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#' Smith normal form
#'
#' For an integer matrix M, this computes the matrices D, P, and Q such that
#' \emph{D = PMQ}, which can be seen as an analogue of the singular value
#' decomposition. All are integer matrices, and P and Q are unimodular (have
#' determinants +- 1).
#'
#' @param mat a matrix (integer entries)
#' @param code return only the M2 code? (default: \code{FALSE})
#' @name snf
#' @seealso [m2_matrix()]
#' @return a list of \code{m2_matrix} objects with names \code{D}, \code{P}, and
#' \code{Q}
#' @examples
#'
#' \dontrun{ requires Macaulay2
#'
#' ##### basic usage
#' ########################################
#'
#' M <- matrix(c(
#' 2, 4, 4,
#' -6, 6, 12,
#' 10, -4, -16
#' ), nrow = 3, byrow = TRUE)
#'
#' snf(M)
#'
#' (mats <- snf(M))
#' P <- mats$P; D <- mats$D; Q <- mats$Q
#'
#' P %*% M %*% Q # = D
#' solve(P) %*% D %*% solve(Q) # = M
#'
#' det(P)
#' det(Q)
#'
#'
#' M <- matrix(c(
#' 1, 2, 3,
#' 1, 34, 45,
#' 2213, 1123, 6543,
#' 0, 0, 0
#' ), nrow = 4, byrow = TRUE)
#' (mats <- snf(M))
#' P <- mats$P; D <- mats$D; Q <- mats$Q
#' P %*% M %*% Q # = D
#'
#'
#'
#' ##### understanding lattices
#' ########################################
#'
#'
#'
#'
#' # cols of m generate the lattice L
#' M <- matrix(c(2,-1,1,3), nrow = 2)
#' row.names(M) <- c("x", "y")
#' M
#'
#' # plot lattice
#' df <- expand.grid(x = -20:20, y = -20:20)
#' pts <- t(apply(df, 1, function(v) M %*% v))
#' w <- c(-15, 15)
#' plot(pts, xlim = w, ylim = w)
#'
#' # decompose m
#' (mats <- snf(M))
#' P <- mats$P; D <- mats$D; Q <- mats$Q
#'
#' # PMQ = D, the columns of MQ = P^(-1) D are a simpler basis of
#' # the lattice generated by (the cols of) M
#' (basis <- solve(P) %*% D)
#'
#' # plot lattice generated by new basis
#' pts2 <- t(apply(df, 1, function(v) basis %*% v))
#' points(pts2, pch = "*", col = "red")
#'
#'
#'
#' ##### other options
#' ########################################
#'
#' snf.(M)
#' snf(M, code = TRUE)
#'
#'
#'
#'
#' }
#'
#' @rdname snf
#' @export
snf <- function (mat, code = FALSE) {
# run m2
args <- as.list(match.call())[-1]
eargs <- lapply(args, eval, envir = parent.frame())
pointer <- do.call(snf., eargs)
if(code) return(invisible(pointer))
# parse output
parsed_out <- m2_parse(pointer)
# list and out
list(
D = parsed_out[[1]],
P = parsed_out[[2]],
Q = parsed_out[[3]]
)
}
#' @rdname snf
#' @export
snf. <- function (mat, code = FALSE) {
# arg checking
# if (is.m2_matrix(mat)) mat <- mat$rmatrix
if (is.m2_pointer(mat)) {
param <- m2_name(mat)
} else {
if (!is.integer(mat)) stopifnot(all(mat == as.integer(mat)))
param <- paste0("matrix", listify_mat(mat))
}
# create code and message
m2_code <- sprintf("smithNormalForm %s", param)
if(code) { message(m2_code); return(invisible(m2_code)) }
# run m2 and return pointer
m2.(m2_code)
}
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