# basis/test/Ansatz functions: difference?

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

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
@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