C++用分治法求解矩阵乘法

矩阵乘法可以使用分治法来解决，其基本思路是将两个矩阵分别划分为四个子矩阵，并且递归地计算这些子矩阵的乘积。下面是一个简单的C语言实现：

#include <stdio.h>

// 矩阵乘法
void matrix_multiply(int n, int A[][n], int B[][n], int C[][n]) {
    if (n == 1) {
        C[0][0] = A[0][0] * B[0][0];
        return;
    }
    int i, j;
    int m = n / 2;
    int A11[m][m], A12[m][m], A21[m][m], A22[m][m];
    int B11[m][m], B12[m][m], B21[m][m], B22[m][m];
    int C11[m][m], C12[m][m], C21[m][m], C22[m][m];
    int P1[m][m], P2[m][m], P3[m][m], P4[m][m], P5[m][m], P6[m][m], P7[m][m];

    // 将A和B分成四个子矩阵
    for (i = 0; i < m; i++) {
        for (j = 0; j < m; j++) {
            A11[i][j] = A[i][j];
            A12[i][j] = A[i][j + m];
            A21[i][j] = A[i + m][j];
            A22[i][j] = A[i + m][j + m];
            B11[i][j] = B[i][j];
            B12[i][j] = B[i][j + m];
            B21[i][j] = B[i + m][j];
            B22[i][j] = B[i + m][j + m];
        }
    }

    // 计算七个子问题
    matrix_multiply(m, A11, B11, P1);
    matrix_multiply(m, A12, B21, P2);
    matrix_multiply(m, A11, B12, P3);
    matrix_multiply(m, A12, B22, P4);
    matrix_multiply(m, A21, B11, P5);
    matrix_multiply(m, A22, B21, P6);
    matrix_multiply(m, A21, B12, P7);
    matrix_multiply(m, A22, B22, C22);

    // 计算C的四个子矩阵
    for (i = 0; i < m; i++) {
        for (j = 0; j < m; j++) {
            C11[i][j] = P1[i][j] + P2[i][j];
            C12[i][j] = P3[i][j] + P4[i][j];
            C21[i][j] = P5[i][j] + P6[i][j];
            C22[i][j] = P1[i][j] + P4[i][j] - P5[i][j] + P7[i][j];
        }
    }

    // 将四个子矩阵合并
    for (i = 0; i < m; i++) {
        for (j = 0; j < m; j++) {
            C[i][j] = C11[i][j];
            C[i][j + m] = C12[i][j];
            C[i + m][j] = C21[i][j];
            C[i + m][j + m] = C22[i][j];
        }
    }
}

int main() {
    int n = 4;
    int A[][4] = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}, {13, 14, 15, 16}};
    int B[][4] = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}, {13, 14, 15, 16}};
    int C[][4] = {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}};
    matrix_multiply(n, A, B, C);
    int i, j;
    for (i = 0; i < n; i++) {
        for (j = 0; j < n; j++) {
            printf("%d ", C[i][j]);
        }
        printf("\n");
    }
    return 0;
}

该实现中，矩阵乘法函数`matrix_multiply()`使用了递归的思想，将两个矩阵分别划分为四个子矩阵，并且递归地计算这些子矩阵的乘积。在计算子矩阵乘积的过程中，还涉及到了计算七个子问题，以及将四个子矩阵合并成一个矩阵的操作。