tricky memoization

Question

Let's say I have the NDSolve example for documentation involving splitting 2nd order into set of 1st order ODEs:

NDSolve[{x'[t] == y[t], y'[t] == -x[t], x[0] == 1, y[0] == 0}, {x, 
y}, {t, 0, 10}]

I need to solve my second order system like this for various reasons.

My ODE is more complicated, and depends on some other parameters call them $(a,b)$, and so when I solved it as a second order system I used memoization in the form below, so that if a solution for a given parameter set had already been computed, Mathematica wouldn't redo it all again. I would like to do this, or something like it for the first order system and where NDSolve is giving the two variables $x[a,b],y[a,b]$.

x[a_,b_]:=
 x[a,b] = x /.NDSolve[{eq[a,b],x[0]== init[a,b],x'[0]==dinit[a,b],x,{r, 0, 10}][[1]];

Answer 1

When I see this right, you have one single function call (NDSolve) depending on two parameters which results in a vector (the interp. functions). Here is an equivalent minimal example function for this call

f[a_, b_] := {b, a}

The underlying issue is now, that you always get both answers from f (x and y) even if you need only x for a specific choice of a and b. You can solve this by memorizing both, x and y, no matter which function you call.

x[a_, b_] := {x[a, b], y[a, b]} = f[a, b];
y[a_, b_] := {x[a, b], y[a, b]} = f[a, b];

A quit check shows you that you memorize the values for x and y even if you compute only x

Table[x[a, b], {a, 0, 2}, {b, 0, 2}];
?x
?y

Lets try this with your NDSolve example. I only wrapped Module around to localize x and y.

ClearAll[x, y, f];
f[a_, b_] := Module[{x, y}, {x, y} /. 
   First@NDSolve[{x'[t] == y[t], y'[t] == -x[t], x[0] == a, y[0] == b}, 
    {x, y}, {t, 0, 10}]];

x[a_, b_] := {x[a, b], y[a, b]} = f[a, b];
y[a_, b_] := {x[a, b], y[a, b]} = f[a, b];

Table[x[a, b], {a, 0, 2}, {b, 0, 2}];

?y

Finally, the repetition of the exact same function definition for x and y looks kind of like we are not experts. Lets improve this. The only thing which is different is the head of the left side in the :=. So we write the line one time and substitute x and y resp.

SetDelayed[#[a_, b_], {x[a, b], y[a, b]} = f[a, b]] & /@ {x, y}