Strassen算法于1969年由德国数学家Strassen提出，该方法引入七个中间变量，每个中间变量都只需要进行一次乘法运算。而朴素算法却需要进行8次乘法运算。
原理
Strassen算法的原理如下所示，使用sympy验证Strassen算法的正确性
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import sympy as s
A = s.Symbol( "A" ) B = s.Symbol( "B" ) C = s.Symbol( "C" ) D = s.Symbol( "D" ) E = s.Symbol( "E" ) F = s.Symbol( "F" ) G = s.Symbol( "G" ) H = s.Symbol( "H" ) 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)
print(A * E + B * G, (p5 + p4 - p2 + p6).simplify()) print(A * F + B * H, (p1 + p2).simplify()) print(C * E + D * G, (p3 + p4).simplify()) print(C * F + D * H, (p1 + p5 - p3 - p7).simplify()) |
复杂度分析
$$f(N)=7\times f(\frac{N}{2})=7^2\times f(\frac{N}{4})=...=7^k\times f(\frac{N}{2^k})$$
最终复杂度为$7^{log_2 N}=N^{log_2 7}$
java矩阵乘法（Strassen算法）
代码如下，可以看看数据结构的定义，时间换空间。
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public class Matrix { private final Matrix[] _matrixArray; private final int n; private int element; public Matrix( int n) { this .n = n; if (n != 1 ) { this ._matrixArray = new Matrix[ 4 ]; for ( int i = 0 ; i < 4 ; i++) { this ._matrixArray[i] = new Matrix(n / 2 ); } } else { this ._matrixArray = null ; } } private Matrix( int n, boolean needInit) { this .n = n; if (n != 1 ) { this ._matrixArray = new Matrix[ 4 ]; } else { this ._matrixArray = null ; } } public void set( int i, int j, int a) { if (n == 1 ) { element = a; } else { int size = n / 2 ; this ._matrixArray[(i / size) * 2 + (j / size)].set(i % size, j % size, a); } } public Matrix multi(Matrix m) { Matrix result = null ; if (n == 1 ) { result = new Matrix( 1 ); result.set( 0 , 0 , (element * m.element)); } else { result = new Matrix(n, false ); result._matrixArray[ 0 ] = P5(m).add(P4(m)).minus(P2(m)).add(P6(m)); result._matrixArray[ 1 ] = P1(m).add(P2(m)); result._matrixArray[ 2 ] = P3(m).add(P4(m)); result._matrixArray[ 3 ] = P5(m).add(P1(m)).minus(P3(m)).minus(P7(m)); } return result; } public Matrix add(Matrix m) { Matrix result = null ; if (n == 1 ) { result = new Matrix( 1 ); result.set( 0 , 0 , (element + m.element)); } else { result = new Matrix(n, false ); result._matrixArray[ 0 ] = this ._matrixArray[ 0 ].add(m._matrixArray[ 0 ]); result._matrixArray[ 1 ] = this ._matrixArray[ 1 ].add(m._matrixArray[ 1 ]); result._matrixArray[ 2 ] = this ._matrixArray[ 2 ].add(m._matrixArray[ 2 ]); result._matrixArray[ 3 ] = this ._matrixArray[ 3 ].add(m._matrixArray[ 3 ]);; } return result; } public Matrix minus(Matrix m) { Matrix result = null ; if (n == 1 ) { result = new Matrix( 1 ); result.set( 0 , 0 , (element - m.element)); } else { result = new Matrix(n, false ); result._matrixArray[ 0 ] = this ._matrixArray[ 0 ].minus(m._matrixArray[ 0 ]); result._matrixArray[ 1 ] = this ._matrixArray[ 1 ].minus(m._matrixArray[ 1 ]); result._matrixArray[ 2 ] = this ._matrixArray[ 2 ].minus(m._matrixArray[ 2 ]); result._matrixArray[ 3 ] = this ._matrixArray[ 3 ].minus(m._matrixArray[ 3 ]);; } return result; } protected Matrix P1(Matrix m) { return _matrixArray[ 0 ].multi(m._matrixArray[ 1 ]).minus(_matrixArray[ 0 ].multi(m._matrixArray[ 3 ])); } protected Matrix P2(Matrix m) { return _matrixArray[ 0 ].multi(m._matrixArray[ 3 ]).add(_matrixArray[ 1 ].multi(m._matrixArray[ 3 ])); } protected Matrix P3(Matrix m) { return _matrixArray[ 2 ].multi(m._matrixArray[ 0 ]).add(_matrixArray[ 3 ].multi(m._matrixArray[ 0 ])); } protected Matrix P4(Matrix m) { return _matrixArray[ 3 ].multi(m._matrixArray[ 2 ]).minus(_matrixArray[ 3 ].multi(m._matrixArray[ 0 ])); } protected Matrix P5(Matrix m) { return (_matrixArray[ 0 ].add(_matrixArray[ 3 ])).multi(m._matrixArray[ 0 ].add(m._matrixArray[ 3 ])); } protected Matrix P6(Matrix m) { return (_matrixArray[ 1 ].minus(_matrixArray[ 3 ])).multi(m._matrixArray[ 2 ].add(m._matrixArray[ 3 ])); } protected Matrix P7(Matrix m) { return (_matrixArray[ 0 ].minus(_matrixArray[ 2 ])).multi(m._matrixArray[ 0 ].add(m._matrixArray[ 1 ])); } public int get( int i, int j) { if (n == 1 ) { return element; } else { int size = n / 2 ; return this ._matrixArray[(i / size) * 2 + (j / size)].get(i % size, j % size); } } public void display() { for ( int i = 0 ; i < n; i++) { for ( int j = 0 ; j < n; j++) { System.out.print(get(i, j)); System.out.print( " " ); } System.out.println(); } }
public static void main(String[] args) { Matrix m = new Matrix( 2 ); Matrix n = new Matrix( 2 ); m.set( 0 , 0 , 1 ); m.set( 0 , 1 , 3 ); m.set( 1 , 0 , 5 ); m.set( 1 , 1 , 7 ); n.set( 0 , 0 , 8 ); n.set( 0 , 1 , 4 ); n.set( 1 , 0 , 6 ); n.set( 1 , 1 , 2 ); Matrix res = m.multi(n); res.display(); }
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总结
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原文链接：https://blog.csdn.net/wj310298/article/details/44857175
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