# Functional Iteration

Long used in its simplest form in mathematics, functional iteration is an elegant way to represent repeated operations. The Wolfram Language's symbolic architecture makes powerful general forms of functional iteration immediately accessible.

NestList successively nest a function: {x,f[x],f[f[x]],f[f[f[x]]],}

Nest give the result of nesting a function: f[f[f[x]]] etc.

NestGraph give the graph of nesting a function

FoldList successively fold in a list of values: {x,f[x,1],f[f[x,1],2],f[f[f [x,1],2],3],}

Fold give the result of folding in a list of values: f[f[f [x,1],2],3] etc.

SequenceFold, SequenceFoldList fold allowing sequences of previous values

FoldPair, FoldPairList give a result and maintain a state at each step (e.g. quotient-remainder)

FoldWhile, FoldWhileList fold while a condition is satisfied

FixedPoint, FixedPointList nest until a fixed point is reached

NestWhile, NestWhileList nest while a condition is satisfied

TakeWhile take from a list while a condition is satisfied

LengthWhile the length while a condition is satisfied