高斯消元法
节约内存的算法：
例1：用高斯消元法求解线性方程组：
# -*- coding: utf-8 -*-
"""
求解线性方程组：
[12 -3 3] [x1] = [15]
[18 -3 1] [x2] = [15]
[-1 2 1] [x3] = [ 6]
@athor: morxio
"""
import sys
import numpy as np
def GassianEliminationSolver(a):
n = a.shape[0]
for k in range(0,n-1): #0,1,2,...,n-2
print(a)
if np.abs(a[k,k]) < 1e-6:
print("主元素过小")
sys.exit(1)
for i in range(k+1,n): #k+1, k+2, ..., n-1
a[i,k] /= a[k,k]
for j in range(k+1,n+1):
a[i,j] -= a[i,k] * a[k,j]
#a[n-1,n] /= a[n-1][n-1]
for i in range(n-1, -1, -1):
sum = 0
for j in range(i+1,n):
sum += a[i,j] * a[j,n]
a[i,n] = (a[i,n] - sum) / a[i,i]
return a[:,n]
def main():
a = np.array([[12., -3., 3., 15.],
[18., -3., 1., 15.],
[-1., 2., 1., 6.]])
x = GassianEliminationSolver(a)
print(x)
if __name__ == '__main__':
main()
结果：[[12. -3. 3. 15.]
[18. -3. 1. 15.]
[-1. 2. 1. 6.]]
[[12. -3. 3. 15. ]
[ 1.5 1.5 -3.5 -7.5 ]
[-0.08333333 1.75 1.25 7.25 ]]
[1. 2. 3.]
