高斯消元法求解线性方程组（分数精确表示）
原理可以参考
https://zh.wikipedia.org/wiki/%E9%AB%98%E6%96%AF%E6%B6%88%E5%8E%BB%E6%B3%95
这里给出使用分数表示计算过程的高斯消元法写法
github地址：https://github.com/YIWANFENG/Algorithm-github
#!/usr/bin/env python3.5
# -*-coding: utf-8 -*-
"""
Created on 2018-6-9
@author: YWF
高斯消元法求解线性方程，
数字采用分数精确表示，无近似
"""
import numpy as np
def gcd(a,b):
if a < b:
c = a
a = b
b = c
while a % b != 0:
c = a % b
a = b
b = c
return b
def lcm(a,b):
return a * b / gcd(a,b)
class Fractional:
def __init__(self,num,den):
self.numerator = num
if(den == 0):
print("Error den")
self.denominator = den
pass
def Simplify(self):
if self.numerator == 0:
return Fractional(0,1)
gcd_ = gcd(self.denominator,self.numerator)
if gcd_ != 1:
self.denominator /= gcd_
self.numerator /= gcd_
return self
def __sub__(self, other):
lcm_ = lcm(self.denominator,other.denominator)
self.numerator *= lcm_ / self.denominator
num = self.numerator - other.numerator * lcm_ / other.denominator
return Fractional(num,lcm_).Simplify()
def __add__(self, other):
lcm_ = lcm(self.denominator,other.denominator)
self.numerator *= lcm_ / self.denominator
num = self.numerator + other.numerator * lcm_ / other.denominator
return Fractional(num,lcm_).Simplify()
def __truediv__(self, other):
return Fractional(self.numerator*other.denominator,
self.denominator*other.numerator).Simplify()
def __mul__(self, other):
return Fractional(self.numerator*other.numerator,
self.denominator*other.denominator).Simplify()
def __gt__(self, other):
item = self - other
if item.numerator > 0 and item.denominator > 0:
return True
return False
def __lt__(self, other):
return not self > other
def __eq__(self, other):
if self.Simplify().numerator == other.Simplify().numerator:
return True
return False
def __ne__(self, other):
return not self == other
def Decimal(self):
return self.numerator / self.denominator
def __repr__(self):
return ("%d/%d = %f"%(self.numerator,self.denominator,self.Decimal()))
def list_is_all_zero(a):
for x in a:
if x != Fractional(0,1):
return False
return True
def Guass(A,row,num_var):
'''
求解 AX = b
:param A: 增广矩阵(为python中的list)
:param row: 矩阵行数
:param num_var: 方程所含变量数
:return: status, x_out, x_base
status = 0 无解, 1 唯一解, -1 无穷解
x_out 为唯一解 or 特解
x_base 当无穷解时为基础解析，以及自由变元编号
'''
col = num_var + 1
r = 0
c = 0
x_free =[]
x_out = []
x_base = []
status = 0
###生成阶梯矩阵
while c < num_var and r < row:
#当前列中且行号不小于列号的数的最大的行号
#为了后续计算时可以简化（即continue）
Max_r = r
for max_r_in_c in range(r+1, row):
if A[max_r_in_c][c] > A[Max_r][c]:
Max_r = max_r_in_c
A[r], A[Max_r] = A[Max_r], A[r]
if A[r][c] == Fractional(0, 1):
x_free.append(c) #回收自由变元
c += 1
continue
div = A[r][c]
for c_cur in range(c, col):
A[r][c_cur] = A[r][c_cur] / div
for r_cur in range(r+1, row):
if A[r_cur][c] == Fractional(0, 1):
continue
div = A[r_cur][c] / A[r][c]
for c_cur in range(c, col):
A[r_cur][c_cur] = A[r_cur][c_cur] - div * A[r][c_cur]
c += 1
r += 1
pass
# 回收自由变元 (当方程数不足,自由变元较多时)
for x in range(c,num_var):
x_free.append(x)
#print("check",x_free)
#show(A)
# 查看有效方程数
row_valid = 0
for x in A:
if not list_is_all_zero(x):
row_valid += 1
#print("有效的方程数", row_valid)
### 检查解的个数
# 无解
for r_cur in A[0:row_valid]:
if r_cur[col-1] != Fractional(0,1) and list_is_all_zero(r_cur[0:col-1]):
status = 0
return status,x_out,x_base
####有解时生成最简矩阵(可选)
row_staris = row_valid - 1 #最低（当前处理）阶梯所在行
for c_cur in range(col-2, -1, -1): #从最后一列开始消除
if c_cur not in x_free and row_staris != 0:
for r_cur in range(row_staris-1, -1, -1): #从下向上每行依次减
div = A[r_cur][c_cur] / A[row_staris][c_cur]
for x in range(c_cur, col): #处理该行
A[r_cur][x] -= A[row_staris][x] * div
pass
row_staris -= 1
pass
### 唯一解
if row_valid == num_var:
for r in range(0, row):
x_out.append(A[r][col-1])
status = 1
return status,x_out,x_base
### 无穷解
b = []
for xx in range(row_valid):
b.append(A[xx][col-1])
x_out = [Fractional(0,1) for i in range(num_var)]
#获得基础解析x_base与特解x_out
for x in x_free: #针对每个自由变元
item = [Fractional(0,1) for i in range(num_var)]
item[x] = Fractional(1,1)
cnt = 0
for xx in range(num_var):
if xx not in x_free:
item[xx] = b[cnt] - A[cnt][x]
x_out[xx] = b[cnt]
cnt += 1
x_base.append(item)
x_base.append(["自由变元编号"] + x_free)
status = -1
return status, x_out, x_base
def show(A):
for i in range(0,len(A)):
print(A[i])
print("\n")
if __name__ == '__main__':
# Test
f = Fractional
A = [[f(1,1),f(2,1),f(3,1),f(4,1)],
[f(1,2),f(2,2),f(3,2),f(4,2)],
[f(4,2),f(3,3),f(1,3),f(3,2)]
]
B = [[f(1,1),f(2,1),f(3,1),f(4,1)],
[f(0,1),f(0,1),f(2,1),f(4,1)],
[f(0,1),f(0,1),f(2,1),f(4,1)]
]
C = [[f(1,1),f(2,1),f(3,1),f(1,2),f(4,1)],
[f(0,1),f(0,1),f(2,1),f(1,2),f(4,1)],
]
D = [[f(1,1),f(2,1),f(0,1),f(4,1),f(3,1)],
[f(0,1),f(0,1),f(1,1),f(2,1),f(2,1)],
]
x = []
x_base = []
#show(C)
#status,x_out,x_base = Guass(C,2,4)
#show(B)
#status,x_out,x_base = Guass(B,3,3)
#show(A)
#status,x_out,x_base = Guass(A,3,3)
show(D)
status,x_out,x_base = Guass(D,2,4)
print(status)
print(x_base)
print(x_out)