def do_gaussian_elimination_and_extract_solution(self):
# 计算方程组的最简化的梯阵形式，然后看看是否有矛盾的等式
rref = self.compute_rref()
# 0=k的形式，或者太多主变量，如果有则抛出异常
rref.raise_exception_if_contradictory_equation()
rref.raise_exception_if_too_few_pivots()
# 如果未有异常，则返回一个向量，向量坐标是方程的常量项
num_variables = rref.dimension
solution_coordinates = [
rref.planes[i].constant_term for i in xrange(num_variables)
]
return Vector(solution_coordinates)
def __init__(self, normal_vector=None, constant_term=None):
self.dimension = 3
if not normal_vector:
all_zeros = ['0'] * self.dimension
normal_vector = Vector(all_zeros)
self.normal_vector = normal_vector
if not constant_term:
constant_term = Decimal('0')
self.constant_term = Decimal(constant_term)
self.set_basepoint()
def extract_basepoint_for_parametrization(self):
num_variables = self.dimension
pivot_indices = self.indices_of_first_nonzero_terms_in_each_row()
basepoint_coords = [0] * num_variables
for i, p in enumerate(self.planes):
pivot_var = pivot_indices[i]
if pivot_var < 0:
break
basepoint_coords[pivot_var] = p.constant_term
return Vector(basepoint_coords)
def __init__(self, normal_vector=None, constant_term=None):
self.dimension = 2
# 直线等式的常量
if not normal_vector:
all_zeros = ['0'] * self.dimension
normal_vector = Vector(all_zeros)
self.normal_vector = normal_vector
# 直线法向量
if not constant_term:
constant_term = Decimal('0')
self.constant_term = Decimal(constant_term)
# 在二维空间里，通过法向量，可以快速获得直线的方向向量，法向量更容易类推到多维空间
# 选择一个系数不为零的变量，并将另一个变量设为零，快速算出基准点
self.set_basepoint()
def set_basepoint(self):
try:
n = self.normal_vector
c = self.constant_term
basepoint_coords = ['0'] * self.dimension
initial_index = Plane.first_nonzero_index(n)
initial_coefficient = n[initial_index]
basepoint_coords[initial_index] = c / initial_coefficient
self.basepoint = Vector(basepoint_coords)
except Exception as e:
if str(e) == Plane.NO_NONZERO_ELTS_FOUND_MSG:
self.basepoint = None
else:
raise e
def intersection_with(self, ell):
try:
A, B = self.normal_vector.coordinates
C, D = ell.normal_vector.coordinates
k1 = self.constant_term
k2 = ell.constant_term
x_numerator = D * k1 - B * k2
y_numerator = -C * k1 + A * k2
one_over_denom = Decimal('1') / (A * D - B * C)
return Vector([x_numerator,
y_numerator]).times_scalar(one_over_denom)
except ZeroDivisionError:
# 除数为零的错误
if self == ell:
return self
else:
return None
def extract_direction_vectors_for_parametrization(self):
# 通过不是主变量确定自由变量，然后对每个自由变量构建一个方向向量
num_variables = self.dimension
pivot_indices = self.indices_of_first_nonzero_terms_in_each_row()
free_variable_indices = set(range(num_variables)) - set(pivot_indices)
direction_vectors = []
for free_var in free_variable_indices:
vector_coords = [0] * num_variables
# 自由变量对应的坐标为1，表示将该变量设为等于它自己
vector_coords[free_var] = 1
# 对方程组中的每个等式确定主变量
for i, p in enumerate(self.planes):
pivot_var = pivot_indices[i]
if pivot_var < 0:
break
# 找到等式中自由变量的系数，然后将主变量对应的坐标设为该系数的倒数
# 因为在参数形式化时，需要从等式两边消去自由变量
vector_coords[pivot_var] = -p.normal_vector[free_var]
direction_vectors.append(Vector(vector_coords))
return direction_vectors
# -*- coding: utf-8 -*-
from line import Line
from my_vector import Vector
ell1 = Line(normal_vector=Vector(['4.046', '2.836']), constant_term='1.21')
ell2 = Line(normal_vector=Vector(['10.115', '7.09']), constant_term='3.025')
# print 'intersection 1:', ell1.intersection_with(ell2)
print('intersection 1:', ell1.intersection_with(ell2))
ell3 = Line(normal_vector=Vector(['7.204', '3.182']), constant_term='8.68')
ell4 = Line(normal_vector=Vector(['8.172', '4.114']), constant_term='9.883')
# print 'intersection 2:', ell3.intersection_with(ell4)
print('intersection 2:', ell3.intersection_with(ell4))
ell5 = Line(normal_vector=Vector(['1.182', '5.562']), constant_term='6.744')
ell6 = Line(normal_vector=Vector(['1.773', '8.343']), constant_term='9.525')
