线性代数-Python-04：线性系统+高斯消元的实现
文章目录
- 1 线性系统
- 2 高斯-jordon消元法的实现
-
-
- 2.1 Matrix
- 2.2 Vector
- 2.3 线性系统
-
- 3 行最简形式
- 4 线性方程组的结构
- 5 线性方程组-通用高斯消元的实现
-
- 5.1 global
- 5.2 Vector-引入is_zero
- 5.3 LinearSystem
- 5.4 main
1 线性系统
2 高斯-jordon消元法的实现
2.1 Matrix
from .Vector import Vector
class Matrix:
def __init__(self, list2d):
self._values = [row[:] for row in list2d]
@classmethod
def zero(cls, r, c):
"""返回一个r行c列的零矩阵"""
return cls([[0] * c for _ in range(r)])
@classmethod
def identity(cls, n):
"""返回一个n行n列的单位矩阵"""
m = [[0]*n for _ in range(n)]
for i in range(n):
m[i][i] = 1;
return cls(m)
def T(self):
"""返回矩阵的转置矩阵"""
return Matrix([[e for e in self.col_vector(i)]
for i in range(self.col_num())])
def __add__(self, another):
"""返回两个矩阵的加法结果"""
assert self.shape() == another.shape(), \
"Error in adding. Shape of matrix must be same."
return Matrix([[a + b for a, b in zip(self.row_vector(i), another.row_vector(i))]
for i in range(self.row_num())])
def __sub__(self, another):
"""返回两个矩阵的减法结果"""
assert self.shape() == another.shape(), \
"Error in subtracting. Shape of matrix must be same."
return Matrix([[a - b for a, b in zip(self.row_vector(i), another.row_vector(i))]
for i in range(self.row_num())])
def dot(self, another):
"""返回矩阵乘法的结果"""
if isinstance(another, Vector):
# 矩阵和向量的乘法
assert self.col_num() == len(another), \
"Error in Matrix-Vector Multiplication."
return Vector([self.row_vector(i).dot(another) for i in range(self.row_num())])
if isinstance(another, Matrix):
# 矩阵和矩阵的乘法
assert self.col_num() == another.row_num(), \
"Error in Matrix-Matrix Multiplication."
return Matrix([[self.row_vector(i).dot(another.col_vector(j)) for j in range(another.col_num())]
for i in range(self.row_num())])
def __mul__(self, k):
"""返回矩阵的数量乘结果: self * k"""
return Matrix([[e * k for e in self.row_vector(i)]
for i in range(self.row_num())])
def __rmul__(self, k):
"""返回矩阵的数量乘结果: k * self"""
return self * k
def __truediv__(self, k):
"""返回数量除法的结果矩阵：self / k"""
return (1 / k) * self
def __pos__(self):
"""返回矩阵取正的结果"""
return 1 * self
def __neg__(self):
"""返回矩阵取负的结果"""
return -1 * self
def row_vector(self, index):
"""返回矩阵的第index个行向量"""
return Vector(self._values[index])
def col_vector(self, index):
"""返回矩阵的第index个列向量"""
return Vector([row[index] for row in self._values])
def __getitem__(self, pos):
"""返回矩阵pos位置的元素"""
r, c = pos
return self._values[r][c]
def size(self):
"""返回矩阵的元素个数"""
r, c = self.shape()
return r * c
def row_num(self):
"""返回矩阵的行数"""
return self.shape()[0]
__len__ = row_num
def col_num(self):
"""返回矩阵的列数"""
return self.shape()[1]
def shape(self):
"""返回矩阵的形状: (行数， 列数)"""
return len(self._values), len(self._values[0])
def __repr__(self):
return "Matrix({})".format(self._values)
__str__ = __repr__
2.2 Vector
import math
from ._globals import EPSILON
class Vector:
def __init__(self, lst):
"""
__init__ 代表类的构造函数
双下划线开头的变量 例如_values,代表类的私有成员
lst是个引用，list(lst)将值复制一遍，防止用户修改值
"""
self._values = list(lst)
def underlying_list(self):
"""返回向量的底层列表"""
return self._values[:]
def dot(self, another):
"""向量点乘，返回结果标量"""
assert len(self) == len(another), \
"Error in dot product. Length of vectors must be same."
