Inverse power method for solving the eigenvalue problem in Python
''' 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'
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