# Updating the qr factorization and the least squares problem

### Updating the qr factorization and the least squares problem - maintaining purity in dating

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\end$$$$\begin \bigl\Vert \bigr\Vert _ & \leq\tilde_ \left \Vert \begin R_ \ G_ \end \right \Vert _ \tilde_(1 \tilde_) \left \Vert \begin R_ \ G_ \end \right \Vert _ \ & \leq\bigl(\tilde_ \tilde _(1 \tilde_)\bigr) \left \Vert \begin R_ \ G_ \end \right \Vert _ \ & \leq2\tilde_ \left \Vert \begin R_ \ G_ \end \right \Vert _, \end$$$$\begin \Vert \hat_ \Vert _&= \bigl\Vert Q_^ Q_^ \hat _ Q_^ e_ \bigr\Vert _ \ &\leq \bigl\Vert Q_^ \bigr\Vert _ \bigl\Vert Q_^ \bigr\Vert _ \Vert \hat_ \Vert _ \bigl\Vert Q_^ \bigr\Vert _ \Vert e_ \Vert _ \ &\leq \Vert \hat_ \Vert _ \Vert e_ \Vert _ \ &\leq\tilde_ \Vert E_ \Vert _ \tilde _ \Vert U_ \Vert _ \ &\leq[\tilde_ \tilde_]\mbox \bigl( \bigr).Algorithms are presented that compute the factorization Ã = ˜ Q ˜ R where Ã is the matrix A = QR after it has had a number of rows or columns added or deleted.This is achieved by updating the factors Q and R, and we show this can be much faster than computing the factorization of Ã from scratch.Therefore, our main concern is to study the error analysis of the updating steps.For others, such as the effect of using the weighting factor, finding the $$\begin \bigl\Vert \bigr\Vert _&= \Vert e_ e_ \Vert _, \ \bigl\Vert e_^ \bigr\Vert _ &\leq \Vert e_ \Vert _ \Vert e_ \Vert _, \ &\leq\tilde_ \left \Vert \begin R_ \ G_ \end \right \Vert _ \tilde_ \left \Vert \begin\hat_\ \hat_ \end \right \Vert _.\end$$$$\begin \Vert e \Vert _& \leq \Vert e_ \Vert _ \Vert \hat_ \Vert _ \ &\leq\tilde_ \left \Vert \begin R_ \ G_ \end \right \Vert _ (\tilde_ \tilde_)\mbox \bigl( \bigr) \ &\leq(\tilde_ \tilde_ \tilde _)\mbox \left(, \left \Vert \begin R_ \ G_ \end \right \Vert _ \right).

\end$$$$\begin \Vert E-\tilde \tilde \Vert _&= \bigl\Vert (E-Q\tilde) \bigl((Q-\tilde)\tilde\bigr) \bigr\Vert _ \ &\leq\sqrt(\tilde_ \tilde_ \tilde_)\max \left(, \left \Vert \begin R_ \ G_ \end \right \Vert _ \right).

biglm has an updating capability when adding observations, but my data are small enough to reside in memory (although I do have a large number of instances to update).

There are ways to do this with bare hands, e.g., to update the QR factorization (see "Updating the QR Factorization and the Least Squares Problem", by Hammarling and Lucas), but I am hoping for an existing implementation.

We also illustrate the implementation and accuracy of the proposed algorithm by providing some numerical experiments with particular emphasis on dense problems. It arises in important applications of science and engineering such as in beam-forming in signal processing, curve fitting, solutions of inequality constrained least squares problems, penalty function methods in nonlinear optimization, electromagnetic data processing and in the analysis of large scale structure [) can be obtained using direct elimination, the nullspace method and method of weighting.

In direct elimination and nullspace methods, the LSE problem is first transformed into unconstrained linear least squares (LLS) problem and then it is solved via normal equations or ].

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