R: QR Decomposition - S4 Methods and Generic

qr-methods {Matrix} | R Documentation

## QR Decomposition – S4 Methods and Generic

### Description

The Matrix package provides methods for the QR decomposition of special classes of matrices. There is a generic function which uses `qr` as default, but methods defined in this package can take extra arguments. In particular there is an option for determining a fill-reducing permutation of the columns of a sparse, rectangular matrix.

### Usage

    qr(x, ...)
    qrR(qr, complete=FALSE, backPermute=TRUE, row.names = TRUE)

### Arguments

`x` | a numeric or complex matrix whose QR decomposition is to be computed. Logical matrices are coerced to numeric.
`qr` | a QR decomposition of the type computed by `qr`.
`complete` | logical indicating whether the \bold{R} matrix is to be completed by binding zero-value rows beneath the square upper triangle.
`backPermute` | logical indicating if the rows of the \bold{R} matrix should be back permuted such that `qrR()`'s result can be used directly to reconstruct the original matrix \bold{X}.
`row.names` | logical indicating if `rownames` should propagated to the result.
`...` | further arguments passed to or from other methods

### Methods

x = "dgCMatrix"

  QR decomposition of a general sparse double-precision matrix with `nrow(x) >= ncol(x)`. Returns an object of class `"sparseQR"`.

x = "sparseMatrix"

  works via `"dgCMatrix"`.

### See Also

`qr`; then, the class documentations, mainly `sparseQR`, and also `dgCMatrix`.

### Examples

    ##------------- example of pivoting -- from base'  qraux.Rd -------------
    X <- cbind(int = 1,
               b1=rep(1:0, each=3), b2=rep(0:1, each=3),
               c1=rep(c(1,0,0), 2), c2=rep(c(0,1,0), 2), c3=rep(c(0,0,1),2))
    rownames(X) <- paste0("r", seq_len(nrow(X)))
    dnX <- dimnames(X)
    bX <- X # [b]ase version of X
    X <- as(bX, "sparseMatrix")
    X # is singular, columns "b2" and "c3" are "extra"
    stopifnot(identical(dimnames(X), dnX))# some versions changed X's dimnames!
    c(rankMatrix(X)) # = 4 (not 6)
    m <- function(.) as(., "matrix")

    ##----- regular case ------------------------------------------
    Xr <- X[ , -c(3,6)] # the "regular" (non-singular) version of X
    stopifnot(rankMatrix(Xr) == ncol(Xr))
    Y <- cbind(y <- setNames(1:6, paste0("y", 1:6)))
    ## regular case:
    qXr   <- qr(  Xr)
    qxr   <- qr(m(Xr))
    qxrLA <- qr(m(Xr), LAPACK=TRUE) # => qr.fitted(), qr.resid() not supported
    qcfXy <- qr.coef (qXr, y) # vector
    qcfXY <- qr.coef (qXr, Y) # 4x1 dgeMatrix
    cf <- c(int=6, b1=-3, c1=-2, c2=-1)
    doExtras <- interactive() || nzchar(Sys.getenv("R_MATRIX_CHECK_EXTRA")) ||
        identical("true", unname(Sys.getenv("R_PKG_CHECKING_doExtras")))
    tolE <- if(doExtras) 1e-15 else 1e-13
    stopifnot(exprs = {
      all.equal(qr.coef(qxr,  y),   cf,  tol=tolE)
      all.equal(qr.coef(qxrLA,y),   cf,  tol=tolE)
      all.equal(qr.coef(qxr,  Y), m(cf), tol=tolE)
      all.equal(  qcfXy,    cf, tol=tolE)
      all.equal(m(qcfXY), m(cf), tol=tolE)
      all.equal(y, qr.fitted(qxr, y), tol=2*tolE)
      all.equal(y, qr.fitted(qXr, y), tol=2*tolE)
      all.equal(m(qr.fitted(qXr, Y)), qr.fitted(qxr, Y), tol=tolE)
      all.equal(  qr.resid (qXr, y),  qr.resid (qxr, y), tol=tolE)
      all.equal(m(qr.resid (qXr, Y)), qr.resid (qxr, Y), tol=tolE)
    })

    ##----- rank-deficient ("singular") case ------------------------------------

    (qX <- qr(  X))           # both @p and @q are non-trivial permutations
     qx <- qr(m(X)) ; str(qx) # $pivot is non-trivial, too

    drop0(R. <- qr.R(qX), tol=tolE) # columns *permuted*: c3 b1 ..
    Q. <- qr.Q(qX)
    qI <- sort.list(qX@q) # the inverse 'q' permutation
    (X. <- drop0(Q. %*% R.[, qI], tol=tolE))## just = X, incl. correct colnames
    stopifnot(all(X - X.) < 8*.Machine$double.eps,
              ## qrR(.) returns R already "back permuted" (as with qI):
              identical(R.[, qI], qrR(qX)) )
    ##
    ## In this sense, classical qr.coef() is fine:
    cfqx <- qr.coef(qx, y) # quite different from
    nna <- !is.na(cfqx)
    stopifnot(all.equal(unname(qr.fitted(qx,y)),
                        as.numeric(X[,nna] %*% cfqx[nna])))
    ## FIXME: do these make *any* sense? --- should give warnings !
    qr.coef(qX, y)
    qr.coef(qX, Y)
    rm(m)

[Package Matrix version 1.3-4 Index]