Using a variable vector in a loop

Hello, I am trying to use a large number of vectors (up to 60) for orthogonalization. Say, we have column vectors, v1, v2, v3,..., vk. To orthogonalize them using the standard method, one can proceed with

u1=v1;

u2=v2-[dot(v2,u1)/dot(u1,u1)]*u1;

u3=v3-[dot(v3,u1)/dot(u1,u1)]*u1-[dot(v3,u2)/dot(u2,u2)]*u2;

and so on. However, I want to use the generalized formula when the number of vectors gets large.

What would be best way to implement the above general formula rather? I was wondering if a "for" loop might work, as follows, but how to define vk? I get an error. vk undefined. Thanks.

for k=1:60;

j=1:k-1;

uk=0;

uk=GS+vk-[dot(vk,uj)/dot(uj,uj)]*uj;

end

Answer 1

[James Tursa]

Suppose you store the column vectors in a matrix called V. Then vk would simply be V(:,k). And if the uj are stored in a matrix U as well, then each uj would be U(:,j). So taking your code and replacing these, along with coding the inner part as a loop, would be something like:

V = your matrix of vk column vectors
[m,n] = size(V);
U = zeros(m,n);
for k=1:n
    p = zeros(m,1);
    for j=1:k-1
        p = p + (dot(V(:,k),U(:,j))/dot(U(:,j),U(:,j))) * U(:,j);
    end
    U(:,k) = V(:,k) - p;
end

[FW]

Thank you. I will try this approach. I gather that the columns of U will be populated in each cycle? We need to generate orthogonal uk vectors from given vk vectors.

Answer 2

[Matt J]

You should not use the classical Gram-Schmidt procedure, as it is numerically unstable. Use orth() instead.

[FW]

Thanks for letting me know about the orth option. I will try both.

[FW]

The output of the orth is completely different from the classical GS. However the vectors are orthonormal.