Why is this line so inefficient

I have a very simple equation that I'm coding in MATLAB that I wouldn't think should be taking as much time as it is. The initial line is in a for loop so I vectorized the problem and that turned out to be even slower (which I was pretty surprised by). Does anyone see anything obvious that I'm not considering for why this is so slow? Typically values for n and m are between 0-10

%    Rnm = zeros(size(rho));
%    MaximumExponent = (n-m)/2;
%    s = 0:MaximumExponent;
%    numerator = (-1) .^ s .* factorial(n-s);
%    denominator = factorial(s) .* factorial((n+m) / 2-s) .* factorial((n-m) / 2-s);
%    Rnm = sum((numerator ./ denominator) .* rho .^ (n-2 .* s), 2);

   Rnm2 = zeros(size(rho));
   for s=0:(n-m)/2
     numerator = (-1)^s * factorial(n-s);
     denominator = factorial(s)*factorial((n+m)/2-s)*factorial((n-m)/2-s);
     Rnm2 = Rnm2 + (numerator / denominator) * rho.^(n-2*s);  
   end % for s statement

Here is a profile for the runtime...

EDIT: I should probably also add a couple more details on some things. rho will be a column vector of ~500,000 elements. I imagine the size of rho is alone the reason that most of this takes so long, but it still seems a bit abnormally long to me.

EDIT 2: Should probably include my system details...running Windows 10 Pro, 11th Gen Intel Core i5-1135G7 @ 2.4GHz, 16GB RAM. Nothing to fantastic, but not the worst PC in the world. That's why I'd think this should be running faster...

[dpb]

What's rho?

[John Gilmore]

Please see my edit.

Answer 1

[David Hill]

tic;
rho=rand(500000,1);n=9;m=3;
s=0:(n-m)/2;
numerator = (-1).^s .* factorial(n-s);
denominator = factorial(s).*factorial((n+m)/2-s).*factorial((n-m)/2-s);
Rnm=sum((numerator./denominator).*rho.^(n-2*s),2); 
toc;

Elapsed time is 0.109334 seconds.

[John Gilmore]

Now that's interesting...this is sort of what my question is implying I suppose. for you this takes ~30 seconds for 280 function calls. Which is about 300% faster than mine. Is that drastic of a speed increase really just do to our different systems?

EDIT: See original post EDIT2 for system specs

[dpb]

>> tic;
rho=rand(500000,1);n=9;m=3;
s=0:(n-m)/2;
numerator = (-1).^s .* factorial(n-s);
denominator = factorial(s).*factorial((n+m)/2-s).*factorial((n-m)/2-s);
Rnm=sum((numerator./denominator).*rho.^(n-2*s),2); 
toc;
Elapsed time is 0.094052 seconds.
>> 

What else do you have in memory? Must be something else going on for it to be more than a fraction of a second with above parameters.

What does

whos

show?

Processor Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz, 3601 Mhz, 4 Core(s), 8 Logical Processor(s)

[John Gilmore]

While the script isn't running? Running that command shows nothing.

[David Hill]

tic;
rho=rand(500000,1);n=9;m=3;
f=factorial(0:n);%could do lookup table for factorials (might help speed slightly)
s=0:(n-m)/2;
numerator = (-1).^s .* f(n-s+1);
denominator = f(s+1).*f((n+m)/2-s+1).*f((n-m)/2-s+1);
Rnm=sum((numerator./denominator).*rho.^(n-2*s),2); 
toc;

Elapsed time is 0.102554 seconds.