Problems with DumpSave and memoization

Question

I defined a function that takes some time to compute using memoization:

y[a_, b_, c_] := y[a, b, c] = First[$y /. NDSolve[.........] 

so that if I call y[a,b,c] a second time it doesn't do the NDSolve computation all over again if it has already done it.

How can I save the result of this to file? I tried DumpSave["test.mx", y[3/10, 2, 0]] after calling y[3/10, 2, 0] (i.e. it is now an InterpolatingFunction; the result of NDSolve), but I get the error

"DumpSave::bsnosym: y[3/10,2,0] is not defined as a symbol or a context. >>"

If I call y[3/10,2,0][2] etc I get the correct numeric result.

Is there some reason this is not working?

Answer 1

The reason this isn't working is that DumpSave expects a symbol as the second argument. The doc page says:

DumpSave["file.mx",symbol] writes definitions associated with a symbol to a file in internal Mathematica format.

With

ClearAll[y]
y[a_, b_, c_] := y[a, b, c] =  NDSolve[{y''[x] == a x, y[0] == b, y'[0] == c}, y[x], {x, 0, 10}]

and

y[3/10, 2, 0]

{{y[x] -> InterpolatingFunction[][x]}}

you can see that y is a symbol

y // Head

Symbol

whereas y[3/10, 2, 0] is not:

 y[3/10, 2, 0] // Head

List

It is the expression {{y[x] -> InterpolatingFunction[][x]}} as we have seen above.

To save expressions, such as a single instantiations of your memoized function, you can use Put (>>).

y[3/10, 2, 0] >> "test.mx"

Clear[y]
<<"test.mx"

{{y[x] -> InterpolatingFunction[][x]}}

To save the whole of y, basic definition and all memoized versions, you can DumpSave y itself:

DumpSave["test.mx", y];
Clear[y]
<<"test.mx"
?y

y[3/10,2,0]={{y[x]->InterpolatingFunction[{{0.,10.}},{4,23,2,{27},{4},0,0,0,0,Automatic},{{0.,0.0002208643237,<<...>>,9.481521054,10.}},{Developer`PackedArrayForm,{0,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81},{2.,<<...>>,3.}},{Automatic}][x]}}

y[a_,b_,c_]:=y[a,b,c]=NDSolve[{(y^[Prime][Prime])[x]==a x,y[0]==b,(y^[Prime])[0]==c},y[x],{x,0,10}]

An extensive treatment of all ways to save data for posterity can be found here.

---

EDIT

To conveniently select a specific memoized definition you could use the following:

dumpSaveSelect[y_] :=
 DynamicModule[{sol, file},
  Manipulate[
   (DownValues[y][[i, 1]]) /. 
    HoldPattern -> HoldForm, {{i, 1, "DownValue:"}, 1, 
    Length[DownValues[y]], 1, ControlType -> SetterBar},
   Button["DumpSave",
    sol = DownValues[y][[i, 1]];
    file = SystemDialogInput["FileSave", "myfile.mx"];
    DumpSave[file, sol],
    Method -> "Queued"
    ],
   ControlPlacement -> {Top, Bottom}
   ]
  ]

dumpSaveSelect[y]

Answer 2

Enter the following

In[1]:= sol=NDSolve[{y′[x]==y[x],y[1] == 2},y,{x, 0, 3}][[1, 1]]

Out[1]= y->InterpolatingFunction[{{0.,3.}},<>]

In[2]:= DumpSave["nds.mx",sol]

Out[2]= {y->InterpolatingFunction[{{0.,3.}},<>]}

Exit Mathematica

Start Mathematica, and run this

In[1]:= Get["nds.mx"] (* or use two < and no quotes and nds.mx *)

In[2]:= y[1]/.sol

Out[2]= 2.