简单解释一下strassen矩阵乘法

Strassen矩阵乘法是一种基于分治思想的矩阵乘法算法，它可以将两个矩阵相乘的时间复杂度从传统的O(n^3)降低到O(n^log7)。它的核心思想是将两个矩阵分成四个子矩阵，然后通过一些数学运算来计算出它们的乘积，最后将这些乘积组合起来得到最终的结果。

相关问题

strassen矩阵乘法c++

以下是使用C语言实现的Strassen标准矩阵乘法算法的代码：

#include <stdio.h>

#define N 4 // 矩阵大小
#define THRESHOLD 2 // 阈值

// 矩阵加法
void add_matrix(int a[][N], int b[][N], int c[][N])
{
    int i, j;
    for (i = 0; i < N; i++) {
        for (j = 0; j < N; j++) {
            c[i][j] = a[i][j] + b[i][j];
        }
    }
}

// 矩阵减法
void sub_matrix(int a[][N], int b[][N], int c[][N])
{
    int i, j;
    for (i = 0; i < N; i++) {
        for (j = 0; j < N; j++) {
            c[i][j] = a[i][j] - b[i][j];
        }
    }
}

// Strassen矩阵乘法
void strassen_mul(int a[][N], int b[][N], int c[][N])
{
    // 达到阈值，使用标准矩阵乘法
    if (N <= THRESHOLD) {
        int i, j, k;
        for (i = 0; i < N; i++) {
            for (j = 0; j < N; j++) {
                c[i][j] = 0;
                for (k = 0; k < N; k++) {
                    c[i][j] += a[i][k] * b[k][j];
                }
            }
        }
        return;
    }

    // 处理矩阵的大小并向上取整
    int size = N / 2;
    if (N % 2 != 0) {
        size += 1;
    }

    int A[size][size], B[size][size], C[size][size], D[size][size];
    int E[size][size], F[size][size], G[size][size], H[size][size];
    int P1[size][size], P2[size][size], P3[size][size], P4[size][size], P5[size][size], P6[size][size], P7[size][size];
    int tmp1[size][size], tmp2[size][size];

    // 拆分矩阵
    int i, j;
    for (i = 0; i < size; i++) {
        for (j = 0; j < size; j++) {
            A[i][j] = a[i][j];
            B[i][j] = a[i][j + size];
            C[i][j] = a[i + size][j];
            D[i][j] = a[i + size][j + size];
            E[i][j] = b[i][j];
            F[i][j] = b[i][j + size];
            G[i][j] = b[i + size][j];
            H[i][j] = b[i + size][j + size];
        }
    }

    // 计算P1到P7
    sub_matrix(F, H, tmp1);
    strassen_mul(A, tmp1, P1);

    add_matrix(A, B, tmp1);
    strassen_mul(tmp1, H, P2);

    add_matrix(C, D, tmp1);
    strassen_mul(tmp1, E, P3);

    sub_matrix(G, E, tmp1);
    strassen_mul(D, tmp1, P4);

    add_matrix(A, D, tmp1);
    add_matrix(E, H, tmp2);
    strassen_mul(tmp1, tmp2, P5);

    sub_matrix(B, D, tmp1);
    add_matrix(G, H, tmp2);
    strassen_mul(tmp1, tmp2, P6);

    sub_matrix(A, C, tmp1);
    add_matrix(E, F, tmp2);
    strassen_mul(tmp1, tmp2, P7);

    // 计算结果矩阵
    add_matrix(P5, P4, tmp1);
    sub_matrix(tmp1, P2, tmp2);
    add_matrix(tmp2, P6, c[0]);

    add_matrix(P1, P2, c[1]);
    add_matrix(P3, P4, c[2]);

    add_matrix(P5, P1, tmp1);
    sub_matrix(tmp1, P3, tmp2);
    sub_matrix(tmp2, P7, c[3]);

