#' @title Rayleigh quotient iteration
#' @description Rayleigh quotient iteration algorithm to computing the maximal eigenpair of
#' matrix Q.
#'
#' @param Q The input matrix to find the maximal eigenpair.
#' @param mu A vector.
#' @param v0_tilde The unnormalized initial vector \eqn{\tilde{v0}}.
#' @param zstart The initial \eqn{z_0} as an approximation of \eqn{\rho(Q)}.
#' @param digit.thresh The precise level of output results.
#' @return A list of eigenpair object are returned, with components '\eqn{z}' and '\eqn{v}'.
#' \item{z}{The approximating sequence of the maximal eigenvalue.}
#' \item{v}{The approximating sequence of the corresponding eigenvector.}
#'
#' @examples
#' Q = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3)
#' ray.quot(Q, mu=rep(1,dim(Q)[1]), v0_tilde=rep(1,dim(Q)[1]), zstart=6,
#' digit.thresh = 6)
#' @export
eff.ray.quot = function(Q, mu, v0_tilde, zstart, digit.thresh = 6) {
z = list()
rz = list()
v = list()
w = list()
v[[1]] = v0_tilde/sqrt(sum(v0_tilde^2 * mu))
z[[1]] = zstart
rz[[1]] = round(zstart, digit.thresh)
ratio = 1
iter = 0
while (ratio >= 10^(-digit.thresh)) {
iter = iter + 1
w = append(w, list(solve(-Q - z[[iter]] * diag(1, length(v0_tilde)),
v[[iter]], tol = 1e-100)))
v = append(v, list(w[[iter]]/sqrt(sum(w[[iter]]^2 * mu))))
z = append(z, list(sum(v[[iter + 1]] * (-Q %*% v[[iter +
1]]) * mu)))
ratio = abs(round(z[[iter + 1]], digit.thresh) - round(z[[iter]],
digit.thresh))
rz[[iter + 1]] = round(z[[iter + 1]], digit.thresh)
}
if (ratio == 0) {
v = v[-(iter + 1)]
rz = rz[-(iter + 1)]
}
return(list(v = v, z = rz))
}
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