R/ray.quot.general.R

In EfficientMaxEigenpair: Efficient Initials for Computing the Maximal Eigenpair

#' @title Rayleigh quotient iteration
#' @description Rayleigh quotient iteration algorithm to computing the maximal eigenpair of
#' general matrix \eqn{A}.
#'
#' @param A 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}, \eqn{v} and \eqn{iter}.
#' \item{z}{The approximating sequence of the maximal eigenvalue.}
#' \item{v}{The approximating eigenfunction of the corresponding eigenvector.}
#' \item{iter}{The number of iterations.}
#' 
#' @examples
#' A = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3)
#' ray.quot.general(A, mu=rep(1,dim(A)[1]), v0_tilde=rep(1,dim(A)[1]), zstart=6,
#'  digit.thresh = 6)

#' @export
ray.quot.general = function(A, 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
        
        wk = solve(-A - z[[iter]] * diag(1, length(v0_tilde)), v[[iter]], tol = 1e-100)
        
        if (sum(sign(wk)) == length(wk) | sum(sign(wk)) == -length(wk)) {
            w = append(w, list(wk))
        } else {
            w = append(w, list(solve(-A - z[[1]] * 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]] * (-A %*% 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)]
        
        iter = iter - 1
    }
    
    return(list(v = v[iter + 1], z = rz, iter = iter))
}