python分治法矩阵乘法
时间: 2023-07-06 07:08:32 浏览: 63
矩阵乘法是计算机科学中的一个基本操作,分治法可以用于优化矩阵乘法的时间复杂度。下面是使用分治法实现矩阵乘法的 Python 代码:
```python
def matrix_multiply_divide_and_conquer(a, b):
n = len(a)
if n == 1:
return [[a[0][0] * b[0][0]]]
else:
# 将矩阵 a 和 b 分成四个子矩阵
a11, a12, a21, a22 = divide_matrix(a)
b11, b12, b21, b22 = divide_matrix(b)
# 递归计算子矩阵的乘积
c11 = add_matrix(matrix_multiply_divide_and_conquer(a11, b11), matrix_multiply_divide_and_conquer(a12, b21))
c12 = add_matrix(matrix_multiply_divide_and_conquer(a11, b12), matrix_multiply_divide_and_conquer(a12, b22))
c21 = add_matrix(matrix_multiply_divide_and_conquer(a21, b11), matrix_multiply_divide_and_conquer(a22, b21))
c22 = add_matrix(matrix_multiply_divide_and_conquer(a21, b12), matrix_multiply_divide_and_conquer(a22, b22))
# 将子矩阵的乘积组合成结果矩阵
c = [[0] * n for _ in range(n)]
for i in range(n//2):
for j in range(n//2):
c[i][j] = c11[i][j]
c[i][j+n//2] = c12[i][j]
c[i+n//2][j] = c21[i][j]
c[i+n//2][j+n//2] = c22[i][j]
return c
def divide_matrix(matrix):
n = len(matrix)
m = n // 2
a11 = [[matrix[i][j] for j in range(m)] for i in range(m)]
a12 = [[matrix[i][j] for j in range(m, n)] for i in range(m)]
a21 = [[matrix[i][j] for j in range(m)] for i in range(m, n)]
a22 = [[matrix[i][j] for j in range(m, n)] for i in range(m, n)]
return a11, a12, a21, a22
def add_matrix(a, b):
n = len(a)
c = [[0] * n for _ in range(n)]
for i in range(n):
for j in range(n):
c[i][j] = a[i][j] + b[i][j]
return c
```
这个算法的时间复杂度为 $O(n^3)$,但由于使用了分治法,实际运行速度可能会比普通的矩阵乘法快。