def qr_fact_givens(matrix):
m, n = matrix.shape
Q = np.identity(m, dtype=float)
R = np.copy(matrix)
for i in range(m):
for j in range(min(i, n)):
if R[i, j] != 0:
x = R[j][j]
y = R[i][j]
norm = util.vector_norm([x, y])
cos = x * 1.0 / norm
sin = -y * 1.0 / norm
G = np.identity(m, dtype=float)
G[i, i] = cos
G[i, j] = sin
G[j, i] = -sin
G[j, j] = cos
Q = util.multiply_matrix(Q, G.T)
R = util.multiply_matrix(G, R)
error = util.matrix_error(util.multiply_matrix(Q, R), matrix)
return Q, R, error
def iterate(data, sInv, T, bVec, maxIt=100, tol=0.00005):
for i in range(maxIt):
previousData = data
temp = Util.multiply_matrix(sInv, T * -1)
data = Util.multiply_matrix(temp, previousData)
data += Util.multiply_matrix(sInv, bVec)
deltaData = np.absolute(previousData - data)
if np.less_equal(deltaData, np.ones(data.shape) * tol).all():
# print "interation: ", i, "\n", data
return data, i
return None
def qr_fact_househ(matrix):
y, x = matrix.shape
Q = house_holder_compute_h(matrix) # start building Q from h_1
R = util.multiply_matrix(Q, matrix) # start building R from h_1 * A
for i in range(y - 2):
h = house_holder_reconstruct(house_holder_compute_h(util.get_sub_matrix(R, i + 1)), y) # get h_n
Q = util.multiply_matrix(Q, h) # (h_1 * ... h_(n-1)) * h_n
R = util.multiply_matrix(h, R) # h_n * (.... h_1 * A)
error = util.matrix_error(util.multiply_matrix(Q, R), matrix)
return Q, R, error
def solve_qr_b(q, r, b):
m, n = q.shape
if b.shape[0] != m:
return None
x = util.multiply_matrix(q.T, b) # x = Q^tb
# upper triangle loop, ,calculate x with dp, backward
for i in reversed(range(m)):
for j in range(i + 1, m):
x[i] -= r[i, j] * x[j]
x[i] /= r[i, i]
return x
def problem_1d(n=(2, 13)):
low_bound, high_bound = n
file = open('problem_1d.txt', 'w')
n, lu_error, lu_px_b_error, qr_househ_error, qr_px_b_househ_error, qr_givens_error, qr_px_b_givens_error, inverse_px_b_error = [], [], [], [], [], [], [], []
for size in range(low_bound, high_bound):
print >> file, '============[ n =', size, ']====================='
n.append(size)
p = generate_pascal_matrix(size).astype(int)
print >> file, 'P ='
print >> file, p
b = generate_b_matrix(size)
print >> file, 'b ='
print >> file, b
print >> file, '============LU factorization==============='
l, u, error = util.lu_fact(p)
print >> file, 'L ='
print >> file, l.round(6)
print >> file, 'U ='
print >> file, u.round(6)
print >> file, '||LU - P||_inf (error) =', error
lu_error.append(error)
x = solve_lu_b(l, u, b)
print >> file, 'x_sol ='
print >> file, x
error = util.matrix_error(util.multiply_matrix(p, x), b)
print >> file, '||Px - b||_inf (error) =', error
lu_px_b_error.append(error)
print >> file, '============QR factorization==============='
print >> file, '(Householder)'
q, r, error = qr_fact_househ(p)
print >> file, 'Q ='
print >> file, q.round(6)
print >> file, 'R ='
print >> file, r.round(6)
print >> file, 'error =', error
qr_househ_error.append(error)
x = solve_qr_b(q, r, b)
print >> file, 'x_sol ='
print >> file, x
error = util.matrix_error(util.multiply_matrix(p, x), b)
print >> file, '||QR - P||_inf (error) =', error
qr_px_b_househ_error.append(error)
print >> file, '============QR factorization==============='
print >> file, '(Givens)'
q, r, error = qr_fact_givens(p)
print >> file, 'Q ='
print >> file, q.round(6)
print >> file, 'R ='
print >> file, r.round(6)
print >> file, '||QR - P||_inf (error) =', error
qr_givens_error.append(error)
x = solve_qr_b(q, r, b)
print >> file, 'x_sol ='
print >> file, x
error = util.matrix_error(util.multiply_matrix(p, x), b)
print >> file, '||Px - b||_inf (error) =', error
qr_px_b_givens_error.append(error)
print >> file
print >> file
# ============Solve system Eq through inversion===============
# Computationally long. Enable only if needed
# x = multiply_matrix(matrix_inverse(p), b)
# px_minus_b = multiply_matrix(p, x) - b
# error = vector_error(px_minus_b)
# print error
f, ((ax1, ax2), (ax3, ax4)) = pyplot.subplots(2, 2, sharey=True, sharex=True)
ax1.set_title('LU')
ax1.plot(n, lu_error, label='LU Error', c='blue')
ax1.plot(n, lu_px_b_error, label='LU Px_b Error', c='black')
ax2.set_title('QR Householder')
ax2.plot(n, qr_househ_error, label='QR Householder Error', c='blue')
ax2.plot(n, qr_px_b_househ_error, label='QR Householder Px_b Error', c='black')
ax3.set_title('QR Givens')
ax3.plot(n, qr_givens_error, label='QR Givens Error', c='blue')
ax3.plot(n, qr_px_b_givens_error, label='QR Givens Px_b Error', c='black')
# Draw legend
error_line, = ax4.plot([], label='Error (on P)', c='blue')
px_b_error_line, = ax4.plot([], label='Error (on b)', c='black')
ax4.legend(handles=[error_line, px_b_error_line])
# Draw Error on P lines
fig = pyplot.figure()
pyplot.xlabel('size (n x n pascal matrix)')
pyplot.ylabel('||LU or QR - P||_inf (error)')
pyplot.title('Error on P')
lue_plot, = pyplot.plot(n, lu_error, label='LU', c='red')
qrhhe_plot, = pyplot.plot(n, qr_househ_error, label='QR Householder', c='green')
qrge_plot, = pyplot.plot(n, qr_givens_error, label='QR Givens', c='blue')
pyplot.legend(handles=[lue_plot, qrhhe_plot, qrge_plot])
# Draw Error on b lines
fig = pyplot.figure()
pyplot.xlabel('size (n x n pascal matrix)')
pyplot.ylabel('||Px - b||_inf (error)')
pyplot.title('Error on b')
lupxbe_plot, = pyplot.plot(n, lu_px_b_error, label='LU', c='red')
qrhhpxbe_plot, = pyplot.plot(n, qr_px_b_househ_error, label='QR Householder', c='green')
qrgpxbe_plot, = pyplot.plot(n, qr_px_b_givens_error, label='QR Givens', c='blue')
pyplot.legend(handles=[lupxbe_plot, qrhhpxbe_plot, qrgpxbe_plot])
pyplot.show()
