eff.ini.maxeig.general {EfficientMaxEigenpair}

General matrix maximal eigenpair

Description

Calculate the maximal eigenpair for the general matrix.

Usage

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 \rho(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 \rho(Q) when z_0='numeric'.

xi

The coefficient used to form the convex combination of \delta_1^{-1} and (v_0,-Q*v_0)_\mu, 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.

See Also

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.

Examples

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)