# Defining a list of functions

I'm using polynomial mappings such as

s[{x_,y_}] := {x + y^2, y -x}


and playing with compositions of such mapping, e.g.

s[s[x,y]]


My problem is I need to generate a list of these mappings (or an array, or what have you, I'm not sure what's the best data structure in this case) inside a loop, and later to compose them. So I would like to do something like

s[][s[][x,y]]


the first element in the list of mappings, composed on the second element. Can someone help me with that ? Also, in case there's a better way to define my polynomial mappings, please let me know.

• Your input s[s[x,y]] doesn't return output other than s[{x + y^2, -x + y}] - is that expected, or did you want {x + y^2 + (-x + y)^2, -2 x + y - y^2} as the output? – Jason B. Dec 15 '15 at 8:55
• s[{x_, y_}] := (* stuff *) would be a way to go. – J. M.'s ennui Dec 15 '15 at 8:56
• Thanks. That's actually what I meant. I'll correct it. – Teddy Dec 15 '15 at 9:01
• @Teddy, So I don't really understand - you want to create a list of functions, that is easy enough. But then what are you doing with these compositions? Say the functions are {s1, s2, s3} - what are you looking to return? Something like  {s1[ s2[ s3[x,y] ] ], s2[ s3[ x,y] ], s3[ x,y] }? – Jason B. Dec 15 '15 at 9:25
• @J.M. I'm trying to create approximations for a certain mapping, call it $A[x,y]$, using polynomial mappings. This is done by analysing various parts of $A$, and generating polynomial mappings accordingly. The desired approximation will be the composition of all those polynomial mappings. – Teddy Dec 15 '15 at 9:30

I'm also not sure whether I fully understand what you are after. But whenever one is using functions more in a mathematical sense than in a programming one, I think it is worth thinking about using pure functions to represent them. Here is a list of pure functions which each does a polynomial mapping as you describe:

polynomials={
Function[{x, y}, {x + y^2, y - x}],
Function[{x, y}, {x - y^2, y + x}]
}


you can now almost do what you want:

polynomials[][x,y]
polynomials[][x,y]


only the composition is syntactically somewhat complicated (because you can't use the pattern matcher to unpack/destructure the list arguments for Functions, so we have to use @@ / Apply):

polynomials[] @@ polynomials[][x, y]


but fortunately it is simple enough to provide a helper function which adds some syntactic sugar so handling the entries in the list is less involved:

poly[i_Integer][x_, y_] := polynomials[[i]][x, y]
poly[i_Integer][{x_, y_}] := polynomials[[i]][x, y]


now you can use the various entries like this:

poly[x, y]
poly[x, y]


and composition is also easier:

poly[poly[{x, y}]]


And you can also use Composition, as suggested in other comments and answers:

(Composition @@ {poly,poly,poly})[x, y]


Depending on what the entries in polynomials actually are it might make sense to turn that list into an Association which then would allow to access the entries by non-integer keys which might make your code more readable and less error prone...