Question

I want to vectorize this double for loop because it is a bottleneck in my code. Since Matlab is a based-one indexing language I have to create an additional term for M = 0.

R,r,lambda are constants

Slm(L,M),Clm(L,M) are matrices 70×70

Plm(L,M) is a matrix 70×71

Cl(L),Pl(L) are vectors 70×1

% function dU_r
  s1 = 0;
  for L = 2:70
     s1 = s1 + ((R/r)^L)*(L+1)*Pl(L)*Cl(L);
     for m = 1:L
        s1 = s1 + ((R/r)^L)*(L+1)*Plm(L,M)*(Clm(L,M)*...
cos(M*lambda) + Slm(L,M)*sin(M*lambda));
     end
  end

  dU_r = -(mu_p/(r^2))*s1;


  % function dU_phi

  s2=0;
  for L = 2:70
      s2 = s2 + ((R/r)^L))*Plm(L,1)*Cl(L);
          for m = 1:l
          s2 = s2 + ((R/r)^L)*(Plm(L,M+1)-M*tan(phi)*Plm(L,M))*...
  (Clm(L,M)*cos(M*lambda) + Slm(L,M)*sin(M*lambda));
          end;
  end;
  dU_phi = (mu_p/r)*s2;

  % function dU_lambda

  s3=0;
      for L=2:70
          for m=1:L
          s3 = s3 + ((R/r)^L)*M*Plm(L,M)*(Slm(L,M)*cos(M*lambda)...
              - Clm(L,M)*sin(M*lambda));
          end;
  dU_lambda = (mu_p/r)*s3;

Answer 1

I think the following code may be a part of the solution (only partially vectorized), for a full vectorization you might want to look at meshgrid, ndgrid, bsxfun and/or repmat, I suppose.

s2     = 0;
L      = 2:70;
PlCl   = Pl.*Cl;
PlmClm = Plm.*Clm;
Rrl    = (R/r).^(L).*(L+1);
COS    = cos((1:70)*lambda);
SIN    = sin((1:70)*lambda);

for l=L
    M  = 1:l;
    s1 = sum(Rrl(l-1)*PlmClm(l,M).*(COS(M) + Slm(l,M).*SIN(M)));
    s2 = s2 + s1;
end;
s2 = s2 + sum(Rrl(L).*PlCl(L));

I haven’t tried to run this code, so if it may complain about some dimensions. You should be able to figure that out (just some transposes here or there may do).

Just a note on the side: try to keep away from short variable names (and certainly ambiguous ones like l (that can be misread for a 1, a I or an l). Also, it might be easier to vectorize your code if you have actual (i.e. not coded) expressions to start with.

edit: applied corections suggested by gnovice