# print 'intersection 3:', ell5.intersection_with(ell6)
print('intersection 3:', ell5.intersection_with(ell6))
# -*- coding: utf-8 -*-
from my_vector import Vector
from plane import Plane
from linsys import LinearSystem
from decimal import Decimal
p0 = Plane(normal_vector=Vector(['1', '1', '1']), constant_term='1')
p1 = Plane(normal_vector=Vector(['0', '1', '0']), constant_term='2')
p2 = Plane(normal_vector=Vector(['1', '1', '-1']), constant_term='3')
p3 = Plane(normal_vector=Vector(['1', '0', '-2']), constant_term='2')
s = LinearSystem([p0, p1, p2, p3])
s.swap_rows(0, 1)
if not (s[0] == p1 and s[1] == p0 and s[2] == p2 and s[3] == p3):
print 'test case 1 failed'
s.swap_rows(1, 3)
if not (s[0] == p1 and s[1] == p3 and s[2] == p2 and s[3] == p0):
print 'test case 2 failed'
s.swap_rows(3, 1)
if not (s[0] == p1 and s[1] == p0 and s[2] == p2 and s[3] == p3):
print 'test case 3 failed'
s.multiply_coefficient_and_row(1, 0)
if not (s[0] == p1 and s[1] == p0 and s[2] == p2 and s[3] == p3):
print 'test case 4 failed'
s.multiply_coefficient_and_row(-1, 2)
if not (s[0] == p1 and s[1] == p0
and s[2] == Plane(normal_vector=Vector(['-1', '-1', '1']),
# -*- coding: utf-8 -*-
from plane import Plane
from my_vector import Vector
p1 = Plane(normal_vector=Vector(['-0.412','3.806','0.728']), constant_term='-3.46')
p2 = Plane(normal_vector=Vector(['1.03','-9.515','-1.82']), constant_term='8.65')
print 'first pair of planes are parallel?: {}'.format(p1.is_parallel_to(p2))
print 'first pair of planes are equal?: {}'.format(p1 == p2)
p3 = Plane(normal_vector=Vector(['2.611','5.528','0.283']), constant_term='4.6')
p4 = Plane(normal_vector=Vector(['7.715','8.306','5.342']), constant_term='3.76')
print 'second pair of planes are parallel?: {}'.format(p3.is_parallel_to(p4))
print 'second pair of planes are equal?: {}'.format(p3 == p4)
p5 = Plane(normal_vector=Vector(['-7.926','8.625','-7.212']), constant_term='-7.95')
p6 = Plane(normal_vector=Vector(['-2.642','2.875','-2.404']), constant_term='-2.44')
print 'third pair of planes are parallel?: {}'.format(p5.is_parallel_to(p6))
print 'third pair of planes are equal?: {}'.format(p5 == p6)
# -*- coding: utf-8 -*- from my_vector import Vector v1 = Vector([8.218, -9.341]) w1 = Vector([-1.129, 2.111]) print(v1.plus(w1)) v2 = Vector([7.119, 8.215]) w2 = Vector([-8.223, 0.878]) print(v2.minus(w2)) v3 = Vector([1.671, -1.012, -0.318]) c = 7.41 print(v3.times_scalar(c)) v4 = Vector([-0.221, 7.437]) print(v4.magnitude()) v5 = Vector([8.813, -1.331, -6.247]) print(v5.magnitude()) v6 = Vector([5.581, -2.136]) print(v6.normalized()) v7 = Vector([1.996, 3.108, -4.554]) print(v7.normalized()) v8 = Vector([7.887, 4.138]) w8 = Vector([-8.802, 6.776]) print(v8.dot(w8))
self.direction_vectors = direction_vectors
self.dimension = self.basepoint.dimension
try:
for v in direction_vectors:
assert v.dimension == self.dimension
except AssertionError:
raise Exception(BASEPT_AND_DIR_VECTORS_MUST_BE_IN_SAME_DIM_MSG)
class MyDecimal(Decimal):
def is_near_zero(self, eps=1e-10):
return abs(self) < eps
p0 = Plane(normal_vector=Vector(['1', '1', '1']), constant_term='1')
p1 = Plane(normal_vector=Vector(['0', '1', '0']), constant_term='2')
p2 = Plane(normal_vector=Vector(['1', '1', '-1']), constant_term='3')
p3 = Plane(normal_vector=Vector(['1', '0', '-2']), constant_term='2')
s = LinearSystem([p0, p1, p2, p3])
print s.indices_of_first_nonzero_terms_in_each_row()
print '{},{},{},{}'.format(s[0], s[1], s[2], s[3])
print len(s)
print s
s[0] = p1
print s
print MyDecimal('1e-9').is_near_zero()
from my_vector import Vector
if __name__ == "__main__":
my_v = Vector([0.221, 7.437])
print(my_v.magnitude())
my_v = Vector([8.813, -1.331, -6.247])
print(my_v.magnitude())
#--------------------------
my_v = Vector([5.581, -2.136])
print(my_v.normalized())
my_v = Vector([1.996, 3.108, -4.554])
print(my_v.normalized())