strassen算法

strassen算法：

     第一步：分解输入矩阵A、B和输出矩阵C为n/2xn/2的子矩阵。A--->A11、A12、A21、A22， B---->B11、B12、B21、B22，  C---->C11，C12，C21，C22

     第二步：创建十个矩阵

                                           S1=B12 - B22

                                           S2=A11 + A12

                                           S3 = A21 + A22

                                           S4 = B21 - B11

                                           S5 = A11 + A22

                                           S6 = B11 + B22

                                           S7 = A12 + A22

                                           S8 = B21 + B22

                                           S9 = A11 - A21

                                           S10 = B11 + B22

      第三步：递归七个矩阵积

                                           P1 = A11 * S1

                                           P2 = B22 * S2

                                           P3 = B11 * S3

                                           P4 = A22 * S4

                                           P5 = S5 * S6

                                           P6 = S7 * S8

                                           P7 = S9 * S10

      第四步：计算出结果矩阵的子矩阵

                                            C11 = P5 + P4 - P2 + P6

                                            C12 = P1 + P2

                                            C21 = P3 + P4

                                            C22 = P5 + P1 - P3 - P7

Strassn算法运行时间T(n)的递归式：

                                                            T(n) = 7 * T(n / 2) + theta(n^2)      当 n > 1

                                                             T(n) = theta(1)               当n = 1

由主定理，f(n) = theta(n^2) = O(n^log2(7))，所以T(n) = theta(n^log2(7))，约等于theta(n^log2(2.807))。对于一般的矩阵乘法来说，时间复杂度的最小下界是theta(n^2)，因为有结果矩阵由n^2个元素。

参见《算法导论》