# eff.ini.maxeig.general: General matrix maximal eigenpair In EfficientMaxEigenpair: Efficient Initials for Computing the Maximal Eigenpair

## Description

Calculate the maximal eigenpair for the general matrix.

## Usage

 1 2 eff.ini.maxeig.general(A, v0_tilde = NULL, z0 = NULL, z0numeric, xi = 1, digit.thresh = 6) 

## Arguments

 A The input general matrix. v0_tilde The unnormalized initial vector \tilde{v0}. z0 The type of initial z_0 used to calculate the approximation of ρ(Q). There are three types: 'fixed', 'Auto' and 'numeric' corresponding to three choices of z_0 in paper. z0numeric The numerical value assigned to initial z_0 as an approximation of ρ(Q) when z_0='numeric'. xi The coefficient used to form the convex combination of δ_1^{-1} and (v_0,-Q*v_0)_μ, it should between 0 and 1. digit.thresh The precise level of output results.

## Value

A list of eigenpair object are returned, with components z, v and iter.

 z The approximating sequence of the maximal eigenvalue. v The approximating eigenfunction of the corresponding eigenvector. iter The number of iterations.

eff.ini.maxeig.tri for the tridiagonal matrix maximal eigenpair by rayleigh quotient iteration algorithm. eff.ini.maxeig.shift.inv.tri for the tridiagonal matrix maximal eigenpair by shifted inverse iteration algorithm.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 A = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3) eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = 'fixed') A = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3) eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = 'Auto') ##Symmetrizing A converge to second largest eigenvalue A = matrix(c(1, 3, 9, 5, 2, 14, 10, 6, 0, 11, 11, 7, 0, 0, 1, 8), 4, 4) S = (t(A) + A)/2 N = dim(S)[1] a = diag(S[-1, -N]) b = diag(S[-N, -1]) c = rep(NA, N) c[1] = -diag(S)[1] - b[1] c[2:(N - 1)] = -diag(S)[2:(N - 1)] - b[2:(N - 1)] - a[1:(N - 2)] c[N] = -diag(S)[N] - a[N - 1] z0ini = eff.ini.maxeig.tri(a, b, c, xi = 7/8)\$z[1] eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = 'numeric', z0numeric = 28 - z0ini)