To add to what Daniel has already written, consider these lines:
function obj=lu_apply(A)
[L,U]=lu(A);
obj.f=@(X) U\(L\X);
end
The L and U matrices get created inside the function lu_apply. They are local to the function and will get destroyed once the function returns. The following line:
obj.f=@(X) U\(L\X);
creates a function handle using the local L and U variables as part of the function definition. At that time, shared data copies of L and U are created and become embedded in the data area of the function handle ... i.e., they essentially become part of the function handle variable itself. Basically these shared data copies of L and U become snapshots of the current state of L and U at the time of the function handle creation and will remain as constants as far as the function handle is concerned.
Whatever happens to L and U downstream doesn't affect this function handle anymore. In particular, when the function returns, the local variables L and U are destroyed, but the shared data copies of them that were created earlier still live on (and use memory) as part of the function handle. The only way to clear this memory is to clear the function handle itself. I don't know of any way to directly interrogate a function handle to determine the sizes of the embedded variables it contains, since I am not familiar with how function handle definitions are encoded in the data area of the variable.
