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())
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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.****.net/wj310298/article/details/44857175