QR decomposition (Givens)

import numpy as np
import math
from numpy import dot
from scipy.linalg import norm
np.set_printoptions(suppress=True, linewidth=160, precision=15)


def transposition(n, i, j):
    trans = np.identity(n)
    trans[[i, j]] = trans[[j, i]]
    return trans


def givens(n, i, j, x, y):
    g_matrix = np.identity(n)
    square = math.sqrt(x ** 2 + y ** 2)
    cosinus = 1.0 * x / square
    sinus = 1.0 * y / square
    g_matrix[i, i] = cosinus
    g_matrix[j, j] = cosinus
    g_matrix[i, j] = sinus
    g_matrix[j, i] = -sinus
    #g_matrix[[i, j], [i, j]] = np.array([[math.cos(alpha), math.sin(alpha)],[-math.sin(alpha), math.cos(alpha)]])
    return g_matrix


def givens_2(x, y):
    g_matrix = np.zeros([2, 2])
    square = math.sqrt(x ** 2 + y ** 2)
    cosinus = 1.0 * x / square
    sinus = 1.0 * y / square

    g_matrix[0, 0] = cosinus
    g_matrix[1, 1] = cosinus
    g_matrix[0, 1] = sinus
    g_matrix[1, 0] = -sinus
    return g_matrix


def max_column_index(A):
    return np.argmax(np.fromiter((norm(col) for col in A.T), dtype=np.float))


def qr_choise(A, tolerance=1e-16):
    m, n = A.shape
    Q = np.identity(m)
    P = np.identity(n)
    MIN_SIZE = min(m, n)
    rank = MIN_SIZE
    for j in xrange(MIN_SIZE):
        max_index = max_column_index(A[j:, j:]) + j
        if norm(A[j:, j:].T[max_index - j]) < tolerance:
            print 'Break at rank', str(j-1)
            rank = j - 1
            break
        trans = transposition(n, j, max_index)
        A[:] = dot(A, trans)
        P[:] = dot(trans, P)
        for i in xrange(m - 1, j, -1):
            givens2 = givens_2(A[j, j], A[i, j])
            A[[j, i], j:n] = dot(givens2, A[[j, i], j:n])
            Q[[j, i], :] = dot(givens2, Q[[j, i], :])
    return Q, P, rank

def qr_simple(A):
    m, n = A.shape
    Q = np.identity(m)
    MIN_SIZE = min(m, n)
    for j in xrange(MIN_SIZE):
        for i in xrange(m-1, j, -1):
            givens2 = givens_2(A[j, j], A[i, j])
            A[[j, i], j:n] = dot(givens2, A[[j, i], j:n])
            Q[[j, i], :] = dot(givens2, Q[[j, i], :])
    return Q

# From citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.379.8623&rep=rep1&type=pdf

def practical_qr(A, tolerance=1e-16):
    m, n = A.shape
    Q = np.identity(m)
    P = np.identity(n)
    MIN_SIZE = min(m, n)
    matrix_norm = norm(A)
    # make column norms
    norms = np.fromiter((norm(col)**2 for col in A.T), dtype=np.float)
    for j in xrange(MIN_SIZE):
        max_index = np.argmax(norms[j:])
        if norms[max_index] < tolerance * matrix_norm:
            print 'Break at rank', j-1
            break
        trans = transposition(n, j, max_index + j)
        norms[[j, max_index + j]] = norms[[max_index + j, j]]
        A[:] = dot(A, trans)
        P[:] = dot(trans, P)
        for i in xrange(m - 1, j, -1):
            givens2 = givens_2(A[j, j], A[i, j])
            giv = givens(m, i, j, A[j, j], A[i, j])
            A[[j, i], j:n] = dot(givens2, A[[j, i], j:n])
            Q = dot(Q, giv)
        norms[j+1:] -= A[j, j+1: n]**2
    return Q, P



A = np.random.randn(5, 6)
A[:, 4] = A[:, 1] + 5*A[:, 2]
A[:, 3] = 3*A[:, 1]
A[:, 5] = 2*A[:, 1] + 4*A[:, 2]
A[:, 0] = 6*A[:, 2] + 5*A[:, 1]
print 'matrix rank of A:'
print np.linalg.matrix_rank(A)
#A = np.loadtxt('matrix.txt')

M = A.copy()

Q, P, rank = qr_choise(A)

R = A.copy()

U = qr_simple(A[:rank, :].T)
T_full = np.zeros(M.shape)
T_full[:rank, :] = A[:rank, :]
print 'R matrix with rank ', rank, ':'
print R
print 'T matrix:'
print T_full
print 'U:'
print U
print 'UR^t:'
print U.shape, R.T.shape
print dot(U.T, R.T)
print '*'*80
print 'Q matrix:'
print Q
print '*'*80
print 'Prove that Q is orthogonal : Q*Q^t'
print dot(Q, Q.T)
print '*'*80
print 'permutation matrix P:'
print P
print '*'*80
print 'Matrix A:'
print M
print 'QR:'

print dot(dot(Q, R), P)
print 'Difference:'
print dot(dot(Q.T, R), P) - M
print dot(dot(Q.T, dot(T_full, U)), P) - M