= Modified Gram-Schmidt
{c}
{title2=$w\leftarrow w-\langle w,q_i\rangle q_i$}
<Modified Gram-Schmidt> starts with $w=v_j$ and, for each retained <orthonormal> <vector> $q_i$ in order, replaces $w$ by $w-\langle w,q_i\rangle q_i$. Each <orthogonal projection> <coefficient> uses the current remainder. In exact arithmetic, subtracting earlier <orthogonal> directions does not change the <inner product> with $q_i$, so the final $w$ equals the <Classical Gram-Schmidt> remainder $u_j$; then divide by $\|w\|$ if it is nonzero.
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