【算法导论】矩阵乘法strassen算法
矩阵运算在做科学运算时是必不可少的，如果采用matlab来计算，这倒也容易。但是如果是自己写c或者c++代码，一般而言，需要做三次循环，其时间复杂度就是O(n^3)。
上图给出了我们一般会采用的方法，就是对应元素相乘和相加。如果把C=A*B进行分解，可以看出，这里需要进行8次的乘法运算：
分别是：
r = a * e + b * g ;
s = a * f + b * h ;
t = c * e + d * g;
u = c * f + d * h;
本文介绍的算法就是strassen提出的，可以将8次乘法降为7次乘法，虽然只是一次乘法，但是其实一次算法耗时要比加减法多很多。处理的方法是写成：
p1 = a * ( f - h )
p2 = ( a + b ) * h
p3 = ( c +d ) * e
p4 = d * ( g - e )
p5 = ( a + d ) * ( e + h )
p6 = ( b - d ) * ( g + h )
p7 = ( a - c ) * ( e + f )
那么只需要计算p1,p2,p3,p4,p5,p6,p7，然后
r = p5 + p4 + p6 - p2
s = p1 + p2
t = p3 + p4
u = p5 + p1 - p3 - p7
这样，八次的乘法就变成了7次乘法和一次加减法，最终达到降低复杂度为O( n^lg7 ) ~= O( n^2.81 );
c++代码如下：
/*
Strassen Algorithm Implementation in C++
Coded By: Seyyed Hossein Hasan Pour MatiKolaee in May 5 2010 .
Mazandaran University of Science and Technology,Babol,Mazandaran,Iran
--------------------------------------------
Email : [email protected]
YM : [email protected]
Updated may 09 2010.
*/
#include
#include
#include
#include
#include
using namespace std;
int Strassen(int n, int** MatrixA, int ** MatrixB, int ** MatrixC);//Multiplies Two Matrices recrusively.
int ADD(int** MatrixA, int** MatrixB, int** MatrixResult, int length );//Adds two Matrices, and places the result in another Matrix
int SUB(int** MatrixA, int** MatrixB, int** MatrixResult, int length );//subtracts two Matrices , and places the result in another Matrix
int MUL(int** MatrixA, int** MatrixB, int** MatrixResult, int length );//Multiplies two matrices in conventional way.
void FillMatrix( int** matrix1, int** matrix2, int length);//Fills Matrices with random numbers.
void PrintMatrix( int **MatrixA, int MatrixSize );//prints the Matrix content.
int main()
{
int MatrixSize = 0;
int** MatrixA;
int** MatrixB;
int** MatrixC;
clock_t startTime_For_Normal_Multipilication ;
clock_t endTime_For_Normal_Multipilication ;
clock_t startTime_For_Strassen ;
clock_t endTime_For_Strassen ;
time_t start,end;
srand(time(0));
cout<>MatrixSize;
int N = MatrixSize;//for readiblity.
MatrixA = new int *[MatrixSize];
MatrixB = new int *[MatrixSize];
MatrixC = new int *[MatrixSize];
for (int i = 0; i < MatrixSize; i++)
{
MatrixA[i] = new int [MatrixSize];
MatrixB[i] = new int [MatrixSize];
MatrixC[i] = new int [MatrixSize];
}
FillMatrix(MatrixA,MatrixB,MatrixSize);
//*******************conventional multiplication test
cout<<"Phase I started: "<< (startTime_For_Normal_Multipilication = clock());
MUL(MatrixA,MatrixB,MatrixC,MatrixSize);
cout<<"\nPhase I ended: "<< (endTime_For_Normal_Multipilication = clock());
cout<<"\nMatrix Result... \n";
PrintMatrix(MatrixC,MatrixSize);
//*******************Strassen multiplication test
cout<<"\nMultiplication started: "<< (startTime_For_Strassen = clock());
Strassen( N, MatrixA, MatrixB, MatrixC );
cout<<"\nMultiplication: "<<(endTime_For_Strassen = clock());
cout<<"\nMatrix Result... \n";
PrintMatrix(MatrixC,MatrixSize);
cout<<"Matrix size "<