I’m a little curious about this. Does J have a notion of the relationship between certain functions and their inverse? What is it that enables “under” in J which makes it impossible in Matlab?
Yes. Many built-in words have inverses assigned, and you can assign inverse functions to your own words with :. https://code.jsoftware.com/wiki/Vocabulary/codot
EDIT: and here's a table with predefined inverses: https://code.jsoftware.com/wiki/Vocabulary/Inverses
But interesting nonetheless.
>> syms x
>> f = @(x) log(sqrt(x)).^2
f = function_handle with value:
@(x)log(sqrt(x)).^2
>> f(x)
ans = log(x^(1/2))^2
>> finverse(f(x))
ans = exp(2*x^(1/2))
function u = under(f, g)
syms x
g_inv = matlabFunction(finverse(g(x)));
u = @(x) g_inv(f(g(x)));
end function y=odddouble(a,b)
y=2*x+1
endfunction
>>> h = under(@(x) 1/x, odddouble)
>>> h(3)
It is definitely not as elegant as the built-in facility in J, but definitely doable and usable in Matlab. In fact, I think any language with flexible enough function overloading should be able to implement such a feature.
f=:(1+*&2)
f 1 2 3 4
3 5 7 9
(f^:_1)f 1 2 3 4
1 2 3 4
(f^:_1) 1 2 3 4
0 0.5 1 1.5