1.采用递归分治的方法进行矩阵相乘，代码如下：
def matrix_divide(mat):
"""将n*n分解为4个n/2*n/2的四个矩阵"""
n = len(mat) / 2
a11 = [[0 for i in range(n)]for j in range(n)]
a12 = [[0 for i in range(n)] for j in range(n)]
a21 = [[0 for i in range(n)] for j in range(n)]
a22 = [[0 for i in range(n)] for j in range(n)]
for i in range(n):
for j in range(n):
a11[i][j] = mat[i][j]
a12[i][j] = mat[i][n+j]
a21[i][j] = mat[n+i][j]
a22[i][j] = mat[n+i][n+j]
return a11, a12, a21, a22
def matrix_add(a, b):
"""将两个矩阵相加"""
n = len(a)
c = [[0 for i in range(n)]for j in range(n)]
for i in range(n):
for j in range(n):
c[i][j] = a[i][j] + b[i][j]
return c
def matrix_merge(a11, a12, a21, a22):
"""将子矩阵合并成一大矩阵"""
n = len(a11)
a = [[0 for i in range(2*n)]for j in range(2*n)]
for i in range(2*n):
for j in range(2 * n):
if i <= n-1 and j <= n-1:
a[i][j] = a11[i][j]
elif i <= n-1 and j > n-1:
a[i][j] = a12[i][j-n]
elif i > n-1 and j <= n-1:
a[i][j] = a21[i-n][j]
else:
a[i][j] = a22[i-n][j-n]
return a
def matrix_multiply(a, b):
"""两个矩阵递归相乘"""
n = len(a)
c = [[0 for i in range(n)] for j in range(n)]
if n == 1:
c[0][0] = a[0][0] * b[0][0]
else:
a11, a12, a21, a22 = matrix_divide(a)
b11, b12, b21, b22 = matrix_divide(b)
c11, c12, c21, c22 = matrix_divide(c)
c11 = matrix_add(matrix_multiply(a11, b11), matrix_multiply(a12, b21))
c12 = matrix_add(matrix_multiply(a11, b12), matrix_multiply(a12, b22))
c21 = matrix_add(matrix_multiply(a21, b11), matrix_multiply(a22, b21))
c22 = matrix_add(matrix_multiply(a21, b12), matrix_multiply(a22, b22))
c = matrix_merge(c11, c12, c21, c22)
return c
if __name__ == "__main__":
a = [[1, 1, 1, 1], [1, 1, 1, 1], [2, 2, 2, 2], [2, 2, 2, 2]]
b = a
print matrix_multiply(a,b)2.Strassen算法实现
def matrix_divide(mat):
"""将n*n分解为4个n/2*n/2的四个矩阵"""
n = len(mat) / 2
a11 = [[0 for i in range(n)]for j in range(n)]
a12 = [[0 for i in range(n)] for j in range(n)]
a21 = [[0 for i in range(n)] for j in range(n)]
a22 = [[0 for i in range(n)] for j in range(n)]
for i in range(n):
for j in range(n):
a11[i][j] = mat[i][j]
a12[i][j] = mat[i][n+j]
a21[i][j] = mat[n+i][j]
a22[i][j] = mat[n+i][n+j]
return a11, a12, a21, a22
def matrix_merge(a11, a12, a21, a22):
"""将子矩阵合并成一大矩阵"""
n = len(a11)
a = [[0 for i in range(2*n)]for j in range(2*n)]
for i in range(2*n):
for j in range(2 * n):
if i <= n-1 and j <= n-1:
a[i][j] = a11[i][j]
elif i <= n-1 and j > n-1:
a[i][j] = a12[i][j-n]
elif i > n-1 and j <= n-1:
a[i][j] = a21[i-n][j]
else:
a[i][j] = a22[i-n][j-n]
return a
def matrix_add(a, b):
"""将两个矩阵相加"""
n = len(a)
c = [[0 for i in range(n)]for j in range(n)]
for i in range(n):
for j in range(n):
c[i][j] = a[i][j] + b[i][j]
return c
def matrix_minus(a, b):
"""将两个矩阵相加"""
n = len(a)
c = [[0 for i in range(n)]for j in range(n)]
for i in range(n):
for j in range(n):
c[i][j] = a[i][j] - b[i][j]
return c
def strassen_multiply(a, b):
n = len(a)
c = [[0 for i in range(n)]for j in range(n)]
if n == 1:
c[0][0] = a[0][0] * b[0][0]
else:
a11, a12, a21, a22 = matrix_divide(a)
b11, b12, b21, b22 = matrix_divide(b)
c11, c12, c21, c22 = matrix_divide(c)
s1 = matrix_minus(b12, b22)
s2 = matrix_add(a11, a12)
s3 = matrix_add(a21, a22)
s4 = matrix_minus(b21, b11)
