how to vectorze this kind of loop for array - matlab

Question

I'm newbie to Matlab. I have some questions.
How can I vectorise this loop:
epsilon = 0.45;
t = -3.033;
n = 100;
I = ones(n,n);
%// diagonal block 1
DB1 = gallery('tridiag',ones(1,n-1),ones(1,n),ones(1,n-1));
for k = 1:n
DB1(k,k) = epsilon;
end
for k = 1:n-1
DB1(k,k+1) = t*heaviside((-1)^(k+1));
end
for k = 2:n
DB1(k,k-1) = t*heaviside((-1)^k);
and this loop with LRG, R2, R1 are 3D arrays
for k = 2:N
LRG(:,:,k) = inv(R(:,:,k) - R2(:,:,k-1)*LRG(:,:,k-1)*R1(:,:,k-1));
end
Is there any way to handle the third dimension of an array (page) without writing (:,:,...) many times?

Answer 1

You don't need to call gallery: this will just initialize DB1 to a specific sparse tridiagonal matrix, which you manually overwrite afterwards. With n=100, I suspect that you don't actually want to work on sparse matrices. Here's the solution:
epsilon = 0.45; t = -3.033;
n = 100;
DB1 = zeros(n); %initialize to 0
%diagonal indices
inds=sub2ind([n n],1:n,1:n);
DB1(inds) = epsilon;
%superdiagonal
inds=sub2ind([n n],1:n-1,2:n);
DB1(inds) = t*heaviside((-1).^(2:n));
%subdiagonal
inds=sub2ind([n n],2:n,1:n-1);
DB1(inds) = t*heaviside((-1).^(2:n));
Here sub2ind lets you compute linear indices from your matrix indices, allowing you to access "subvectors" in your matrix. Watch out for the exponentialization in the heaviside: you need .^ instead of ^ to perform an element-wise operation using the vector 2:n.
If you insist on getting a sparse matrix, you could either convert this matrix to sparse by calling
DB1=sparse(DB1);
which is the easiest option if your matrix is small. If n was larger and you would really need sparse matrices, then you could set up the sparse matrix itself by
DB1new=sparse([1:n 1:n-1 2:n],[1:n 2:n 1:n-1],...
[epsilon*ones(1,n) t*heaviside((-1).^(2:n)) t*heaviside((-1).^(2:n))],n,n);
This call sets the elements of DB1new one-by-one according to its vector arguments: it takes the first index from the first input argument, the seconf index from the second, and the corresponding element from the third vector (see help sparse in MATLAB).
As for your second question: I'm not sure I understand correctly. Like in my example, sub2ind can be used to transform the multidimensional indices into linear ones, see help sub2ind. But personally, I think using the array notation is far more transparent and less prone to typos. And I think you can't vectorize that loop: you have to explicitly calculate the inverse matrix for each k.