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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 ...

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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
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@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! – DeepYellow Sep 13 '11 at 20:07
@Neil I have. You can google it, too, if you want. – Daniel δ Sep 13 '11 at 22:11

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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.

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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 δ Sep 13 '11 at 18:11
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Each iteration is correct. Iterate is a verb, and iteration is a noun.

Iterate

Iteration

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Apparently iterate pronounced "EYE-ter-itt" is a noun. – Daniel δ 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 δ Sep 13 '11 at 17:43
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I'm really curious: when did you last see "iterate" used as a noun? – Neil Coffey Sep 13 '11 at 18:28
@Neil: 'iterate' sounds wrong to me as a noun, and of many online sources, only dictionary.com shows it as a noun. But it -is- used commonly enough in the technical language of math/CS to refer to one item in a sequence. – Mitch Sep 13 '11 at 19:56
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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!

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