how to Vectorize iterative '' coordinates transformation ''loop operation for speed

I want to transform local coordinates(tip positions) of a rotating blade into corresponding global coords. Tip coordinates and angular distance changes in time. which means there is a unique transformation matrix for every time step. hence i used loop to multiply transformation matrix (3*3) with local coords to get me global coords each time step. for accuracy reasons i need more than million transformations,which take more than 5 hours with my core i5 computer.its a 2D transformation. i need to vectorize only following part of the code.Please check attachment for complete code.

for i = 1: vec_size

M=[cos(theta_inst(i)) -sin(theta_inst(i)) dx(i);...
       sin(theta_inst(i)) cos(theta_inst(i)) dy(i); ...
         0 0 1];               %transformation matrix

glbl(1:3,i)= M*local(:,i);   % dx,dy ~=0;  

end

Answer 1

[Matt J]

In non-homogeneous coordinates,

c=cos(theta_inst(:)).'; s=sin(theta_inst(:)).';
dx=dx(:).'; dy=dy(:).';
L=local(1:2,:);

glbl = [ sum([c;-s].*L) ; sum([+s;c].*L) ]+[dx;dy] ;

[kamal wisal]

Perfect...Thank you @Matt J. I compared results of both loop and vectorized operations. there is a difference of order 2^-16 to 2^-18 in both results. Is it because of using SUM ftn??? may be sum ftn will round off digits and hence creating a difference?

[Matt J]

Relative error?

[kamal wisal]

yes. difference of outputs (global coords) as calculated by iterative process (for loop) and by vectorized operation using sum function.

[Matt J]

If it's relative error, it is likely due simply to floating point errors.