How to speed this up - large variable

Hello,

Is there a way to speed this up? This if within a function and i can use codegen - and I do. But I need to speed this up more. I think the problem is that one variable is huge and the access to it takes long? What should be done in such a case?

xElements = 1201;
maxN = 100;
umnHolder = complex(zeros(maxN + 1, maxN + 1));
betaSumSq1 = zeros(xElements, xElements); % preallocate

besselsFisher = zeros(1201, 1201, 101); % just to show the size     LARGE, ~780 MB
XY = zeros(xElements, xElements);       % just to show the size
acosContainer = XY;                     % just to show the size

parfor i = 1 : xElements
    for j = 1 : xElements
        umn = umnHolder;
        for n = 0:maxN
            mm = 1;
            for m = -n:2:n
                nn = n + 1; % for indexing

                  if m > 0
                      umn(nn, mm) = sqrt(n+1) * XY(i, j) * besselsFisher(i, j, nn) * cos( abs(m)*acosContainer(i, j) );
                  end

                  if m < 0
                      umn(nn, mm) = sqrt(n+1) * XY(i, j) * besselsFisher(i, j, nn) * sin( abs(m)*sign(x(i))*acosContainer(i, j) );
                  end

                  if m == 0
                      umn(nn, mm) = sqrt(n+1) * XY(i, j) * besselsFisher(i, j, nn);
                  end  
                  mm = mm + 1;

              end % m
          end % n

          beta1 = sum(sum(Aj1.*umn));
          betaSumSq1(i, j) = abs(beta1).^2;

          beta2 = sum(sum(Aj2.*umn));
          betaSumSq2(i, j) = abs(beta2).^2;

      end % j
  end % i

Best regards, Alex

[Walter Roberson]

Any hope for a meaningful speed-up is probably going to depend upon the extent to which the unknown function besselsFisher can be vectorized. I do not find that routine anywhere the internet.

[Alex Kurek]

besselsFisher i something I wrote. It is precomputed before, so in the loop I only refer to already computed values.

[Thorsten]

The first step before optimising would be to identify where most of the time is spent using profile.

[Alex Kurek]

Most of the time is spent here:

 umn(nn, mm) = sqrt(n+1) * XY(i, j) * besselsFisher(i, j, nn) * cos( abs(m)*acosContainer(i, j) );

 umn(nn, mm) = sqrt(n+1) * XY(i, j) * besselsFisher(i, j, nn) * sin( abs(m)*sign(x(i))*acosContainer(i, j) );

This makes sense, since if you multiply

 besselsFisher = zeros(1201, 1201, 101);

by e.g. 2 it takes ~1.6 seconds.

Answer 1

[Walter Roberson]

You should factor out common sub-expressions. acosContainer(i, j) is the same for all m and n so assign it to a variable outside the m loop. Taking abs(m) is a waste of time when you know that m > 0 . sign(x(1)) is the same for all j, m, n so assign it to a variable. Multiplying by sign(x(1)) is done for the vector -n to -1 so you can vectorize to precalculate, sin((-n : 2 : -1) .* sign(x(1)) .* acosContainer(i, j)); you can probably vectorize the rest of that case as well.

All of the cases for any one n are multiplied by sqrt(n+1) so hold off on that multiplication until you have done the entire set of m values, and then multiply them all by sqrt(n+1) to get economy of scale.

And so on.