Modified Gram-Schmidt ($w\leftarrow w-⟨w,q_i⟩q_i$)

Modified Gram-Schmidt starts with $w=v_j$ and, for each retained orthonormal vector $q_i$ in order, replaces $w$ by $w-⟨w,q_i⟩q_i$. 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 $\Vert w\Vert$ if it is nonzero.