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Literature on numerical analysis using approximation of functions via projection into finite-base function space uses terms test function, Ansatz function, basis function. What is the difference?

My understanding is that test $\equiv$ Ansatz functions are those which are in chosen finite basis, while basis function is a function in any basis (by definition). In another words, the former are subset of the latter. Is that right?

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If a basis function was a function in any basis, then every function was a basis function. –  Rasmus Oct 14 '11 at 7:28
    
The formulation was not precise, I meant that test function is a function in finite basis, while basis function is function in (possibly infinite) basis. –  eudoxos Oct 14 '11 at 7:46
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The terms "test function" and "ansatz" are more general. In my experience, the term "test function" is largely synonymous with "trial function" and is often used in connection with variational methods such as the Ritz method. A typical usage is one where the test function is a linear combination of basis functions, but the term is not restricted to this case and can refer to any parameterized function that selects a subset of the function space of interest. The term "ansatz" is largely synonymous, except perhaps it's less strongly associated with variational methods and refers more generally to any attempt to write down a solution in parametrized form. The literal meaning of the German word "Ansatz" from which this term is derived is something like "approach"; that is, an ansatz is a certain way of approaching the problem by choosing a particular functional form.

The term "basis function", by contrast, is only used in the linear case and refers to one of the functions from whose linear combinations the test function is formed.

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I actually encoutered most of those terms in literature on Ritz/Galerkin methods. I checked the Ritz's article (referenced from wikipedia), he does not give any name to the function $\psi_i$ he uses in there, though he calls the $a1\psi_1+s2\psi_2+\cdots$ Ansatz (perhaps just in the sense of "formulation" or "expression"). –  eudoxos Oct 14 '11 at 7:56
    
When you say "test function is a linear combination of basis functions", do you then imply that Ansatz=test=trial(=blending?) functions refer to basis of a selected subspace? –  eudoxos Oct 14 '11 at 7:59
    
@eudoxos: I don't think I understand that question. If I do, I don't think I have anything to add to what I wrote in the answer. The linear combinations of a set of functions necessarily form a linear subspace, and if the functions in the set are linearly independent, the set is necessarily a basis for that linear subspace. If you're trying to ask anything beyond that, I don't understand it; in that case please try rephrasing your question. –  joriki Oct 14 '11 at 8:09
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@joriki, nice post. Let me mention that in physics, Ansatz appears in Bethe Ansatz, where it may refer (and where it does in most of the cases I know) to some unspecified approximation procedure. Typically, imagine that to compute exactly an $n$-point function requires to know the $(n+1)$-point function, for every $n$, then one cheats by approximating, for a given $n$, the latter by a functional of the former, and this allows to get an autonomous (approximate) equation involving the former only. (And the mathematician would want to compare this (fake) solution .../... –  Did Oct 14 '11 at 8:20
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.../... to the (unknown) solution of the original system, while the physicist would not care...) But I understand from what you say, that Ansatz is not a priori related to this notion of approximation. Question: would you say the meaning of the word evolved to the point of including (at least in some circles) this nuance which it did not have at first? –  Did Oct 14 '11 at 8:21
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