def correction(self, H, h, measurements, estimated_cov_matrix):
H_hat = multiply(H, estimated_cov_matrix, transpose(H))
K = multiply(estimated_cov_matrix, transpose(H), invert(H_hat + self.Ez))
self.estimated_position += multiply(K, (measurements - h))
self.cov_matrix = multiply((np.eye(N=4) - multiply(K, H)), estimated_cov_matrix)
self.prediction_sequence.append(transpose(self.estimated_position))
def selectBestPositions(self, estimated_cov_matrix, estimated_position):
estimated_cov_matrix_inv = invert(estimated_cov_matrix)
distances = np.ones(self.sensor_size) * -1
for i in range(self.sensor_size):
extended_basestation_pos = np.append(self.basestations[i].position, np.array([0, 0]))
difference = transpose(estimated_position) - extended_basestation_pos
distances[i] = multiply(difference, estimated_cov_matrix_inv, transpose(difference))
valid_distances = self.sortWithIndeces(distances)
return [valid_distances[i][0] for i in range(0, min(3, len(valid_distances)))]
def main():
A = build_matrix()
U, Es = gauss(A)
E = multiply(multiply(Es[2], Es[1]), Es[0])
L = gauss_jordan(E)
T = transpose(A)
D, new_U = factorize_D(U)
assert multiply(E, A) == U
assert multiply(E, A) != multiply(A, E)
assert multiply(L, U) == A
assert multiply(multiply(L, D), new_U) == A
assert transpose(T) == A
def backward_substitution(a): # currently supports only (n x n+1) matrix n = a.shape[0] p = zeros(n) sum_of_others = 0 # last variable p[-1] = a[-1][-1] / a[-1][-2]; # other variables for i in xrange(n-2,-1,-1): for j in xrange(n-1,i,-1): sum_of_others += a[i][j]*p[j] p[i] = (a[i][n] - sum_of_others) / a[i][i] sum_of_others = 0 return mo.transpose(p)
def backward_substitution(a):
# currently supports only (n x n+1) matrix
n = a.shape[0]
p = zeros(n)
sum_of_others = 0
# last variable
p[-1] = a[-1][-1] / a[-1][-2]
# other variables
for i in xrange(n - 2, -1, -1):
for j in xrange(n - 1, i, -1):
sum_of_others += a[i][j] * p[j]
p[i] = (a[i][n] - sum_of_others) / a[i][i]
sum_of_others = 0
return mo.transpose(p)
def prediction(self):
self.estimated_position = multiply(self.F, self.estimated_position)
estimated_cov_matrix = multiply(self.F, self.cov_matrix, transpose(self.F)) + self.Ex
measurements = self.selectiveMeasurements(estimated_cov_matrix)
h = np.zeros(self.sensor_size)
for i in range(0, self.sensor_size):
if measurements[i] != 0:
h[i] = self.model.spaceToValue(self.estimated_position[0:2] - self.basestations[i].position)
H = np.empty((0, 4))
for i in range(0, len(measurements)):
if measurements[i] != 0:
dh_dx, dh_dy = self.model.derivative(self.estimated_position[0:2] - self.basestations[i].position)
else:
dh_dx, dh_dy = 0.0, 0.0
H = np.append(H, np.array([[dh_dx, dh_dy, 0.0, 0.0]]), axis=0)
return H, h, measurements, estimated_cov_matrix
print i
for j in xrange(0, i):
factor = a[j][i] / a[i][i]
for k in xrange(0, n + 1):
a[j][k] -= factor * a[i][k]
a = ge.row_scaling(a, False)
return a
# solve a set of equation
def solve(a, b):
print "Coefficients:\n", a
print "Values:\n", b
p = mo.augmented_matrix(a, b)
print 'Augmented:\n', p
p = ge.row_scaling(p)
print 'Row scaled:\n', p
# p = partial_pivoting(p)
# print "Partially pivoted:\n", p
p = transform_to_identity(p)
print 'Transformed to identity form:\n', p
q = ge.backward_substitution(p)
print "On backward substitution:\n", q
# test
A = array([[2, 3, -1], [4, 4, -3], [-2, 1, -1]])
B = mo.transpose(array([5, 3, -3]))
# A = array([[4,1],[1,3]])
# B = mo.transpose(array([1,2]))
solve(A, B)
n = a.shape[0] for i in xrange(1,n): print i for j in xrange(0,i): factor = a[j][i]/a[i][i] for k in xrange(0,n+1): a[j][k] -= factor*a[i][k] a = ge.row_scaling(a, False) return a # solve a set of equation def solve(a,b): print "Coefficients:\n", a print "Values:\n", b p = mo.augmented_matrix(a,b) print 'Augmented:\n', p p = ge.row_scaling(p) print 'Row scaled:\n', p # p = partial_pivoting(p) # print "Partially pivoted:\n", p p = transform_to_identity(p) print 'Transformed to identity form:\n', p q = ge.backward_substitution(p) print "On backward substitution:\n", q # test A = array([[2,3,-1],[4,4,-3],[-2,1,-1]]) B = mo.transpose(array([5,3,-3])) # A = array([[4,1],[1,3]]) # B = mo.transpose(array([1,2])) solve(A,B)
# Lesson6, Task1
# import own module
import matrix_operations as m_op
# import numpy module
import numpy as np
# get machine epsilone
u = np.finfo(float).eps
# set initial data
A = m_op.hilbert_matrix(5)
b = m_op.transpose([[1, 2, 3, 4, 5]])
# print initial data
print("Matrix A:")
m_op.print_matrix(m_op.matrix_in_fractions(A))
print()
print("Matrix b:")
m_op.print_matrix(b)
# first calculations with LU method
x = m_op.solve_equations_by_LU(A, b)
r = m_op.subtract_two_dimensional_matricies(b, \
m_op.matrix_multiplication(A,x))
# print first results
print("First x:")
m_op.print_matrix(x)