MATLAB: How can i vectorize this for loop? please help.

vectorization

t = 0:dt:T;
x = xl:dx:xr;
u = zeros(xn+1,tn+1);
for j = 1:tn;
  tm = j*dt;
  if j == 1; 
    for i = 2:xn;
      T1 = ((gamma(2-bta))/((gamma(j-1+2))*(gamma(1-bta-j+1))))*(u(i+1,1)-2*u(i,1)+u(i-1,1));
      T2 = (2.5*(tm.^1.5)+(gamma(3.5)/gamma(3.5-bta))*(tm.^(2.5-bta)))*sin(x(i));
      u(i,j+1)=u(i,j)+mu.*T1+dt.*(T2);
    end
  end
end

Best Answer

Start with moving all repeated but constant expressions outside the loops:
t = 0:dt:T;
x = xl:dx:xr;
u = zeros(xn+1,tn+1);
c1 = gamma(2-bta);
c4 = gamma(3.5) / gamma(3.5-bta);
for j = 1:tn;
  tm = j*dt;
  c5 = tm.^(2.5-bta);
  c6 = tm.^1.5;
  c7 = 2.5*c6 + c4*c5;
  if j == 1   % ??? Why do you use a loop over j, if only j==1 is considered? 
    c2 = gamma(j-1+2);
    c3 = gamma(1-bta-j+1);
    c8 = (c1 / (c2*c3));
    for i = 2:xn
      T1 = c8 * (u(i+1,1) - 2*u(i,1) + u(i-1,1));
      T2 = c7 * sin(x(i));
      u(i,j+1) = u(i,j) + mu.*T1+dt.*T2;
    end
  end
end
Now the innerloop can be vectorized by moving the index from the for loop insice the calculations: Instead of the block "for i... end"
        T1 = c8 .* (u(3:xn+1,1) - 2 * u(2:xn,1) + u(1:xn-1,1));
        T2 = c7 .* sin(x(2:xn));
        u(2:xn,j+1) = u(2:xn,j) + mu .* T1 + dt .* T2;
I cannot test the code, because some variables are missing.
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