Using a variable vector in a loop

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FW
FW on 3 Dec 2021
Commented: FW on 4 Dec 2021
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

Accepted Answer

James Tursa
James Tursa on 3 Dec 2021
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
  1 Comment
FW
FW on 4 Dec 2021
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.

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More Answers (1)

Matt J
Matt J on 4 Dec 2021
You should not use the classical Gram-Schmidt procedure, as it is numerically unstable. Use orth() instead.
  2 Comments
FW
FW on 4 Dec 2021
The output of the orth is completely different from the classical GS. However the vectors are orthonormal.

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