# Fast updating of the R matrix updates the R factorization when $m \geq 1$ rows or columns are added to the matrix $X \in \mathbb{R}^{n \times p}$, where $n > p$. The R factorization of $X$ produces an upper triangular matrix $R \in \mathbb{R}^{p \times p}$ such that $X^\top X = R^\top R$. For more details on this method, refer to Golub and Van Loan (2013). Columns can only be added in positions $p+1$ through $p+m$, while the position of added rows does not need to be specified. ## Arguments - `R`: a $p\times p$ upper triangular matrix. - `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. - `fast`: fast mode: disable to check whether the provided matrices are valid inputs. Default is FALSE. ## Returns 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 <- fastQR::qr(X, complete = TRUE) Q <- output$Q R <- output$R R1 <- R[1:p,] ## create column to be added u <- matrix(rnorm(n), n, 1) X1 <- cbind(X, u) ## update the R decomposition R2 <- fastQR::rupdate(X = X, R = R1, U = u, fast = FALSE, type = "column") ## check max(abs(crossprod(R2) - crossprod(X1))) ## Add m columns ## generate sample data 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 R1 <- R[1:p,] ## create the matrix of columns to be added m <- 2 U <- matrix(rnorm(n*m), n, m) X1 <- cbind(X, U) # QR update R2 <- fastQR::rupdate(X = X, R = R1, U = U, fast = FALSE, type = "column") ## check max(abs(crossprod(R2) - crossprod(X1))) ## Add one row ## generate sample data 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 <- 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 R decomposition R2 <- fastQR::rupdate(R = R1, X = X, U = u, type = "row") ## check max(abs(crossprod(R2) - crossprod(X1))) ## Add m rows ## generate sample data 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 to be added 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 R decomposition R2 <- fastQR::rupdate(R = R1, X = X, U = U, fast = FALSE, type = "row") ## check max(abs(crossprod(R2) - crossprod(X1))) ``` ## References \insertRef golub_van_loan.2013fastQR \insertRef bjorck.2015fastQR \insertRef bjorck.2024fastQR \insertRef bernardi_etal.2024fastQR

rupdate function