
# coding: utf-8

# In[14]:

import pprint
import scipy
import scipy.linalg   # SciPy Linear Algebra Library

def cmp(a, b):
    return (a > b) - (a < b) 

A = scipy.array([[12, -51, 4], [6, 167, -68], [-4, 24, -41]])  # From the Wikipedia Article on QR Decomposition
Q, R = scipy.linalg.qr(A)

print ("A:")
pprint.pprint(A)

print ("Q:")
pprint.pprint(Q)

print ("R:")
pprint.pprint(R)


# In[15]:

import numpy as np
 
def qr(A):
    m, n = A.shape
    Q = np.eye(m)
    for i in range(n - (m == n)):
        H = np.eye(m)
        H[i:, i:] = make_householder(A[i:, i])
        Q = np.dot(Q, H)
        A = np.dot(H, A)
    return Q, A
 
def make_householder(a):
    v = a / (a[0] + np.copysign(np.linalg.norm(a), a[0]))
    v[0] = 1
    H = np.eye(a.shape[0])
    H -= (2 / np.dot(v, v)) * np.dot(v[:, None], v[None, :])
    return H
 
# task 1: show qr decomp of wp example
a = np.array(((
    (12, -51,   4),
    ( 6, 167, -68),
    (-4,  24, -41),
)))
 
q, r = qr(a)
print('q:\n', q.round(6))
print('r:\n', r.round(6))
 
# task 2: use qr decomp for polynomial regression example
def polyfit(x, y, n):
    return lsqr(x[:, None]**np.arange(n + 1), y.T)
 
def lsqr(a, b):
    q, r = qr(a)
    _, n = r.shape
    return np.linalg.solve(r[:n, :], np.dot(q.T, b)[:n])
 
x = np.array((0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10))
y = np.array((1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321))
 
print('\npolyfit:\n', polyfit(x, y, 2))

