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单词 RecurrenceRelation
释义

recurrence relation


A recurrence relationMathworldPlanetmath is an equation which gives the value of an element of a sequenceMathworldPlanetmath in terms of the values of the sequence for smaller values of the position index and the position index itself. If the position n of a sequence s is denoted by sn, then the next value of the sequence expressed as a recurrence relation would be of the form

sn+1=f(s1,s2,,sn-1,sn,n)

where f is any function.

If k is a positive integer, then a sequence s satisfies a kth order recurrence relation if sn+1 can be written in terms of sn,,sn-k+1 whenever n+1>k. In other words, the recurrence relation for s is of the form

sn-1=f(sn-k+1,,sn-1,sn,n)

for some function f.

An example of a recurrence relation is

sn+1=sn+(n+1),

which is the recurrence relation for the sum of the integers from 1 to n+1. This could also be expressed as

sn=sn-1+n

keeping in mind that, as long as we set the proper initial values of the sequence, the recurrence relation indices can have any constant amount added or subtracted. Note that this is a first order recurrence relation.

As another example of a recurrence relation, the Fibonacci sequenceMathworldPlanetmath satisfies the recurrence relation

sn+1=sn+sn-1.

Note that this is a second order recurrence relation.

Titlerecurrence relation
Canonical nameRecurrenceRelation
Date of creation2013-03-22 11:56:04
Last modified on2013-03-22 11:56:04
Ownerrspuzio (6075)
Last modified byrspuzio (6075)
Numerical id13
Authorrspuzio (6075)
Entry typeDefinition
Classificationmsc 03D20
Classificationmsc 11B37
Synonymdifference equation
Related topicBerlekampMasseyAlgorithm
Related topicEquation
Related topicFiniteDifference
Definesfirst order
Definessecond order
Defineskth order
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更新时间:2025/5/4 3:13:10