s5 = matrix_add(a11, a22)
s6 = matrix_add(b11, b22)
s7 = matrix_minus(a12, a22)
s8 = matrix_add(b21, b22)
s9 = matrix_minus(a11, a21)
s10 = matrix_add(b11, b12)
p1 = strassen_multiply(a11, s1)
p2 = strassen_multiply(s2, b22)
p3 = strassen_multiply(s3, b11)
p4 = strassen_multiply(a22, s4)
p5 = strassen_multiply(s5, s6)
p6 = strassen_multiply(s7, s8)
p7 = strassen_multiply(s9, s10)
c11 = matrix_add(matrix_minus(matrix_add(p5, p4), p2), p6)
c12 = matrix_add(p1, p2)
c21 = matrix_add(p3, p4)
c22 = matrix_minus(matrix_minus(matrix_add(p5, p1), p3), p7)
c = matrix_merge(c11, c12, c21, c22)
return c
if __name__ == "__main__":
a = [[1, 1, 1, 1], [1, 1, 1, 1], [2, 2, 2, 2], [2, 2, 2, 2]]
b = a
print(strassen_multiply(a, b))3.前两个都需要矩阵是2的n次方，且必须是方阵，如果要求任意矩阵，可以考虑先用零填充到2的n次方，最后再删除到原大小。
# -*- coding:utf-8 -*-
#_author_ = 'xuqn'
from math import *
def matrix_divide(mat):
"""将n*n分解为4个n/2*n/2的四个矩阵"""
n = len(mat) / 2
a11 = [[0 for i in range(n)]for j in range(n)]
a12 = [[0 for i in range(n)] for j in range(n)]
a21 = [[0 for i in range(n)] for j in range(n)]
a22 = [[0 for i in range(n)] for j in range(n)]
for i in range(n):
for j in range(n):
a11[i][j] = mat[i][j]
a12[i][j] = mat[i][n+j]
a21[i][j] = mat[n+i][j]
a22[i][j] = mat[n+i][n+j]
return a11, a12, a21, a22
def matrix_add(a, b):
"""将两个矩阵相加"""
n = len(a)
c = [[0 for i in range(n)]for j in range(n)]
for i in range(n):
for j in range(n):
c[i][j] = a[i][j] + b[i][j]
return c
def matrix_merge(a11, a12, a21, a22):
"""将子矩阵合并成一大矩阵"""
n = len(a11)
a = [[0 for i in range(2*n)]for j in range(2*n)]
for i in range(2*n):
for j in range(2 * n):
if i <= n-1 and j <= n-1:
a[i][j] = a11[i][j]
elif i <= n-1 and j > n-1:
a[i][j] = a12[i][j-n]
elif i > n-1 and j <= n-1:
a[i][j] = a21[i-n][j]
else:
a[i][j] = a22[i-n][j-n]
return a
def matrix_expand(mat):
"""将a扩展成2的n次方，0填充"""
n = max(len(mat),len(mat[0]))
m = ceil(log(n, 2))
n_expand = int(2 ** m)
c = [[0 for i in range(n_expand)] for j in range(n_expand)]
for i in range(n_expand):
for j in range(n_expand):
if i < len(mat) and j < len(mat[0]):
c[i][j] = mat[i][j]
else:
c[i][j] = 0
return c
def matrix_shrink(mat, high, weight):
"""将a中之前填充的0去掉，输入需要降到的维度"""
c = [[0 for i in range(weight)] for j in range(high)]
for i in range(high):
for j in range(weight):
c[i][j] = mat[i][j]
return c
def matrix_multiply(a, b):
"""两个矩阵递归相乘"""
n = len(a)
c = [[0 for i in range(n)] for j in range(n)]
if n == 1:
c[0][0] = a[0][0] * b[0][0]
else:
a11, a12, a21, a22 = matrix_divide(a)
b11, b12, b21, b22 = matrix_divide(b)
c11, c12, c21, c22 = matrix_divide(c)
c11 = matrix_add(matrix_multiply(a11, b11), matrix_multiply(a12, b21))
c12 = matrix_add(matrix_multiply(a11, b12), matrix_multiply(a12, b22))
c21 = matrix_add(matrix_multiply(a21, b11), matrix_multiply(a22, b21))
c22 = matrix_add(matrix_multiply(a21, b12), matrix_multiply(a22, b22))
c = matrix_merge(c11, c12, c21, c22)
return c
if __name__ == "__main__":
a = [[1, 1, 1, 1], [1, 1, 1, 1], [2, 2, 2, 2]]
b = [[1, 1], [2, 2], [2, 1], [1, 1]]
print(matrix_shrink(matrix_multiply(matrix_expand(a), matrix_expand(b)), len(a), len(b[0])))
编程时看了网友的程序，在此表示感谢：
https://blog.csdn.net/weixin_42206504/article/details/81042798