Modified Gram-Schmidt

Orthogonalizing by row instead of column

Algorithm

Instead of processing each original vector $a_j$ entirely, MGS computes $q_i$ and immediately projects it out of all remaining vectors $v_j$ ( $j > i$ ). This updating of the remaining pool of vectors improves numerical stability compared to Classical Gram-Schmidt.

Step-by-Step Construction

Step 0 of 6

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Initial vectors

v_1=a_1

v_2=a_2

v_3=a_3

R Matrix Construction

R = \begin{bmatrix} \hphantom{r_{11}}\llap{0} & \hphantom{r_{12}}\llap{0} & \hphantom{r_{13}}\llap{0} \\[0.5em] \hphantom{r_{21}}\llap{0} & \hphantom{r_{22}}\llap{0} & \hphantom{r_{23}}\llap{0} \\[0.5em] \hphantom{r_{31}}\llap{0} & \hphantom{r_{32}}\llap{0} & \hphantom{r_{33}}\llap{0} \end{bmatrix}

Entries $r_{ij} = q_i^T v_j^{(i)}$ and $r_{ii} = \|v_i^{(i)}\|$

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