memoization based on function expression?

Question

I want to create a function $f$ which takes as a parameter a function $h$, and calculates its integral between $a$ and $b$.

But I want to apply memoization to this, by only calculating the integral the first time it is called for a particular function of $h$.

If we have a normal function that takes a real number (instead of another function), then we simply do:

g[x_]:=g[x]=...

How do we do this memoization for $f$, where the argument is the expression for a function?

Answer 1

As a pure function:

f1[h_, a_, b_] := f1[h, a, b] = Integrate[h[x], {x, a, b}]
f1[#^2 &, 1, 2]

As a function of x:

f2[h_, a_, b_] := f2[h, a, b] = Integrate[h, {x, a, b}]
f2[x^2, 1, 2]

As a function of any variable var:

f3[h_, var_, a_, b_] := f3[h, var, a, b] = Integrate[h, {var, a, b}]
f3[y^2, y, 1, 2]

All give

7/3

Answer 2

Implicitly, this works best if you express $h$ as a pure function, but it's relatively straightforwards. If you are interested in evaluating the integral from $a$ to $b$ precisely once, then this should work:

f[h_] := f[h] = Function[{a,b},Evaluate[Integrate[h[x],{x,a,b}]]];

This defines f as a function which takes a parameter h (which will need to be a function itself to work properly), and then memoizes its result. Its result is a function representing the evaluated integral of h[x], to within Mathematica's abilities.

For example, f[Sin] (note that this is not f[Sin[x]]) will return:

Function[{a$,b$}, Cos[a$] - Cos[b$]]

This represents a function of a$ and b$ and contains the evaluated integral of Sin. From here f[Sin][1,3] will equal Cos[1]-Cos[3], which is equal to Integrate[Sin[x],{x,1,3}].

To evaluate a more arbitrary function, using pure functions is prudent. For example, to find the integral of $x^2$ once, write it as a pure function:

f[#^2 &]
(* Function[{a$,b$},-a$^3/3+b$^3/3] *)

f[#^2 &][1,2]
(* 7/3 *)