cholesky.py
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__author__ = 'chunk'
import numpy as np
import math
def calc_upper(U, b):
n = len(U)
assert len(b) == n
x = [0.0] * n
for i in reversed(range(n)):
if U[i][i] == 0:
raise Exception('u_ii == 0')
x[i] = b[i]
for j in reversed(range(i + 1, n)):
x[i] -= U[i][j] * x[j]
x[i] /= U[i][i]
return x
def calc_lower(L, b):
n = len(L)
assert len(b) == n
x = [0.0] * n
for i in range(n):
if L[i][i] == 0:
raise Exception('l_ii == 0')
x[i] = b[i]
for j in range(i):
x[i] -= L[i][j] * x[j]
x[i] /= L[i][i]
return x
def gen_hilbert(n):
return np.array([1.0 / (i + j + 1) for i in range(n) for j in range(n)]).reshape((n, -1))
def gen_b(n, x=None):
if x == None:
x = [1] * n
return np.dot(gen_hilbert(n), x)
def cholesky(A):
n = len(A)
for j in range(n):
for k in range(j):
A[j][j] -= (A[j][k] * A[j][k])
A[j][j] = math.sqrt(A[j][j])
for i in range(j + 1, n):
for k in range(j):
A[i][j] -= (A[i][k] * A[j][k])
A[i][j] /= A[j][j]
return np.array(A)
def calculate(n):
H = gen_hilbert(n)
b = gen_b(n)
L = cholesky(H)
y = calc_lower(L, b)
x = calc_upper(L.T, y)
# validate:
# print np.dot(H, x) - b
return x, b - np.dot(H, x), [x[i] - 1.0 for i in range(n)]
def test():
print gen_hilbert(3)
print gen_b(5)
U = [[1, -0.7, 0], [0, 1, -60], [0, 0, 155]]
b = [0.7, -61, 155]
print calc_upper(U, b)
L = [[1, 0, 0], [-60, 1, 0], [0, -0.7, 1]]
b = [1, -61, 0.7]
print calc_lower(L, b)
A = [[5, -1, -1], [-1, 3, -1], [-1, -1, 5]]
print cholesky(A)
x, r, delta = calculate(10)
print "%s\n%s\n%s\n" % (x, r, delta)
if __name__ == '__main__':
test()
pass