIT Press.
ISBN
0-262-12274-X
.
Bibliography
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]
- Batchelder, Paul M. (1967).
An introduction to linear difference equations
. Dover Publications.
- Miller, Kenneth S. (1968).
Linear difference equations
. W. A. Benjamin.
- Fillmore, Jay P.; Marx, Morris L. (1968). "Linear recursive sequences".
SIAM Rev
. Vol. 10, no. 3. pp. 324?353.
JSTOR
2027658
.
- Brousseau, Alfred (1971).
Linear Recursion and Fibonacci Sequences
. Fibonacci Association.
- Thomas H. Cormen
,
Charles E. Leiserson
,
Ronald L. Rivest
, and
Clifford Stein
.
Introduction to Algorithms
, Second Edition. MIT Press and McGraw-Hill, 1990.
ISBN
0-262-03293-7
. Chapter 4: Recurrences, pp. 62?90.
- Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994).
Concrete Mathematics: A Foundation for Computer Science
(2 ed.). Addison-Wesley.
ISBN
0-201-55802-5
.
- Enders, Walter (2010).
Applied Econometric Times Series
(3 ed.). Archived from
the original
on 2014-11-10.
- Cull, Paul;
Flahive, Mary
; Robson, Robbie (2005).
Difference Equations: From Rabbits to Chaos
. Springer.
ISBN
0-387-23234-6
.
chapter 7.
- Jacques, Ian (2006).
Mathematics for Economics and Business
(Fifth ed.). Prentice Hall. pp.
551
?568.
ISBN
0-273-70195-9
.
Chapter 9.1: Difference Equations.
- Minh, Tang; Van To, Tan (2006).
"Using generating functions to solve linear inhomogeneous recurrence equations"
(PDF)
.
Proc. Int. Conf. Simulation, Modelling and Optimization, SMO'06
. pp. 399?404. Archived from
the original
(PDF)
on 2016-03-04
. Retrieved
2014-08-07
.
- Polyanin, Andrei D.
"Difference and Functional Equations: Exact Solutions"
.
at EqWorld - The World of Mathematical Equations.
- Polyanin, Andrei D.
"Difference and Functional Equations: Methods"
.
at EqWorld - The World of Mathematical Equations.
- Wang, Xiang-Sheng; Wong, Roderick (2012). "Asymptotics of orthogonal polynomials via recurrence relations".
Anal. Appl
.
10
(2): 215?235.
arXiv
:
1101.4371
.
doi
:
10.1142/S0219530512500108
.
S2CID
28828175
.
External links
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