Create vector-valued function with arbitrary components

Question

Good evening everyone,

I want to create a function

f(x) = [f1(x), f2(x), ... , fn(x)]

in MatLab, with an arbitrary form and number for the fi. In my current case they are meant to be basis elements for a finite-dimensional function space, so for example a number of multi variable polynomials. I want to able to be able to set form (e.g. hermite/lagrange polynomials, ...) and number via arguments in some sort of "function creating" function, so I would like to solve this for arbitrary functions fi.
Assume for now that the fi are fi:R^d -> R, so vector input to scalar output. This means the result from f should be a n-dim vector containing the output of all n functions. The number of functions n could be fairly large, as there is permutation involved. I also need to evaluate the resulting function very often, so I hope to do it as efficiently as possible.

---

Currently I see two ways to do this:

1. Create a cell with each fi using a loop, using something like
   funcell{i}=matlabFunction(createpoly(degree, x),'vars',{x})
   and one of the functions from the symbolic toolbox and a symbolic x (vector). It is then possible to create the desired function with cellfun, e.g.
   f=@(x) cellfun(@(v) v(x), funcell)
   This is relatively short, easy and what can be found when doing searches. It even allows extension to vector output using 'UniformOutput',false and cell2mat. On the downside it is very inefficient, first during creation because of matlabFunction and then during evaluation because of cellfun.

2. The other idea I had is to create a string and use eval. One way to do this would be
   stringcell{i}=[char(createpoly(degree, x)),';']
   and then use strjoin. In theory this should yield an efficient function. There are two problems however. The first is the use of eval (mostly on principle), the second is inserting the correct arguments. The symbolic toolbox does not allow symbols of the form x(i), so the resulting string will not contain them either. The only remedy I have so far is some sort of string replacement on the xi that are allowed, but this is also far from elegant.

---

So I do have ways to do what I need right now, but I would appreciate any ideas for a better solution.

Answer 1

From my understanding of the problem, you could do the straightforward:

Initialization step:

my_fns = cell(n, 1);  %where n is number of functions
my_fns{1} = @f1;   % Assuming f1 is defined in f1.m etc...
my_fns{2} = @f2;

Evaluation at x:

z = zeros(n, 1);
for i=1:n,
   z(i) = my_fns{i}(x)
end

For example if you put it in my_evaluate.m:

function z = my_evaluate(my_fns, x)
  z = zeros(n, 1);
  for i=1:n,
    z(i) = my_fns{i}(x)
  end      

How might this possibly be sped up?

Depends on if you have special structure than can be exploited.

• Are there calculations common to some subset of f1 through fn that need not be repeated with each function call? Eg. if the common calculation step is costly, you could do y = f_helper(x) and z(i) = fi(x, y).
• Can the functions f1...fn be vector / matrix friendly, allowing evaluation of multiple points with each function call?

The big issue is how fast your function calls f1 through fn are, not how you collect the results from those calls in a vector.