How to vectorize matrixvector multiplications with previous-dependent input?

How to avoid the for loop in the following code, creating a 2D polymer structure with a new random created bending angle of the following segment:

N=10;
K=200;
Xi=50;
l=0.311;
Fi=normrnd(0,sqrt(l/Xi),[1,1,K])
dna=zeros(2,K);
t=ones(2,K);
for i=2:K
    A=[cos(Fi(i)) -sin(Fi(i));sin(Fi(i)) cos(Fi(i))]
    t(:,i)=A*t(:,i-1);
    dna(:,i)=dna(:,i-1)+t(:,i);
end

Answer 1

[Roger Stafford]

The for-loop can be eliminated, but I doubt that it will save you any time.

   K = 200;
   Xi = 50;
   l = 0.311;
   t = [rand(2,1),zeros(2,K-1)]; % <-- You choose t(:,1)
   Fi = normrnd(0,sqrt(l/Xi),[1,1,K])
   Gi = cumsum(reshape(Fi(1,1,2:K),1,[]),2); % cumsum the angles in Fi
   sG = sin(Gi);
   cG = cos(Gi);
   t(1,2:K) = cG*t(1,1)-sG*t(2,1);
   t(2,2:K) = sG*t(1,1)+cG*t(2,1);
   dna = [zeros(2,1),cumsum(t(:,2:K),2)];

Answer 2

[Stephen23]

Most of the operations can be moved out of the loop:

t = ones(2,K);
B = [cos(Fi),-sin(Fi);sin(Fi),cos(Fi)];
for k = 2:K
    t(:,k) = B(:,:,k) * t(:,k-1);
end
t(:,1) = 0;
dna = cumsum(t,2);

but because of the mtimes * the t calculation is easiest to do in a loop.