}

int main()
{
    int a[N][N] = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}, {13, 14, 15, 16}};
    int b[N][N] = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}, {13, 14, 15, 16}};
    int c[N][N];
    int i, j;

    strassen_mul(a, b, c);

    printf("Result:\n");
    for (i = 0; i < N; i++) {
        for (j = 0; j < N; j++) {
            printf("%d ", c[i][j]);
        }
        printf("\n");
    }

    return 0;
}

本代码中定义了THRESHOLD变量，当矩阵大小小于等于阈值时，使用标准矩阵乘法算法计算。简单起见，本代码中矩阵大小固定为4 * 4，可以根据需要修改。

strassen矩阵乘法算法c语言

### 回答1：
Strassen矩阵乘法算法是一种用于计算两个矩阵乘积的高效方法，其基本思想是将原始矩阵划分为较小的子矩阵，并通过递归调用来计算乘积。下面是使用C语言实现Strassen矩阵乘法算法的一个示例：

#include<stdio.h>

void strassen(int n, int A[][n], int B[][n], int C[][n]) {
    if (n == 1) {
        C[0][0] = A[0][0] * B[0][0];
        return;
    }
    
    // 计算矩阵的中间大小
    int half = n / 2;

    // 划分原始矩阵为四个子矩阵
    int A11[half][half], A12[half][half], A21[half][half], A22[half][half];
    int B11[half][half], B12[half][half], B21[half][half], B22[half][half];
    int C11[half][half], C12[half][half], C21[half][half], C22[half][half];
    int P[half][half], Q[half][half], R[half][half], S[half][half], T[half][half], U[half][half], V[half][half];

    // 初始化子矩阵
    for (int i = 0; i < half; i++) {
        for (int j = 0; j < half; j++) {
            A11[i][j] = A[i][j];
            A12[i][j] = A[i][j + half];
            A21[i][j] = A[i + half][j];
            A22[i][j] = A[i + half][j + half];

            B11[i][j] = B[i][j];
            B12[i][j] = B[i][j + half];
            B21[i][j] = B[i + half][j];
            B22[i][j] = B[i + half][j + half];
        }
    }

    // 递归调用计算子矩阵
    strassen(half, A11, B11, P);
    strassen(half, A12, B21, Q);
    strassen(half, A11, B12, R);
    strassen(half, A12, B22, S);
    strassen(half, A21, B11, T);
    strassen(half, A22, B21, U);
    strassen(half, A21, B12, V);

    // 计算结果矩阵的子矩阵
    for (int i = 0; i < half; i++) {
        for (int j = 0; j < half; j++) {
            C11[i][j] = P[i][j] + Q[i][j];
            C12[i][j] = R[i][j] + S[i][j];
            C21[i][j] = T[i][j] + U[i][j];
            C22[i][j] = R[i][j] + T[i][j] + U[i][j] + V[i][j];
        }
    }

    // 将子矩阵组合为结果矩阵
    for (int i = 0; i < half; i++) {
        for (int j = 0; j < half; j++) {
            C[i][j] = C11[i][j];
            C[i][j + half] = C12[i][j];
            C[i + half][j] = C21[i][j];
            C[i + half][j + half] = C22[i][j];
        }
    }
}

int main() {
    int n;
    printf("请输入矩阵维度n：");
    scanf("%d", &n);

    int A[n][n], B[n][n], C[n][n];
    printf("请输入矩阵A：\n");
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            scanf("%d", &A[i][j]);
        }
    }

    printf("请输入矩阵B：\n");
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            scanf("%d", &B[i][j]);
        }
    }

    strassen(n, A, B, C);

    printf("结果矩阵C：\n");
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            printf("%d ", C[i][j]);
        }
        printf("\n");
    }

