# “Iterate” and “iteration” as nouns

What are the differences between iterate and iteration as nouns?

I don't quite understand the definition of iterate as noun:

A quantity arrived at by iteration

For example, in computer programming, there are language features for iterations, such as for loop, while loop. Which one is correct: each iterate or each iteration?

I saw a lot of scientific literature use each iterate, such as this one:

Since each iterate is contained within an interval ...

• Has anybody ever ever ever seen the word "iterate" used as a noun? I don't mean in a dictionary: I mean, actually ever seen it used in a real-life programming book/article/presentation? – Neil Coffey Sep 13 '11 at 18:27
• @Neil: I can't answer if anybody has seen the word used that way, but I can say that I haven't and I've been around computers for longer than I'm willing to say! – Codie CodeMonkey Sep 13 '11 at 20:07
• @Neil I have. You can google it, too, if you want. – Daniel Sep 13 '11 at 22:11
• I am (mostly) not a programmer, but in mathematics you can find iterate used as a noun. (Pronounced differently than iterate used as a verb.) I may say that sin(sin(sin(x))) is the third iterate of the function sin(x). – GEdgar Nov 29 '13 at 22:31
• I have seen and heard it used as such many times. It might be helpful to note that as a verb the pronunciation is what you probably know — like it-err-ate — whereas the noun is pronounced more like it-err-it. This is much like the distinction between the pronunciations of "conjugate" as a verb or a noun (or adjective). – Mike Nov 1 '18 at 14:21

Each iteration is correct here.

Iterate as a noun means the end result of many iterations. So in an iterative function, each iteration is one loop of calculation, whereas the iterate is the sum of all iterations performed.

An important thing to note here is that iterate can also mean a function that iterates. So if you ever see the noun iterate used, test both definitions to see which makes more sense.

• Thanks! I saw a lot of scientific literature use each iterate, such as this one (google.com/…): "Since each iterate is contained within an interval ..." – Tim Sep 13 '11 at 17:34
• What does "a function that iterates" mean? I don't understand the example in Wikitionary either. – Tim Sep 13 '11 at 17:54
• A function is a mathematical construct which turns the info you give it into the info you want. When a function is iterative, it performs a certain operation on the info, and then performs that operation again on the answer, and then performs it again, and again, until you tell it to stop. Each of the separate calculations is called an iteration, and the answer at the end, when you tell the function to stop, is the iterate. But the entire function is also called an iterate, too. Both meanings are used. – Daniel Sep 13 '11 at 18:11
• The second paragraph of this answer is a little ambiguous, so I'll suggest a clarification. An iteration is specifically the process — the actions happening during one loop of calculation. But an iterate is not just the final result after all of those loops; rather, it is the result after any one of those loops. As such, I think the first sentence of this answer is incorrect: iterate is the correct choice. – Mike Nov 1 '18 at 14:29

In optimization, I think that it's common to use "iterate (noun)" to denote the variable vector at the current iteration. For example, when you try to minimize the function f(x) by iterative methods, you generate a sequence of iterates x_1,x_2,...,x_\infty that ideally converges to a minimum of f(x).

In this context, "iteration" is used to denote the full process of going from x_k to x_{k+1} and can include multiple calculations, evaluations, etc. The points x_k and x_{k+1} themselves are the iterates.

Example sentence: "For a problem having n variables, our algorithm has been proven to generate a sequence of iterates with the distance between the k-th iterate and the minimum x* bounded by |x_k - x* | < B*n/k, with B > 0 constant".

I've personally used "iterate" and "iteration" with this distinction many times in papers I've written, and other authors in my field appear to as well.

As nouns, in many contexts, iterate and iteration are synonymous.

The definition given by Wiktionary for iterate (noun) encompasses not just a quantity, but a mathematical concept:

(mathematics) a function that iterates
f2(x0) is the second iterate of x0 under f.

(PlanetMath has a more detailed mathematical description; they show the nouns iterate and iteration as synonyms for this idea.)

Each iterate refers to each subsequent result of iteration (xk+1, in the paper you mentioned).

So, if an iterate is the result of iteration, what is an iteration?

Wiktionary tells us that it can mean a single repetition, which is what makes the two nearly synonymous. This could be used in math, computer science, or indeed in art or music or writing, meaning:

A variation of a design.

Iteration also means the act or use of repetition, and it has a domain-specific meaning you alluded to:

(computing) the use of repetition in a computer program, especially in the form of a loop

So a computing iteration sounds a lot like a mathematical iterate, except that iteration is much more common; I'll certainly say:

Hmm. It failed on the fourth iteration.
The iteration in my for loop isn't working!

An iterate can also be the state part way through an iteration. For example, in an infinite iteration that converges, you can say that the iteration can be stopped when the distance between the iterates x_n and x_{n+1} drops below a chosen threshold.

Each iteration is correct. Iterate is a verb, and iteration is a noun.

Iterate

Iteration

• Apparently iterate pronounced "EYE-ter-itt" is a noun. – Daniel Sep 13 '11 at 17:21
• @drɱ65: the pronunciation as noun is same as verb. See google.com/… – Tim Sep 13 '11 at 17:39
• You are right! Hmm. Well, my main point is still that there is a noun iterate, so the reasoning in the above answer is fallacious. – Daniel Sep 13 '11 at 17:43
• I'm really curious: when did you last see "iterate" used as a noun? – Neil Coffey Sep 13 '11 at 18:28
• I suspect the dictionaries may be wrong. The mathematicians I know pronounce the noun closer to iterative without the final syllable, so perhaps [/ˈɪtərət/] while the verb is more like iteration without its ending – Henry Dec 6 '13 at 7:42