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

Description Usage Arguments Value See Also Examples

Calculate the maximal eigenpair for the general matrix.

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

`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,-Qv_0)_μ*, it should between 0 and 1.
`digit.thresh`	The precise level of output results.

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.

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)