基于openmp的strassen矩阵乘法随着调用线程个数的增加，运行时间会怎么变化

对于基于OpenMP的Strassen矩阵乘法，随着调用线程个数的增加，运行时间会先减少后增加。当线程个数小于计算机处理器的核心数时，运行时间会随着线程个数的增加而减少。当线程个数达到处理器核心数时，运行时间会趋于稳定，不再减少。如果再增加线程个数，反而会导致运行时间增加，因为线程的开销会占用更多的计算资源，而且出现线程竞争和线程切换等问题，影响计算性能。

相关问题

写出strassen矩阵乘法基于openmp的程序，并输出运行时间

抱歉，我无法为你提供直接的代码和运行时间，因为我没有权限在此平台上运行代码。但是，我可以提供一些基本的思路来编写基于OpenMP的Strassen矩阵乘法程序，这样你就可以自己动手编写了。

首先，需要确定矩阵的大小以及递归的基线情况。然后，可以使用递归的Strassen算法来计算矩阵乘法，这个算法可以将一个大矩阵分解成四个子矩阵，然后递归计算每个子矩阵，最后将结果合并为一个大矩阵。在合并子矩阵时，需要使用OpenMP并行化来提高计算效率。

下面是伪代码示例：

void strassen_mult(int n, double **A, double **B, double **C) {
    if (n <= BASE) {
        // 如果差小于基线，则使用普通矩阵乘法
        matrix_multiply(n, A, B, C);
        return;
    }
    // 计算矩阵大小的一半
    int m = n / 2;

    #pragma omp parallel sections
    {
        // 递归计算四个子矩阵的乘积
        #pragma omp section
        {
            double **A11 = create_matrix(m, m);
            double **B11 = create_matrix(m, m);
            double **C11 = create_matrix(m, m);
            matrix_add(m, A, 0, 0, A, m, m, A11);
            matrix_add(m, B, 0, 0, B, m, m, B11);
            strassen_mult(m, A11, B11, C11)
            free_matrix(m, A11);
            free_matrix(m, B11);
            free_matrix(m, C11);
        }
        #pragma omp section
        {
            double **A12 = create_matrix(m, m);
            double **B21 = create_matrix(m, m);
            double **C12 = create_matrix(m, m);
            matrix_add(m, A, 0, m, A, m, m, A12);
            matrix_add(m, B, m, 0, B, 0, m, B21);
            strassen_mult(m, A12, B21, C12)
            free_matrix(m, A12);
            free_matrix(m, B21);
            free_matrix(m, C12);
        }
        #pragma omp section
        {
            double **A11 = create_matrix(m, m);
            double **B12 = create_matrix(m, m);
            double **C21 = create_matrix(m, m);
            matrix_add(m, A, 0, 0, A, 0, m, A11);
            matrix_add(m, B, 0, m, B, m, m, B12);
            strassen_mult(m, A11, B12, C21)
            free_matrix(m, A11);
            free_matrix(m, B12);
            free_matrix(m, C21);
        }
        #pragma omp section
        {
            double **A12 = create_matrix(m, m);
            double **B22 = create_matrix(m, m);
            double **C22 = create_matrix(m, m);
            matrix_add(m, A, 0, m, A, m, m, A12);
            matrix_add(m, B, m, m, B, m, m, B22);
            strassen_mult(m, A12, B22, C22)
            free_matrix(m, A12);
            free_matrix(m, B22);
            free_matrix(m, C22);
        }
    }

    // 合并子矩阵
    #pragma omp parallel for
    for (int i = 0; i < m; i++) {
        for (int j = 0; j < m; j++) {
            C[i][j] = C11[i][j] + C12[i][j];
            C[i][j + m] = C21[i][j] + C22[i][j];
            C[i + m][j] = C11[i + m][j] + C21[i + m][j];
            C[i + m][j + m] = C12[i + m][j] + C22[i + m][j];
        }
    }
}

在这个示例程序中，使用了两个函数`matrix_add`和`matrix_multiply`计算矩阵的加法和乘法，可以自己编写这些函数。并且，在递归计算四个子矩阵时，使用OpenMP的并行化来提高计算效率。

至于运行时间的输出，可以使用`MPI_Wtime`函数在程序的开始和结束位置分别记录时间，然后计算两者之差即可得到程序的运行时间。

可以写出strassen矩阵乘法基于openmp的程序，并输出运行时间吗

当然可以，以下是Strassen矩阵乘法基于OpenMP的程序示例：

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

void strassen(int n, double **a, double **b, double **c, int p) {
    if (n <= p) { // perform regular matrix multiplication when n is small
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                for (int k = 0; k < n; k++) {
                    c[i][j] += a[i][k] * b[k][j];
                }
            }
        }
        return;
    }

    int m = n / 2;

    double **a11, **a12, **a21, **a22, **b11, **b12, **b21, **b22;
    double **s1, **s2, **s3, **s4, **s5, **s6, **s7, **s8, **s9, **s10;
    double **p1, **p2, **p3, **p4, **p5, **p6, **p7;

    #pragma omp parallel sections num_threads(8)
    {
    #pragma omp section
        a11 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        a12 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        a21 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        a22 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        b11 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        b12 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        b21 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        b22 = (double **) malloc(m * sizeof(double *));
    }

    #pragma omp parallel for collapse(2) schedule(static) num_threads(8)
    for (int i = 0; i < m; i++) {
        for (int 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];
        }
    }

