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Wir versuchen wieder $||r||_2^2$ zu minimieren, mit $r = Ax - b$. -Mithilfe der $QR$-Zerlegung lässt sich ein Lösungsansatz herleiten, der höhere numerische Stabilität aufweist, als die Normalengleichungen. +Mithilfe der $QR$-Zerlegung lässt sich ein Lösungsansatz herleiten, der höhere numerische Stabilität aufweist als die Normalengleichungen. Sei $A = QR = Q \begin{bmatrix} \tilde{R} \\ 0 \end{bmatrix}$. -Dann, nach Umformungen erhalten wir $||Rx - \tilde{b}||_2^2$ mit $\tilde{b} = Q^Hb$. +Dann, nach Umformungen, erhalten wir $||Rx - \tilde{b}||_2^2$ mit $\tilde{b} = Q^Hb$. Nutzt man beispielsweise Housholder-Spiegelungen zur Berechnung der $QR$-Zerlegung, so kann man die Transformationen direkt auf $b$ anwenden und so kann man sich das Abspeichern der Matrix $Q$ komplett sparen.