The Taylor series formula for the cosine(x) function is as follows:

Write, compile, and run this serial cosine(x) code (see the following code, which shows serial cosine(x) code) to get a feel of the program:

/***********************************
* Serial cosine(x)code.          * 
 *                                * 
 * Taylor series representation   * 
 * of the trigonometric cosine(x).*  
 *                                * 
 * Author: Carlos R. Morrison     * 
 *                                * 
 * Date: 1/10/2017                * 
 **********************************/ 
#include <math.h> 
#include <stdio.h> 
 
int main(void) 
{ 
  unsigned int j; 
  unsigned long int k; 
  long long int B,D;  
  int num_loops = 17; 
  float y; 
  double x; 
  double sum0=0,A,C,E; 
/******************************************************/ 
  printf("
"); 
  printf("Enter angle(deg.):
"); 
  printf("
"); 
  scanf("%f",&y); 
  if(y <= 180.0) 
  {       
    x = y*(M_PI/180.0); 
  } 
  else 
  x = -(360.0-y)*(M_PI/180.0); 
/******************************************************/ 
  sum0 = 0; 
  for(k = 0; k < num_loops; k++) 
  {     
    A = (double)pow(-1,k);// (-1^k) 
    B = 2*k; 
    C = (double)pow(x,B);// x^2k 
 
    D = 1; 
    for(j=1; j <= B; j++)// (2k!) 
    { 
      D *= j;  
    } 
 
    E = (A*C)/(double)D; 
 
    sum0 += E; 
  } 
 
  printf("
"); 
  printf("   %.1f deg. = %.3f rads
", y, x); 
  printf("Cosine(%.1f) = %.4f
", y, sum0); 
 
  return 0; 
} 
 
Next, write, compile, and run the MPI version of the preceding serial cosine(x) code (see MPI cosine(x) code below), using one processor from each of the 16 nodes. 
/**********************************  
 * MPI cosine(x) code.            * 
 *                                * 
 * Taylor series representation   * 
 * of the trigonometric cosine(x).*  
 *                                * 
 * Author: Carlos R. Morrison     * 
 *                                * 
 * Date: 1/10/2017                * 
 **********************************/ 
#include<mpi.h>// (Open)MPI library 
#include<math.h>// math library 
#include<stdio.h>// Standard Input/Output library 
 
int main(int argc, char*argv[]) 
{ 
  long long int total_iter,B,D; 
  int n = 17,rank,length,numprocs,i,j; 
  unsigned long int k; 
  double sum,sum0,rank_integral,A,C,E; 
  float y,x; 
  char hostname[MPI_MAX_PROCESSOR_NAME]; 
 
  MPI_Init(&argc, &argv);                    // initiates MPI 
  MPI_Comm_size(MPI_COMM_WORLD, &numprocs);  // acquire number of processes 
  MPI_Comm_rank(MPI_COMM_WORLD, &rank);      // acquire current process id 
  MPI_Get_processor_name(hostname, &length); // acquire hostname 
 
  if (rank == 0) 
  { 
    printf("
"); 
    printf("#######################################################"); 
    printf("

"); 
    printf("*** Number of processes: %d
",numprocs); 
    printf("*** processing capacity: %.1f GHz.
",numprocs*1.2);  
    printf("

"); 
    printf("Master node name: %s
", hostname);  
    printf("
"); 
    printf("Enter angle(deg.):
"); 
    printf("
"); 
    scanf("%f",&y); 
    if(y <= 180.0) 
    {       
      x = y*(M_PI/180.0); 
    } 
    else 
    x = -(360.0-y)*(M_PI/180.0); 
  } 
 
// broadcast to all processes, the number of segments you want 
  MPI_Bcast(&n, 1, MPI_INT, 0, MPI_COMM_WORLD);  
  MPI_Bcast(&x, 1, MPI_INT, 0, MPI_COMM_WORLD); 
 
// this loop increments the maximum number of iterations, thus providing 
// additional work for testing computational speed of the processors    
//for(total_iter = 1; total_iter < n; total_iter++)  
  { 
    sum0 = 0.0; 
//  for(i = rank + 1; i <= total_iter; i += numprocs) 
    for(i = rank + 1; i <= n; i += numprocs) 
    {     
      k = (i-1); 
 
