MATLAB: The Gram-Schmidt algorithm for unknown n

algorithmgrammatrixschmidtvector

Hiya,

I’m trying to understand how to carry out the gram Schmidt algorithm when you have an unknown number of vectors n.

Any help would be appreciated.

Best Answer

V is matrix input where vectors are column vectors. Output U is matrix with replacement vectors as column vectors.

n = size(V,1);
k = size(V,2);
U = zeros(n,k);
U(:,1) = V(:,1)/sqrt(V(:,1)'*V(:,1));
for i = 2:k
    U(:,i) = V(:,i);
    for j = 1:i-1
        U(:,i) = U(:,i) - ( U(:,j)'*U(:,i) )/( U(:,j)'*U(:,j) )*U(:,j);
    end
    U(:,i) = U(:,i)/sqrt(U(:,i)'*U(:,i));
end

Related Solutions

MATLAB: How to vectorize three for loop or fast

Avoid repeated calculations.
Process the data in column order.

Ur = zeros(291,243,1459);
[NR, NC]=size(lat);
for j = 2:NC-1
   latj = lat(j) - lat(j-1)^2;
   for i = 2:NR-1
      loni = lon(i) - lon(i-1)^2;
      for k = 1: length(t)
         Ur(k,i,j) = sqrt((u(i+1,j,k) - 2 * u(i,j,k) + u(i-1,j,k)) ./ loni ...
                        + (v(i,j+1,k) - 2 * v(i,j,k) + v(i,j-1,k)) ./ latj);
      end
   end
end

If you provide all input arguments, e.g. created by some rand() command, we could check the speed. I assume the inner loop is easy to vectorize:

Ur = zeros(291,243,1459);
[NR, NC]=size(lat);
for j = 2:NC-1
   latj = lat(j) - lat(j-1)^2;
   for i = 2:NR-1
      loni = lon(i) - lon(i-1)^2;
      Ur(:,i,j) = sqrt((u(i+1,j,:) - 2 * u(i,j,:) + u(i-1,j,:)) ./ loni + ...
                       (v(i,j+1,:) - 2 * v(i,j,:) + v(i,j-1,:)) ./ latj);
   end
end

A further vectorization is possible, but I hesitate to implement it: It looks strange, that you treat lat as 2D in [NR, NC]=size(lat), but access the first column only by using 1 index later. Does the code really do, what you want?

MATLAB: How to create an orthogonal basis for the column space of a matrix A

Your question sounds like you are looking for:

B(:,i) = B(:,i) - dot(C(:,i),B(:,j))/dot(B(:,j),B(:,j))* B(:,j)
% not    C(:,i)

But it should be possible for you to find the corresponding line in many text books and other implementations. Did you ask Google already?

Btw, the first loop to create C can be omitted:

C = A(:, jb);

Best Answer

Related Solutions

MATLAB: How to vectorize three for loop or fast

MATLAB: How to create an orthogonal basis for the column space of a matrix A

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