使用java写的矩阵乘法实例（Strassen算法）

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

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。

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();
	}
 
}

总结

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