return sum(a * b for a, b in zip(self, another))
def norm(self):
"""返回向量的模"""
return math.sqrt(sum(e**2 for e in self))
def normalize(self):
"""
归一化，规范化
返回向量的单位向量
此处设计到了除法: def __truediv__(self, k):
"""
if self.norm() < EPSILON:
raise ZeroDivisionError("Normalize error! norm is zero.")
return Vector(self._values) / self.norm()
# return 1 / self.norm() * Vector(self._values)
# return Vector([e / self.norm() for e in self])
def __truediv__(self, k):
"""返回数量除法的结果向量：self / k"""
return (1 / k) * self
@classmethod
def zero(cls, dim):
"""返回一个dim维的零向量
@classmethod 修饰符对应的函数不需要实例化，不需要 self 参数，但第一个参数需要是表示自身类的cls参数，可以来调用类的属性，类的方法，实例化对象等。
"""
return cls([0] * dim)
def __add__(self, another):
"""向量加法，返回结果向量"""
assert len(self) == len(another), \
"Error in adding. Length of vectors must be same."
# return Vector([a + b for a, b in zip(self._values, another._values)])
return Vector([a + b for a, b in zip(self, another)])
def __sub__(self, another):
"""向量减法，返回结果向量"""
assert len(self) == len(another), \
"Error in subtracting. Length of vectors must be same."
return Vector([a - b for a, b in zip(self, another)])
def __mul__(self, k):
"""返回数量乘法的结果向量：self * k"""
return Vector([k * e for e in self])
def __rmul__(self, k):
"""
返回数量乘法的结果向量：k * self
self本身就是一个列表
"""
return self * k
def __pos__(self):
"""返回向量取正的结果向量"""
return 1 * self
def __neg__(self):
"""返回向量取负的结果向量"""
return -1 * self
def __iter__(self):
"""返回向量的迭代器"""
return self._values.__iter__()
def __getitem__(self, index):
"""取向量的第index个元素"""
return self._values[index]
def __len__(self):
"""返回向量长度（有多少个元素）"""
return len(self._values)
def __repr__(self):
"""打印显示：Vector([5, 2])"""
return "Vector({})".format(self._values)
def __str__(self):
"""打印显示：(5, 2)"""
return "({})".format(", ".join(str(e) for e in self._values))
2.3 线性系统
from .Matrix import Matrix
from .Vector import Vector
class LinearSystem:
def __init__(self, A, b):
assert A.row_num() == len(b), "row number of A must be equal to the length of b"
self._m = A.row_num()
self._n = A.col_num()
assert self._m == self._n # TODO: no this restriction
self.Ab = [Vector(A.row_vector(i).underlying_list() + [b[i]])
for i in range(self._m)]
def _max_row(self, index_i, index_j, n):
best, ret = abs(self.Ab[index_i][index_j]), index_i
for i in range(index_i + 1, n):
if abs(self.Ab[i][index_j]) > best:
best, ret = abs(self.Ab[i][index_j]), i
return ret
def _forward(self):
n = self._m
for i in range(n):
# Ab[i][i]为主元
max_row = self._max_row(i, i, n)
self.Ab[i], self.Ab[max_row] = self.Ab[max_row], self.Ab[i]
# 将主元归为一
self.Ab[i] = self.Ab[i] / self.Ab[i][i] # TODO: self.Ab[i][i] == 0?