    return 0;
}

这个示例代码实现了一个递归的Strassen矩阵乘法算法。用户需要在运行代码时输入矩阵的维度n，以及矩阵A和B的元素。程序将计算A和B的乘积，并打印结果矩阵C。

### 回答2：
Strassen矩阵乘法算法是一种用于快速计算矩阵乘法的算法，采用分治策略，并且在一些情况下具有比传统算法更高的效率。下面是一个使用C语言实现Strassen矩阵乘法算法的例子：

#include <stdio.h>
#include <stdlib.h>

void strassen(int n, int A[][n], int B[][n], int C[][n]) {
    if (n == 2) { // 基本情况，直接使用传统算法计算
        int P = (A[0][0] + A[1][1]) * (B[0][0] + B[1][1]);
        int Q = (A[1][0] + A[1][1]) * B[0][0];
        int R = A[0][0] * (B[0][1] - B[1][1]);
        int S = A[1][1] * (B[1][0] - B[0][0]);
        int T = (A[0][0] + A[0][1]) * B[1][1];
        int U = (A[1][0] - A[0][0]) * (B[0][0] + B[0][1]);
        int V = (A[0][1] - A[1][1]) * (B[1][0] + B[1][1]);

        C[0][0] = P + S - T + V;
        C[0][1] = R + T;
        C[1][0] = Q + S;
        C[1][1] = P + R - Q + U;
    } else {
        int newSize = n/2;
        int A11[newSize][newSize], A12[newSize][newSize], A21[newSize][newSize], A22[newSize][newSize];
        int B11[newSize][newSize], B12[newSize][newSize], B21[newSize][newSize], B22[newSize][newSize];
        int C11[newSize][newSize], C12[newSize][newSize], C21[newSize][newSize], C22[newSize][newSize];
        int P1[newSize][newSize], P2[newSize][newSize], P3[newSize][newSize], P4[newSize][newSize], P5[newSize][newSize], P6[newSize][newSize], P7[newSize][newSize];

        int i, j;
        for (i = 0; i < newSize; i++) {
            for (j = 0; j < newSize; j++) {
                A11[i][j] = A[i][j];
                A12[i][j] = A[i][j + newSize];
                A21[i][j] = A[i + newSize][j];
                A22[i][j] = A[i + newSize][j + newSize];
                B11[i][j] = B[i][j];
                B12[i][j] = B[i][j + newSize];
                B21[i][j] = B[i + newSize][j];
                B22[i][j] = B[i + newSize][j + newSize];
            }
        }

        strassen(newSize, A11, B11, P1);
        strassen(newSize, A12, B21, P2);
        strassen(newSize, A11, B12, P3);
        strassen(newSize, A12, B22, P4);
        strassen(newSize, A21, B11, P5);
        strassen(newSize, A22, B21, P6);
        strassen(newSize, A21, B12, P7);

        for (i = 0; i < newSize; i++) {
            for (j = 0; j < newSize; j++) {
                C11[i][j] = P1[i][j] + P4[i][j] - P5[i][j] + P7[i][j];
                C12[i][j] = P3[i][j] + P5[i][j];
                C21[i][j] = P2[i][j] + P4[i][j];
                C22[i][j] = P1[i][j] + P3[i][j] - P2[i][j] + P6[i][j];

                C[i][j] = C11[i][j];
                C[i][j + newSize] = C12[i][j];
                C[i + newSize][j] = C21[i][j];
                C[i + newSize][j + newSize] = C22[i][j];
            }
        }
    }
}

int main() {
    int n = 4; // 矩阵维数
    int A[][4] = {{1, 2, 3, 4},
                  {5, 6, 7, 8},
                  {9, 10, 11, 12},
                  {13, 14, 15, 16}};
    int B[][4] = {{17, 18, 19, 20},
                  {21, 22, 23, 24},
                  {25, 26, 27, 28},
                  {29, 30, 31, 32}};
    int C[4][4];

    strassen(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;
}

以上是一个简单的C语言实现的Strassen矩阵乘法算法。在此例子中，我们使用了一个4x4的矩阵作为输入，并打印出计算结果。