def qr_householder(matrix):
rows, cols = g.shape(matrix)
Q = identity_matrix(rows)
for i in range(cols):
u, v = _get_householder_vectors(matrix, i)
if not u or u == v: # 'u' and 'v' may be empty when called for the last column. hint: check for square matrix
continue
w = g.divide_by_scalar(g.subtract(u, v), vector.norm(g.subtract(u, v)))
temp = g.multiply_by_scalar(g.multiply(w, g.transpose(w)), 2)
H = g.subtract(identity_matrix(len(w)), temp)
H = _add_identity_matrix_to_householder(H, i)
Q = g.multiply(H, Q)
matrix = g.multiply(H, matrix)
return g.transpose(Q), matrix
def eigenvalues(matrix, algo='householder'):
# returns a row vector of eigenvalues
if algo == 'householder':
qr_algo = qr_householder
else:
qr_algo = qr_gram_schmidt
size = check_for_square(matrix)
if size == -1:
print("can not find eigenvalues of a non-square matrix!")
return
change = 1
E = diagonal(matrix)
steps = 0
while change > 0.00000001:
if steps > 50000: # eigenvalues are not converging
return None
Eold = E
Q, R = qr_algo(matrix)
matrix = g.multiply(R, Q)
E = diagonal(matrix)
change = vector.norm(g.subtract(E, Eold))
steps += 1
return vector.transform_to_row_vector(diagonal(matrix))
def tridiagonal(matrix):
size = check_for_square(matrix)
if size == -1:
print("can not tridiagonalize a non-square matrix")
return
# H = I - w*transpose(w)
# w = (u - v)/norm(u - v)
# since the matrix is symmetric, we can extract 'u' vector from matrix rows and then transform it to column vector
for i in range(size - 2):
u = vector.transform_to_column_vector([matrix[i][u_i] for u_i in range(i + 1, size)])
v = vector.transform_to_column_vector([vector.norm(u) if v_i == 0 else 0 for v_i in range(len(u))])
w = g.divide_by_scalar(g.subtract(u, v), vector.norm(g.subtract(u, v)))
temp = g.multiply_by_scalar(g.multiply(w, g.transpose(w)), 2)
H = g.subtract(identity_matrix(len(w)), temp)
H = _add_identity_matrix_to_householder(H, size - len(H))
matrix = g.multiply(H, g.multiply(matrix, H))
return matrix
def dot(v1, v2):
return g.multiply(g.transpose(v1), v2)[0][0]
def qr_gram_schmidt(matrix):
Q = gram_schmidt_orthonormalize(matrix)
R = g.multiply(g.transpose(Q), matrix) # as Q is orthonormal matrix, so, transpose(Q) = inverse(Q)
return Q, R