#' Samples uniform from the Stiefel Manifold
#'
#' @param p row dim.
#' @param q col dim.
#' @return `(p, q)` semi-orthogonal matrix
#' @examples
#'  V <- rStiefel(6, 4)
#' @export
rStiefl <- function(p, q) {
    return(qr.Q(qr(matrix(rnorm(p * q, 0, 1), p, q))))
}

#' Null space basis of given matrix `V`
#'
#' @param V `(p, q)` matrix
#' @return Semi-orthogonal `(p, p - q)` matrix spaning the null space of `V`.
#' @keywords internal
#' @export
null <- function(V) {
    tmp <- qr(V)
    set <- if(tmp$rank == 0L) seq_len(ncol(V)) else -seq_len(tmp$rank)
    return(qr.Q(tmp, complete=TRUE)[, set, drop=FALSE])
}

#' Creates a (numeric) matrix where each row contains
#' an element to element matching.
#' @param elements numeric vector of elements to match
#' @return matrix of size `(n * (n - 1) / 2, 2)` for a argument of lenght `n`.
#' @keywords internal
#' @export
elem.pairs <- function(elements) {
    # Number of elements to match.
    n <- length(elements)
    # Create all combinations.
    pairs <- cbind(rep(elements, each=n), rep(elements, n))
    # Select unique combinations without self interaction.
    return(pairs[pairs[, 1] < pairs[, 2], ])
}

#' Apply function to pairs of matrix rows or columns.
#'
#' \code{row.pair.apply} applies \code{FUN} to each unique row pair without self
#' interaction while \code{col.pair.apply} does the same for columns.
#' @param X Matrix.
#' @param FUN Function to apply to each pair.
#' @examples
#'  X <- matrix(seq(12), 4, 3)
#' # matrix containing all row to row differences.
#'  row.pair.apply(X, `-`)
#' @aliases row.pair.apply, col.pair.apply
#' @keywords internal
#' @export
row.pair.apply <- function(X, FUN) {
    pairs <- elem.pairs(seq(nrow(X)))
    return(FUN(X[pairs[, 1], ], X[pairs[, 2], ]))
}
#' @rdname row.pair.apply
#' @keywords internal
#' @export
col.pair.apply <- function(X, FUN) {
    pairs <- elem.pairs(seq(ncol(X)))
    return(FUN(X[, pairs[, 1]], X[, pairs[, 2]]))
}