eff.ini.maxeig.shift.inv.tri: Tridiagonal matrix maximal eigenpair In EfficientMaxEigenpair: Efficient Initials for Computing the Maximal Eigenpair

Description

Calculate the maximal eigenpair for the tridiagonal matrix by shifted inverse iteration algorithm.

Usage

 `1` ```eff.ini.maxeig.shift.inv.tri(a, b, c, xi = 1, digit.thresh = 6) ```

Arguments

 `a` The lower diagonal vector. `b` The upper diagonal vector. `c` The shifted main diagonal vector. The corresponding unshift diagonal vector is -c(b[1] + c[1], a[1:N - 1] + b[2:N] + c[2:N], a[N] + c[N + 1]) where N+1 is the dimension of matrix. `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.general` for the general matrix maximal eigenpair.
 ```1 2 3 4 5``` ```a = c(1:7)^2 b = c(1:7)^2 c = rep(0, length(a) + 1) c[length(a) + 1] = 8^2 eff.ini.maxeig.shift.inv.tri(a, b, c, xi = 1) ```