    #pragma omp parallel sections num_threads(8)
    {
    #pragma omp section
        s1 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        s2 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        s3 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        s4 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        s5 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        s6 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        s7 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        s8 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        s9 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        s10 = (double **) malloc(m * sizeof(double *));
    }

    #pragma omp parallel sections num_threads(8)
    {
    #pragma omp section
        p1 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        p2 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        p3 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        p4 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        p5 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        p6 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        p7 = (double **) malloc(m * sizeof(double *));
    }

    #pragma omp parallel sections num_threads(8)
    {
    #pragma omp section
        double **tmp1 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        double **tmp2 = (double **) malloc(m * sizeof(double *));
    }

    #pragma omp parallel sections num_threads(8)
    {
    #pragma omp section
        double **tmp3 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        double **tmp4 = (double **) malloc(m * sizeof(double *));
    }

    #pragma omp parallel sections num_threads(8)
    {
    #pragma omp section
        double **tmp5 = (double **) malloc(m * sizeof(double *));
    #pragma omp section
        double **tmp6 = (double **) malloc(m * sizeof(double *));
    }

    #pragma omp parallel sections num_threads(8)
    {
    #pragma omp section
        double **tmp7 = (double **) malloc(m * sizeof(double *));
    }

    #pragma omp task
    {
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < m; j++) {
                s1[i][j] = b12[i][j] - b22[i][j];
                s2[i][j] = a11[i][j] + a12[i][j];
                s3[i][j] = a21[i][j] + a22[i][j];
                s4[i][j] = b21[i][j] - b11[i][j];
                s5[i][j] = a11[i][j] + a22[i][j];
                s6[i][j] = b11[i][j] + b22[i][j];
                s7[i][j] = a12[i][j] - a22[i][j];
                s8[i][j] = b21[i][j] + b22[i][j];
                s9[i][j] = a11[i][j] - a21[i][j];
                s10[i][j] = b11[i][j] + b12[i][j];
            }
        }
    }

    #pragma omp taskwait

    #pragma omp task
    {
        strassen(m, a11, s1, tmp1, p);
    }

    #pragma omp task
    {
        strassen(m, s2, b22, tmp2, p);
    }

    #pragma omp task
    {
        strassen(m, s3, b11, tmp3, p);
    }

    #pragma omp task
    {
        strassen(m, a22, s4, tmp4, p);
    }

    #pragma omp task
    {
        strassen(m, s5, s6, tmp5, p);
    }

    #pragma omp task
    {
        strassen(m, s7, s8, tmp6, p);
    }

    #pragma omp task
    {
        strassen(m, s9, s10, tmp7, p);
    }

    #pragma omp taskwait

    #pragma omp parallel for collapse(2) schedule(static) num_threads(8)
    for (int i = 0; i < m; i++) {
        for (int j = 0; j < m; j++) {
            p1[i][j] = tmp5[i][j] + tmp4[i][j] - tmp2[i][j] + a22[i][j] * b22[i][j];
            p2[i][j] = tmp1[i][j] + tmp2[i][j];
            p3[i][j] = tmp3[i][j] + tmp4[i][j];
            p4[i][j] = tmp5[i][j] + tmp1[i][j] - tmp3[i][j] - b11[i][j] * a11[i][j];
            p5[i][j] = tmp5[i][j] + tmp1[i][j] - tmp3[i][j] + b22[i][j] * a22[i][j];
            p6[i][j] = tmp7[i][j] + tmp5[i][j] + tmp6[i][j] + tmp2[i][j];
            p7[i][j] = tmp6[i][j] + tmp4[i][j] - tmp8[i][j] + b12[i][j] * a21[i][j];
        }
    }

    #pragma omp parallel for collapse(2) schedule(static) num_threads(8)
    for (int i = 0; i < m; i++) {
        for (int j = 0; j < m; j++) {
            c[i][j] = p5[i][j] + p4[i][j] - p2[i][j] + p6[i][j];
            c[i][j + m] = p1[i][j] + p2[i][j];
            c[i + m][j] = p3[i][j] + p4[i][j];
            c[i + m][j + m] = p5[i][j] + p1[i][j] - p3[i][j] + p7[i][j];
        }
    }

    return;
}

int main() {
    int n = 1024; // size of the matrices
    int p = 32; // threshold for switching to regular matrix multiplication

    double **a, **b, **c;

    a = (double **) calloc(n, sizeof(double *));
    b = (double **) calloc(n, sizeof(double *));
    c = (double **) calloc(n, sizeof(double *));

    for (int i = 0; i < n; i++) {
        a[i] = (double *) calloc(n, sizeof(double));
        b[i] = (double *) calloc(n, sizeof(double));
        c[i] = (double *) calloc(n, sizeof(double));
    }

    // initialize matrices with random numbers
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            a[i][j] = (double) rand() / RAND_MAX;
            b[i][j] = (double) rand() / RAND_MAX;
            c[i][j] = 0.0;
        }
    }

    double start_time = omp_get_wtime();

    #pragma omp parallel
    {
        #pragma omp single
        strassen(n, a, b, c, p); // call the strassen function
    }

    double end_time = omp_get_wtime();

    printf("Time: %f seconds\n", end_time - start_time);

    // free memory
    for (int i = 0; i < n; i++) {
        free(a[i]);
        free(b[i]);
        free(c[i]);
    }

    free(a);
    free(b);
    free(c);

    return 0;
}

其中，p代表阈值，当矩阵大小小于等于p时，使用普通的矩阵乘法算法。该程序使用OpenMP来并行化计算，运行时间可通过输出start_time和end_time的差值得到。