Editing Gram–Schmidt process (section)

== Algorithm ==
The following [[MATLAB]] algorithm implements classical Gram–Schmidt orthonormalization. The vectors {{math|'''v'''<sub>1</sub>, ..., '''v'''<sub>''k''</sub>}} (columns of matrix <code>V</code>, so that <code>V(:,j)</code> is the <math>j</math>th vector) are replaced by orthonormal vectors (columns of <code>U</code>) which span the same subspace.

<syntaxhighlight lang="matlab" line="1">
function U = gramschmidt(V)
    [n, k] = size(V);
    U = zeros(n,k);
    U(:,1) = V(:,1) / norm(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);
        end
        U(:,i) = U(:,i) / norm(U(:,i));
    end
end
</syntaxhighlight>

The cost of this algorithm is asymptotically {{math|O(''nk''<sup>2</sup>)}} floating point operations, where {{mvar|n}} is the dimensionality of the vectors.{{sfn|Golub|Van Loan|1996|loc=§5.2.8}}