問題描述

分別實現矩陣相乘的3種演算法，比較三種演算法在矩陣大小分別為$2^2*2^2$, $2^3*2^3$, $2^4*2^4$, $2^5*2^5$, $2^6*2^6$, $2^7*2^7$, $2^8*2^8$, $2^9*2^9$時的執行時間與MATLAB自帶的矩陣相乘的執行時間，繪製時間對比圖。

解題方法

本文采用了以下方法進行求值：矩陣計演算法、定義法、分治法和Strassen方法。這裡我們使用Matlab以及Python對這個問題進行處理，比較兩種語言在一樣的條件下，運算速度的差別。

程式語言

Python

具體程式碼

#-*- coding: utf-8 -*-
from matplotlib.font_manager import FontProperties
import numpy as np
import time
import random
import math
import copy
import matplotlib.pyplot as plt

#n = [2**2, 2**3, 2**4, 2**5, 2**6, 2**7, 2**8, 2**9, 2**10, 2**11, 2**12]
n = [2**2, 2**3, 2**4, 2
 
**5, 2**6, 2**7, 2**8, 2**9, 2**10, 2**11]
Sum_time1 = []
Sum_time2 = []
Sum_time3 = []
Sum_time4 = []
for m in n:
    A = np.random.randint(0, 2, [m, m])
    B = np.random.randint(0, 2, [m, m])
    A1 = np.mat(A)
    B1 = np.mat(B)
    time_start = time.time()
    C1 = A1*B1
    time_end = time.time()
    Sum_time1.append(time_end - time_start)

    C2 = np.zeros([m, m], dtype = np.int)
    time_start = time.time()
    for
 
 i in range(m):
        for k in range(m):
            for j in range(m):
                C2[i, j] = C2[i, j] + A[i, k] * B[k, j]
    time_end = time.time()
    Sum_time2.append(time_end - time_start)
    A11 = np.mat(A[0:m//2, 0:m//2])
    A12 = np.mat(A[0:m//2, m//2:m])
    A21 = np.mat(A[m//2:m, 0:m//2])
    A22 = np.mat(A[m//2:m, m//2:m])
    B11 = np.mat(B[0:m//2, 0:m//2])
    B12 = np.mat(B[0:m//2, m//2:m])
    B21 = np.mat(B[m//2:m, 0:m//2])
    B22 = np.mat(B[m//2:m, m//2:m])
    time_start = time.time()
    C11 = A11 * B11 + A12 * B21
    C12 = A11 * B12 + A12 * B22
    C21 = A21 * B11 + A22 * B21
    C22 = A21 * B12 + A22 * B22
    C3 = np.vstack((np.hstack((C11, C12)), np.hstack((C21, C22))))
    time_end = time.time()
    Sum_time3.append(time_end - time_start)
    time_start = time.time()
    M1 = A11 * (B12 - B22)
    M2 = (A11 + A12) * B22
    M3 = (A21 + A22) * B11
    M4 = A22 * (B21 - B11)
    M5 = (A11 + A22) * (B11 + B22)
    M6 = (A12 - A22) * (B21 + B22)
    M7 = (A11 - A21) * (B11 + B12)
    C11 = M5 + M4 - M2 + M6
    C12 = M1 + M2
    C21 = M3 + M4
    C22 = M5 + M1 - M3 - M7
    C4 = np.vstack((np.hstack((C11, C12)), np.hstack((C21, C22))))
    time_end = time.time()
    Sum_time4.append(time_end - time_start)

f1 = open('python_time1.txt', 'w')
for ele in Sum_time1:
    f1.writelines(str(ele) + '\n')
f1.close()

f2 = open('python_time2.txt', 'w')
for ele in Sum_time2:
    f2.writelines(str(ele) + '\n')
f2.close()

f3 = open('python_time3.txt', 'w')
for ele in Sum_time3:
    f3.writelines(str(ele) + '\n')
f3.close()

f4 = open('python_time4.txt', 'w')
for ele in Sum_time4:
    f4.writelines(str(ele) + '\n')
f4.close()

font = FontProperties(fname=r"c:\windows\fonts\simsun.ttc", size=8)
plt.figure(1)
plt.subplot(221)
plt.semilogx(n, Sum_time1, 'r-*')
plt.ylabel(u"時間(s)", fontproperties=font)
plt.xlabel(u"矩陣的維度n", fontproperties=font)
plt.title(u'python自帶的方法', fontproperties=font)
plt.subplot(222)
plt.semilogx(n, Sum_time2, 'b-*')
plt.ylabel(u"時間(s)", fontproperties=font)
plt.xlabel(u"矩陣的維度n", fontproperties=font)
plt.title(u'定義法', fontproperties=font)
plt.subplot(223)
plt.semilogx(n, Sum_time3, 'y-*')
plt.ylabel(u"時間(s)", fontproperties=font)
plt.xlabel(u"矩陣的維度n", fontproperties=font)
plt.title( u'分治法', fontproperties=font)
plt.subplot(224)
plt.semilogx(n, Sum_time4, 'g-*')
plt.ylabel(u"時間(s)", fontproperties=font)
plt.xlabel(u"矩陣的維度n", fontproperties=font)
plt.title( u'Strasses法', fontproperties=font)
plt.figure(2)
plt.semilogx(n, Sum_time1, 'r-*', n, Sum_time2, 'b-+', n, Sum_time3, 'y-o', n, Sum_time4, 'g-^')
#plt.legend(u'python自帶的方法', u'定義法', u'分治法', u'Strasses法', fontproperties=font)
plt.show()