fastQR (version 1.0.0)

rupdate: Fast updating of the R matrix

Description

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.

Value

R the updated R matrix.

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.

References

golub_van_loan.2013fastQR

bjorck.2015fastQR

bjorck.2024fastQR

bernardi_etal.2024fastQR

Examples

## 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)))