# Fast downdating of the R matrix rdowndate provides the update of the thin R matrix of the QR factorization after the deletion of $m\geq 1$ rows or columns to the matrix $X\in\mathbb{R}^{n\times p}$ with $n>p$. The R factorization of the matrix $X$ returns the upper triangular matrix $R\in\mathbb{R}^{p\times p}$ such that $X^\top X=R^\top R$. See Golub and Van Loan (2013) for further details on the method. ```r rdowndate(R, k = NULL, m = NULL, U = NULL, fast = NULL, type = NULL) ``` ## Arguments - `R`: a $p\times p$ upper triangular matrix. - `k`: position where the columns or the rows are removed. - `m`: number of columns or rows to be removed. - `U`: a $p\times m$ matrix of rows to be removed. It should only be provided when rows are being removed. - `fast`: fast mode: disable to check whether the provided matrices are valid inputs. Default is FALSE. - `type`: either 'row' of 'column', for removing rows or columns. ## Returns R the updated R matrix. ## Examples ```r ## Remove one column ## generate sample data set.seed(10) n <- 10 p <- 6 X <- matrix(rnorm(n * p, 1), n, p) ## get the initial QR factorization output <- fastQR::qr(X, type = "householder", nb = NULL, complete = TRUE) Q <- output$Q R <- output$R R1 <- R[1:p,] ## select the column to be deleted from X and update X k <- 2 X1 <- X[, -k] ## downdate the R decomposition R2 <- fastQR::rdowndate(R = R1, k = k, m = 1, type = "column") ## check max(abs(crossprod(R2) - crossprod(X1))) ## Remove m columns ## generate sample data set.seed(10) n <- 10 p <- 6 X <- matrix(rnorm(n * p, 1), n, p) ## get the initial QR factorization output <- fastQR::qr(X, type = "householder", nb = NULL, complete = TRUE) Q <- output$Q R <- output$R R1 <- R[1:p,] ## select the column to be deleted from X and update X k <- 2 X1 <- X[, -c(k,k+1)] ## downdate the R decomposition R2 <- fastQR::rdowndate(R = R1, k = k, m = 2, type = "column") ## check max(abs(crossprod(R2) - crossprod(X1))) ## Remove one row ## generate sample data set.seed(10) n <- 10 p <- 6 X <- matrix(rnorm(n * p, 1), n, p) ## get the initial QR factorization output <- fastQR::qr(X, type = "householder", nb = NULL, complete = TRUE) Q <- output$Q R <- output$R R1 <- R[1:p,] # select the row to be deleted from X and update X k <- 5 X1 <- X[-k,] U <- as.matrix(X[k,], p, 1) ## downdate the R decomposition R2 <- rdowndate(R = R1, k = k, m = 1, U = U, fast = FALSE, type = "row") ## check max(abs(crossprod(R2) - crossprod(X1))) ## Remove m rows ## create data: n > p set.seed(10) n <- 10 p <- 6 X <- matrix(rnorm(n * p, 1), n, p) output <- fastQR::qr(X, type = "householder", nb = NULL, complete = TRUE) Q <- output$Q R <- output$R R1 <- R[1:p,] ## select the rows to be deleted from X and update X k <- 2 m <- 2 X1 <- X[-c(k,k+m-1),] U <- t(X[k:(k+m-1), ]) ## downdate the R decomposition R2 <- rdowndate(R = R1, k = k, m = m, 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

rdowndate function