def exercise_one():
A = np.array([[0, 0, 0, 1], [2, 1, 2, 0], [4, 4, 0, 0], [2, 3, 1, 0]])
b_one = np.array([3, 5, 4, 5])
b_two = np.array([4, 10, 12, 11])
LU, p = zerlegung(A)
pb = permutation(p, b_one)
y = vorwaerts(LU, pb)
print(rueckwaerts(LU, y))
pb = permutation(p, b_two)
y = vorwaerts(LU, pb)
print(rueckwaerts(LU, y))
def exercise_two():
for n in [5, 10, 15, 20]:
A = np.array([[1 / (i + j - 1) for j in range(1, n + 1)]
for i in range(1, n + 1)])
b = np.array([1 / (i + 1) for i in range(1, n + 1)])
LU, p = zerlegung(A)
pb = permutation(p, b)
y = vorwaerts(LU, pb)
print('n:', n, '=>', rueckwaerts(LU, y))
def exercise_b():
for n in range(10, 16):
print(f'n = {n}:')
A, b = create_a_b_exercise_b(n)
LU, p = zerlegung(A, greatest_non_zero)
x = permutation(p, b)
y = vorwaerts(LU, x)
z = rueckwaerts(LU, y)
print(nachiteration(A, b, LU, p, z, nachiterationen))
def hager(B):
n = len(B)
x = np.full(n, 1) / n
LU, p = zerlegung(B, first_non_zero)
while True:
v = vorwaerts_T(LU.T, x)
w = rueckwaerts_T(LU.T, v)
y = permutation_T(p, w)
xi = np.sign(y)
v = vorwaerts(LU, permutation(p, xi))
z = np.array(rueckwaerts(LU, v))
if norm(z, np.inf) <= np.dot(z.T, x):
return norm(y, 1)
j = np.argmax(np.abs(z))
x = np.array([1 if k == j else 0 for k in range(len(x))])
def init_A_b(n):
A = np.identity(n)
for y in range(1, n):
for x in range(y):
A[y, x] = -1
A[:, -1] = 1
b = np.array([2 - n if i == n else 3 - i for i in range(n)])
return A, b
if __name__ == '__main__':
A = np.array([[20, 18, 44], [0, 40, 45], [-15, 24, -108]])
b = np.array([-4, -45, 78])
x = householder(A, b)
print(x)
for n in [40, 50, 60]:
A, b = init_A_b(n)
x = householder(A, b)
print(f'n={n}: {x}')
LU, p = zerlegung(A, greatest_non_zero)
x = permutation(p, b)
y = vorwaerts(LU, x)
res = rueckwaerts(LU, y)
print(res)
return np.add(np.dot(np.absolute(A), np.absolute(x̃)), np.absolute(b));
def calculate_ɛ(r̃, s̃):
return np.amax([abs(r̃[i]) / abs(s̃[i]) for i in range(0, len(r̃))])
def calculate_machine_ɛ():
return (1 / 2) * (2 ** (1 - np.finfo(float).nmant))
δ_s = [10**-8, 10**-10,10**-12];
for δ in δ_s:
print('δ: ', δ)
A, b = init_for_delta(δ)
LU, p = zerlegung(A, greatest_non_zero)
p_b = permutation(p, b)
y = vorwaerts(LU, p_b)
x̃ = rueckwaerts(LU, y)
print(x̃)
print(correct_answer(δ))
r̃ = calculate_r̃(A, b, x̃);
s̃ = calculate_s̃(A, b, x̃);
ɛ = calculate_ɛ(r̃, s̃);
kondition = np.linalg.cond(A);
print(' ɛ: ', ɛ)
print(' ~ɛ: ', kondition * calculate_machine_ɛ())
# -*- coding: utf-8 -*-
import numpy as np
from LUtils import zerlegung, permutation, vorwaerts, rueckwaerts
A = np.array([[0, 0, 0, 1], [2, 1, 2, 0], [4, 4, 0, 0], [2, 3, 1, 0]])
u = np.array([0, 1, 2, 3])
v = np.array([0, 0, 0, 1])
b = np.array([3, 5, 4, 5])
LU, p = zerlegung(A)
per = permutation(p, u)
y = vorwaerts(LU, per)
z = rueckwaerts(LU, y)
val = 1 + np.dot(np.transpose(v), z)
if val != 0:
a = 1 / val
per = permutation(p, b)
y = vorwaerts(LU, per)
z_dach = rueckwaerts(LU, y)
print(z_dach - a * np.dot(np.dot(np.transpose(v), z_dach), z))
def with_pivot_search(n, beta):
A, b = init_for_n(n, beta)
LU, p = zerlegung(A, greatest_non_zero)
x = permutation(p, b)
y = vorwaerts(LU, x)
return rueckwaerts(LU, y)
