Define a family of functions

I want to define a family of functions f(r) := @(x,y) x^2+y^2-r^2 for each r>0.

I can define a sequence of functions f(1), ..., f(n) by creating a cell array of size n*1

f = cell{n}
for k=1:n
    f{k} = @(x,y) x.^2+y.^2-n^2
end

However, it is not possible to create an infinite cell array indexed by the set of positive real numbers. I would like to know an alternative way to define family of functions.

Answer 1

[John D'Errico]

DON'T DO IT THAT WAY. Instead, define ONE function. One function to rule them all. (Sorry. That last part just slipped out.)

Seriously. Define one function.

circfamily = @(x,y,r) x.^2 + y.^2 - r.^2;

Any single function in that family is now accessible simply as:

r = 5;
circ_5 = @(x,y) circfamily(x,y,r);

Or you might just do

circ_5 = @(x,y) circfamily(x,y,5);

In any case, we can now use that member of the family.

fimplicit(circ_5)
axis equal

We can even plot the entire family of such functions as a conic surface.

fimplicit3(circfamily,[-5,5,-5,5,0,5])
xlabel X
ylabel Y
zlabel R
grid on
box on
axis equal

There is no need to define the entire set of members of that family in advance. Anyway, you CANNOT explicitly define infinitely many functions, even if you wanted to do so. But essentially circfamily does exactly that implicitly, yielding infinitely many functions.

Answer 2

[Walter Roberson]

If you could define a "family of functions" in MATLAB, what properties would you want such a thing to have? If there were class familyOfFunctions then what methods would you want the class to have?

For example if you define that particular family of functions without providing a specific value for r, then do you need to be able to integral2() the family over specific bounds and get out a family of solutions that you could then instantiate for a particular r to get back the corresponding numeric integral?... and if so then would it be acceptable to layer this all on top of the Symbolic Toolbox, or does it need to operate without the Symbolic Toolbox?

[Quang Huy Pham]

Thanks for your answer. Yes, I want to integral2 these functions over some domains for random values of r. If I could define a family of functions f(r), I don't need to define a new function f inside the loop and directly use integral2(f{r},-1,1,-1,1). But now I realize that it doesn't matter whether f is defined inside or outside the for loop.

for j = 1:5
    r = rand; 
    f = @(x,y) x.^2+y.^2-r^2;
    integral2(f,-1,1,-1,1);
end