for j in range(i + 1, n):
self.Ab[j] = self.Ab[j] - self.Ab[j][i] * self.Ab[i]
def _backward(self):
n = self._m
for i in range(n - 1, -1, -1):
# Ab[i][i]为主元
for j in range(i - 1, -1, -1):
self.Ab[j] = self.Ab[j] - self.Ab[j][i] * self.Ab[i]
def gauss_jordan_elimination(self):
self._forward()
self._backward()
def fancy_print(self):
for i in range(self._m):
print(" ".join(str(self.Ab[i][j]) for j in range(self._n)), end=" ")
print("|", self.Ab[i][-1])
3 行最简形式
4 线性方程组的结构
5 线性方程组-通用高斯消元的实现
5.1 global
# 包中的变量，但是对包外不可见，因此使用“_”开头
EPSILON = 1e-8
def is_zero(x):
return abs(x) < EPSILON
def is_equal(a, b):
return abs(a - b) < EPSILON
5.2 Vector-引入is_zero
import math
from ._globals import is_zero
class Vector:
def __init__(self, lst):
"""
__init__ 代表类的构造函数
双下划线开头的变量 例如_values,代表类的私有成员
lst是个引用，list(lst)将值复制一遍，防止用户修改值
"""
self._values = list(lst)
def underlying_list(self):
"""返回向量的底层列表"""
return self._values[:]
def dot(self, another):
"""向量点乘，返回结果标量"""
assert len(self) == len(another), \
"Error in dot product. Length of vectors must be same."
return sum(a * b for a, b in zip(self, another))
def norm(self):
"""返回向量的模"""
return math.sqrt(sum(e**2 for e in self))
def normalize(self):
"""
归一化，规范化
返回向量的单位向量
此处设计到了除法: def __truediv__(self, k):
"""
if is_zero(self.norm()):
raise ZeroDivisionError("Normalize error! norm is zero.")
return Vector(self._values) / self.norm()
# return 1 / self.norm() * Vector(self._values)
# return Vector([e / self.norm() for e in self])
def __truediv__(self, k):
"""返回数量除法的结果向量：self / k"""
return (1 / k) * self
@classmethod
def zero(cls, dim):
"""返回一个dim维的零向量
@classmethod 修饰符对应的函数不需要实例化，不需要 self 参数，但第一个参数需要是表示自身类的cls参数，可以来调用类的属性，类的方法，实例化对象等。
"""
return cls([0] * dim)
def __add__(self, another):
"""向量加法，返回结果向量"""
assert len(self) == len(another), \
"Error in adding. Length of vectors must be same."
# return Vector([a + b for a, b in zip(self._values, another._values)])
return Vector([a + b for a, b in zip(self, another)])
def __sub__(self, another):
"""向量减法，返回结果向量"""
assert len(self) == len(another), \
"Error in subtracting. Length of vectors must be same."
return Vector([a - b for a, b in zip(self, another)])
def __mul__(self, k):
"""返回数量乘法的结果向量：self * k"""
return Vector([k * e for e in self])
def __rmul__(self, k):
"""
返回数量乘法的结果向量：k * self
self本身就是一个列表
"""
return self * k
def __pos__(self):
"""返回向量取正的结果向量"""
return 1 * self
def __neg__(self):
"""返回向量取负的结果向量"""
return -1 * self
def __iter__(self):
"""返回向量的迭代器"""
return self._values.__iter__()
def __getitem__(self, index):
"""取向量的第index个元素"""
return self._values[index]
def __len__(self):
"""返回向量长度（有多少个元素）"""
return len(self._values)
def __repr__(self):
"""打印显示：Vector([5, 2])"""
return "Vector({})".format(self._values)
def __str__(self):
"""打印显示：(5, 2)"""
return "({})".format(", ".join(str(e) for e in self._values))
5.3 LinearSystem
from .Matrix import Matrix
from .Vector import Vector
from ._globals import is_zero
class LinearSystem:
def __init__(self, A, b):
assert A.row_num() == len(b), "row number of A must be equal to the length of b"
self._m = A.row_num()
self._n = A.col_num()
# assert self._m == self._n # TODO: no this restriction
self.Ab = [Vector(A.row_vector(i).underlying_list() + [b[i]])
for i in range(self._m)]
self.pivots = []
def _max_row(self, index_i, index_j, n):
best, ret = abs(self.Ab[index_i][index_j]), index_i
for i in range(index_i + 1, n):
if abs(self.Ab[i][index_j]) > best:
best, ret = abs(self.Ab[i][index_j]), i
return ret
def _forward(self):
i, k = 0, 0
while i < self._m and k < self._n:
# 看Ab[i][k]位置是否可以是主元
max_row = self._max_row(i, k, self._m)
self.Ab[i], self.Ab[max_row] = self.Ab[max_row], self.Ab[i]
if is_zero(self.Ab[i][k]):
k += 1
else:
# 将主元归为一
self.Ab[i] = self.Ab[i] / self.Ab[i][k]
for j in range(i + 1, self._m):
self.Ab[j] = self.Ab[j] - self.Ab[j][k] * self.Ab[i]
self.pivots.append(k)
i += 1
def _backward(self):
n = len(self.pivots)
for i in range(n - 1, -1, -1):
k = self.pivots[i]
# Ab[i][k]为主元
for j in range(i - 1, -1, -1):
self.Ab[j] = self.Ab[j] - self.Ab[j][k] * self.Ab[i]
def gauss_jordan_elimination(self):
"""如果有解，返回True；如果没有解，返回False"""
self._forward()
self._backward()
for i in range(len(self.pivots), self._m):
if not is_zero(self.Ab[i][-1]):
return False
return True
def fancy_print(self):
for i in range(self._m):
print(" ".join(str(self.Ab[i][j]) for j in range(self._n)), end=" ")
print("|", self.Ab[i][-1])
5.4 main
from playLA.Matrix import Matrix
from playLA.Vector import Vector
from playLA.LinearSystem import LinearSystem
if __name__ == "__main__":
A = Matrix([[1, 2, 4], [3, 7, 2], [2, 3, 3]])
b = Vector([7, -11, 1])
ls = LinearSystem(A, b)
ls.gauss_jordan_elimination()
ls.fancy_print()
print()
# [-1, -2, 3]
A2 = Matrix([[1, -3, 5], [2, -1, -3], [3, 1, 4]])
b2 = Vector([-9, 19, -13])
ls2 = LinearSystem(A2, b2)
ls2.gauss_jordan_elimination()
ls2.fancy_print()
print()
# [2, -3, -4]
A3 = Matrix([[1, 2, -2], [2, -3, 1], [3, -1, 3]])
b3 = Vector([6, -10, -16])
ls3 = LinearSystem(A3, b3)
ls3.gauss_jordan_elimination()
ls3.fancy_print()
print()
# [-2, 1, -3]
A4 = Matrix([[3, 1, -2], [5, -3, 10], [7, 4, 16]])
b4 = Vector([4, 32, 13])
ls4 = LinearSystem(A4, b4)
ls4.gauss_jordan_elimination()
ls4.fancy_print()
print()
# [3, -4, 0.5]
A5 = Matrix([[6, -3, 2], [5, 1, 12], [8, 5, 1]])
b5 = Vector([31, 36, 11])
ls5 = LinearSystem(A5, b5)
ls5.gauss_jordan_elimination()
ls5.fancy_print()
print()
# [3, -3, 2]
A6 = Matrix([[1, 1, 1], [1, -1, -1], [2, 1, 5]])
b6 = Vector([3, -1, 8])
ls6 = LinearSystem(A6, b6)
ls6.gauss_jordan_elimination()
ls6.fancy_print()
print()
# [1, 1, 1]
A7 = Matrix([[1, -1, 2, 0, 3],
[-1, 1, 0, 2, -5],
[1, -1, 4, 2, 4],
[-2, 2, -5, -1, -3]])
b7 = Vector([1, 5, 13, -1])
ls7 = LinearSystem(A7, b7)
ls7.gauss_jordan_elimination()
ls7.fancy_print()
print()
A8 = Matrix([[2, 2],
[2, 1],
[1, 2]])
b8 = Vector([3, 2.5, 7])
ls8 = LinearSystem(A8, b8)
if not ls8.gauss_jordan_elimination():
print("No Solution!")
ls8.fancy_print()
print()
A9 = Matrix([[2, 0, 1],
[-1, -1, -2],
[-3, 0, 1]])
b9 = Vector([1, 0, 0])
ls9 = LinearSystem(A9, b9)
if not ls9.gauss_jordan_elimination():
print("No Solution!")
ls9.fancy_print()
print()
