fastQR (version 1.1.4)

rdowndate: Fast downdating of the R matrix

Description

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.

Usage

rdowndate(R, k = NULL, m = NULL, U = NULL, fast = NULL, type = NULL)

Value

R the updated R matrix.

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. It is not required when deleting columns. If NULL, it defaults to the number of columns in \(U\).

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.

References

golub_van_loan.2013fastQR

bjorck.2015fastQR

bjorck.2024fastQR

bernardi_etal.2024fastQR

Examples

## 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)))