      A = (double)pow(-1,k);// (-1^k) 
      B = 2*k; 
      C = (double)pow(x,B);// x^2k 
 
      D = 1; 
      for(j=1; j <= B; j++)// (2k!) 
      { 
        D *= j; 
      } 
      E = (A*C)/(double)D; 
 
      sum0 += E;     
    } 
 
    rank_integral = sum0;// Partial sum for a given rank 
 
//  collect and add the partial sum0 values from all processes 
    MPI_Reduce(&rank_integral, &sum, 1, MPI_DOUBLE,MPI_SUM, 0, 
    MPI_COMM_WORLD); 
 
  }// End of for(total_iter = 1; total_iter < n; total_iter++) 
 
  if(rank == 0) 
  { 
    printf("

");     
    printf("    %.1f deg. = %.3f rads
", y, x); 
    printf("Cosine(%.3f) = %.3f
", x, sum); 
  } 
// clean up, done with MPI 
  MPI_Finalize(); 
 
  return 0;   
}// End of int main(int argc, char*argv[]) 
 

The following is the MPI cosine(x) run:

alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
0 
 
   
     0.0 deg. = 0.000 rads 
Cosine(0.000) = 1.000 
 
real 0m5.309s 
user 0m1.280s 
sys  0m0.280s 
 
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
30 
 
    30.0 deg. = 0.524 rads 
Cosine(0.524) = 0.866 
 
 
 
real 0m13.045s 
user 0m1.270s 
sys  0m0.400s 
 
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
60 
 
 
    60.0 deg. = 1.047 rads 
Cosine(1.047) = 0.500 
 
real 0m18.477s 
user 0m1.150s 
sys  0m0.450s 
  
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
90 
 
 
    90.0 deg. = 1.571 rads 
Cosine(1.571) = -0.000 
 
real 0m10.567s 
user 0m1.240s 
sys  0m0.360s 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
120 
 
 
   120.0 deg. = 2.094 rads 
Cosine(2.094) = -0.500 
 
real 0m7.056s 
user 0m1.310s 
sys  0m0.330s 
 
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
150 
 
 
   150.0 deg. = 2.618 rads 
Cosine(2.618) = -0.866 
 
real 0m8.202s 
user 0m1.200s 
sys  0m0.390s 
 
 
 
 
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
180 
 
 
   180.0 deg. = 3.142 rads 
Cosine(3.142) = -1.001 
 
real 0m11.139s 
user 0m1.250s 
sys  0m0.350s 
 
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
210 
 
 
    210.0 deg. = -2.618 rads 
Cosine(-2.618) = -0.866 
 
real 0m6.400s 
user 0m1.290s 
sys  0m0.330s 
 
 
 
 
 
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
240 
 
 
    240.0 deg. = -2.094 rads 
Cosine(-2.094) = -0.500 
 
real 0m11.837s 
user 0m1.320s 
sys  0m0.340s 
 
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
270 
 
 
    270.0 deg. = -1.571 rads 
Cosine(-1.571) = -0.000 
 
real 0m8.228s 
user 0m1.200s 
sys  0m0.360s 
 
 
 
 
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
300 
 
 
    300.0 deg. = -1.047 rads 
Cosine(-1.047) = 0.500 
 
real 0m7.509s 
user 0m1.270s 
sys  0m0.350s 
 
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
330 
 
 
    330.0 deg. = -0.524 rads 
Cosine(-0.524) = 0.866 
 
real 0m7.167s 
user 0m1.180s 
sys  0m0.350s 
 
 
 
 
 
 
alpha@Mst0:/beta/gamma $ time mpiexec -H Mst0,Slv1,Slv2,Slv3,Slv4,Slv5,Slv6,Slv7,Slv8, 
Slv9,Slv10,Slv11,Slv12,Slv13,Slv14,Slv15 MPI_cosine 
####################################################### 
 
*** Number of processes: 16 
*** processing capacity: 19.2 GHz. 
 
 
Master node name: Mst0 
 
Enter angle(deg.): 
 
360 
 
 
    360.0 deg. = -0.000 rads 
Cosine(-0.000) = 1.000 
 
real 0m10.469s 
user 0m1.250s 
sys  0m0.300s 

The following figure is a plot of the MPI cosine(x) run: