线性代数-Python-04：线性系统+高斯消元的实现

大大枫2023-11-11 9:14

文章目录

  • [1 线性系统](#1 线性系统)
  • [2 高斯-jordon消元法的实现](#2 高斯-jordon消元法的实现)
      • [2.1 Matrix](#2.1 Matrix)
      • [2.2 Vector](#2.2 Vector)
      • [2.3 线性系统](#2.3 线性系统)
  • [3 行最简形式](#3 行最简形式)
  • [4 线性方程组的结构](#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 线性方程组-通用高斯消元的实现)
    • [5.1 global](#5.1 global)
    • [5.2 Vector-引入is_zero](#5.2 Vector-引入is_zero)
    • [5.3 LinearSystem](#5.3 LinearSystem)
    • [5.4 main](#5.4 main)

1 线性系统



2 高斯-jordon消元法的实现

2.1 Matrix

c 复制代码 
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

c 复制代码 
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

c 复制代码 
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

c 复制代码 
# 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

c 复制代码 
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

c 复制代码 
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

c 复制代码 
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()
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