# Fast updating of the QR factorization qrupdate provides the update of the QR factorization after the addition of $m>1$ rows or columns to the matrix $X\in\mathbb{R}^{n\times p}$ with $n>p$. The QR factorization of the matrix $X$ returns the matrices $Q\in\mathbb{R}^{n\times n}$ and $R\in\mathbb{R}^{n\times p}$ such that $X=QR$. The $Q$ and $R$ matrices are factorized as $Q=\begin{bmatrix}Q_1&Q_2\end{bmatrix}$ and $R=\begin{bmatrix}R_1\\R_2\end{bmatrix}$, with $Q_1\in\mathbb{R}^{n\times p}$, $Q_2\in\mathbb{R}^{n\times (n-p)}$ such that $Q_1^{\top}Q_2=Q_2^\top Q_1=0$ and $R_1\in\mathbb{R}^{p\times p}$ upper triangular matrix and $R_2\in\mathbb{R}^{(n-p)\times p}$. qrupdate accepts in input the matrices $Q$ and either the complete matrix $R$ or the reduced one, $R_1$. See Golub and Van Loan (2013) for further details on the method. ```r qrupdate(Q, R, k, U, type = NULL, fast = NULL, complete = NULL) ``` ## Arguments - `Q`: a $n\times p$ matrix. - `R`: a $p\times p$ upper triangular matrix. - `k`: position where the columns or the rows are added. - `U`: either a $n\times m$ matrix or a $p\times m$ matrix of columns or rows to be added. - `type`: either 'row' of 'column', for adding rows or columns. Default is 'column'. - `fast`: fast mode: disable to check whether the provided matrices are valid inputs. Default is FALSE. - `complete`: logical expression of length 1. Indicates whether an arbitrary orthogonal completion of the $Q$ matrix is to be made, or whether the $R$ matrix is to be completed by binding zero-value rows beneath the square upper triangle. ## Returns A named list containing - **Q**: the updated Q matrix. - **R**: the updated R matrix. ## Examples ```r ## Add one column ## generate sample data set.seed(1234) n <- 12 p <- 5 X <- matrix(rnorm(n * p, 1), n, p) ## get the initial QR factorization output <- qr(X, complete = TRUE) Q <- output$Q R <- output$R ## create column u to be added k <- p+1 u <- matrix(rnorm(n), n, 1) X1 <- cbind(X, u) ## update the QR decomposition out <- fastQR::qrupdate(Q = Q, R = R, k = k, U = u, type = "column", fast = FALSE, complete = TRUE) ## check round(out$Q %*% out$R - X1, 5) max(abs(out$Q %*% out$R - X1)) ## Add m columns ## create data: n > p set.seed(1234) n <- 10 p <- 5 X <- matrix(rnorm(n * p, 1), n, p) ## get the initial QR factorization output <- fastQR::qr(X, complete = TRUE) Q <- output$Q R <- output$R ## create the matrix of two columns to be added ## in position 2 k <- 2 m <- 2 U <- matrix(rnorm(n*m), n, m) X1 <- cbind(X[,1:(k-1)], U, X[,k:p]) # update the QR decomposition out <- fastQR::qrupdate(Q = Q, R = R, k = k, U = U, type = "column", fast = FALSE, complete = TRUE) ## check round(out$Q %*% out$R - X1, 5) max(abs(out$Q %*% out$R - X1)) ## Add one row ## create data: n > p set.seed(1234) n <- 12 p <- 5 X <- matrix(rnorm(n * p, 1), n, p) ## get the initial QR factorization output <- fastQR::qr(X, complete = TRUE) Q <- output$Q R <- output$R R1 <- R[1:p,] ## create the row u to be added u <- matrix(data = rnorm(p), p, 1) k <- n+1 if (k<=n) { X1 <- rbind(rbind(X[1:(k-1), ], t(u)), X[k:n, ]) } else { X1 <- rbind(rbind(X, t(u))) } ## update the QR decomposition out <- fastQR::qrupdate(Q = Q, R = R, k = k, U = u, type = "row", complete = TRUE) ## check round(out$Q %*% out$R - X1, 5) max(abs(out$Q %*% out$R - X1)) ## Add m rows ## create data: n > p set.seed(1234) n <- 12 p <- 5 X <- matrix(rnorm(n * p, 1), n, p) ## get the initial QR factorization output <- fastQR::qr(X, complete = TRUE) Q <- output$Q R <- output$R R1 <- R[1:p,] ## create the matrix of rows U to be added: ## two rows in position 5 m <- 2 U <- matrix(data = rnorm(p*m), p, m) k <- 5 if (k<=n) { X1 <- rbind(rbind(X[1:(k-1), ], t(U)), X[k:n, ]) } else { X1 <- rbind(rbind(X, t(U))) } ## update the QR decomposition out <- fastQR::qrupdate(Q = Q, R = R, k = k, U = U, type = "row", complete = FALSE) ## check round(out$Q %*% out$R - X1, 5) max(abs(out$Q %*% out$R - X1)) ``` ## References \insertRef golub_van_loan.2013fastQR \insertRef bjorck.2015fastQR \insertRef bjorck.2024fastQR \insertRef bernardi_etal.2024fastQR

qrupdate function