inversePower.py

## module inversePower
''' lam,x = inversePower(a,s,tol=1.0e-6).
    Inverse power method for solving the eigenvalue problem
    [a]{x} = lam{x}. Returns 'lam' closest to 's' and the
    corresponding eigenvector {x}.
'''
from numpy import zeros,dot,identity
from LUdecomp import *
from math import sqrt
from random import random

def inversePower(a,s,tol=1.0e-6):
    n = len(a)
    aStar = a - identity(n)*s   # Form [a*] = [a] - s[I]
    aStar = LUdecomp(aStar)     # Decompose [a*]
    x = zeros(n)
    for i in range(n):          # Seed [x] with random numbers
        x[i] = random()
    xMag = sqrt(dot(x,x))       # Normalize [x]
    x =x/xMag
    for i in range(50):         # Begin iterations      
        xOld = x.copy()         # Save current [x]
        x = LUsolve(aStar,x)    # Solve [a*][x] = [xOld]
        xMag = sqrt(dot(x,x))   # Normalize [x]
        x = x/xMag
        if dot(xOld,x) < 0.0:   # Detect change in sign of [x]
            sign = -1.0
            x = -x
        else: sign = 1.0
        if sqrt(dot(xOld - x,xOld - x)) < tol:
            return s + sign/xMag,x
    print 'Inverse power method did not converge'