NestWhileList

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`NestWhileList`

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NestWhileList[f,expr,test]

generates a list of the results of applying f repeatedly, starting with expr, and continuing until applying test to the result no longer yields True.

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NestWhileList[f,expr,test,m]

supplies the most recent m results as arguments for test at each step.

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NestWhileList[f,expr,test,All]

supplies all results so far as arguments for test at each step.

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NestWhileList[f,expr,test,m,max]

applies f at most max times.

Details

The last element of the list returned by NestWhileList[f,expr,test] is always an expression to which applying test does not yield True.
NestWhileList[f,expr,test,m] at each step evaluates test[res₁,res₂,…,res_m]. It does not put the results res_i in a list.
The res_i are given in the order they are generated, with the most recent coming last.
NestWhileList[f,expr,test,m] does not start applying test until at least m results have been generated. »
NestWhileList[f,expr,test,{m_min,m}] does not start applying test until at least m_min results have been generated. At each step it then supplies as arguments to test as many recent results as possible, up to a maximum of m. »
NestWhileList[f,expr,test,m] is equivalent to NestWhileList[f,expr,test,{m,m}]. »
NestWhileList[f,expr,UnsameQ,2] is equivalent to FixedPointList[f,expr]. »
NestWhileList[f,expr,test,All] is equivalent to NestWhileList[f,expr,test,{1,Infinity}]. »
NestWhileList[f,expr,UnsameQ,All] goes on applying f until the same result first appears more than once.
NestWhileList[f,expr,test,m,max,n] applies f an extra n times, appending the results to the list generated. »
NestWhileList[f,expr,test,m,max,-n] drops the last n elements from the list generated. »

Examples

open allclose all

Basic Examples (2)Summary of the most common use cases

Keep dividing by 2 until the result is no longer an even number:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-x0g

Out[1]=1

Iterate taking logarithms until the result is no longer positive:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-wba

Out[1]=1

Scope (4)Survey of the scope of standard use cases

Start comparisons after 4 iterations, and compare using the 4 last values:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-t2u457

Out[1]=1

Start comparisons after 4 iterations, and compare using the 6 last values:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-zhom6v

Out[1]=1

Always compare all values generated:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-k60s56

Out[1]=1

Stop after at most 4 steps, even if the condition is still True:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-l8lha4

Out[1]=1

Generalizations & Extensions (1)Generalized and extended use cases

Continue until the result is no longer greater than 1:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-0bqu8h

Out[1]=1

Perform one more step after the condition is no longer True:

In[2]:=2

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https://wolfram.com/xid/0dqw6k228vu-8xuhnc

Out[2]=2

Drop the last value generated (for which the test was no longer True):

In[3]:=3

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https://wolfram.com/xid/0dqw6k228vu-ev4bxa

Out[3]=3

Applications (5)Sample problems that can be solved with this function

Find successive integers until a prime is reached:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-k1v

Out[1]=1

Find the multiplicative order of 2 modulo 19:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-ev

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dqw6k228vu-fdhyhu

Out[2]=2

Use MultiplicativeOrder to compute directly:

In[3]:=3

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https://wolfram.com/xid/0dqw6k228vu-c53rl0

Out[3]=3

Find the orbit of under the mapping :

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-fd5

Out[1]=1

Keep applying iterations in the problem until the results repeat:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-rld

Out[1]=1

Exclude the first repeating element from the output:

In[2]:=2

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https://wolfram.com/xid/0dqw6k228vu-jzn4n5

Out[2]=2

Apply Newton iterations for until successive results are within 0.001:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-b4bmxr

Out[1]=1

Properties & Relations (3)Properties of the function, and connections to other functions

These two forms are equivalent:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-pvvciq

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dqw6k228vu-3jni4a

Out[2]=2

NestWhileList returns all intermediate values of NestWhile:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-eg6ff6

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dqw6k228vu-bb0h1c

Out[2]=2

FixedPointList always compares the last two values; these two forms are equivalent:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-q9ips1

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dqw6k228vu-qmo6dq

Out[2]=2

Neat Examples (2)Surprising or curious use cases

Find the digits of a number:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-pfk1xm

Out[1]=1

Distance of two vertices in a graph:

In[1]:=1

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https://wolfram.com/xid/0dqw6k228vu-nk80px

In[2]:=2

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https://wolfram.com/xid/0dqw6k228vu-c4bflh

In[3]:=3

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https://wolfram.com/xid/0dqw6k228vu-ggy70w

Out[3]=3

A plot of the graph:

In[4]:=4

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https://wolfram.com/xid/0dqw6k228vu-em0qx6

Out[4]=4

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