Question #1:

#include <cmath>
#include <math.h>
#include <gsl/gsl_integration.h>
#include <iostream>
#include <iomanip>

using std::setprecision;

//sub function to calculate f
double f (double x, void * params) {
  double f = ((exp(x)-exp(-x))/(exp(x)+exp(-x)))*pow(x,x);
  return f;
}
     
int
main (void)
{
  gsl_integration_workspace * w 
    = gsl_integration_workspace_alloc (1000);
       
  double result, error;
  double alpha = 1.0;
     
  gsl_function F;
  F.function = &f;
  F.params = &alpha;
     
  gsl_integration_qags (&F, 2, 3, 0, 1e-8, 1000,
                        w, &result, &error); 
     
  std::cout<<setprecision(22)<<"the answer is: \t"<<result<<".\n The estimated error is: \t"<<error<<"\n The number of interval: \t"<<w->size<<std::endl;
     
  gsl_integration_workspace_free (w);
     
  return 0;
}

//output is here.

the answer is:  11.54026056218155815714.
The estimated error is:        1.281226298630761953398e-13
The number of interval:        1


Question #2:

gnuplot> set xrange [2:3]
gnuplot> plot ((exp(x)-exp(-x))/(exp(x)+exp(-x)))*x**x
gnuplot> set terminal postscript eps enhanced
gnuplot> set output "myplot.eps"
gnuplot> replot

Question #3:


#include <iostream>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>

int main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;
     
  int i, n = 10;
      
  /* create a generator chosen by the 
     environment variable GSL_RNG_TYPE */

  T = gsl_rng_default;
  r = gsl_rng_alloc (T);
  unsigned long int s = time(0);
  gsl_rng_set(r, s);

     
            
  /* print n random variates chosen from 
     the poisson distribution with mean 
     parameter mu */
     
  for (i = 0; i < n; i++) 
  {
    double k = gsl_rng_uniform (r);
    double j = 25+12*k;
	
    std::cout<<j<<"\t"<<std::endl;
  }
     
  printf ("\n");
  gsl_rng_free (r);
  return 0;
}

//outputs are here.
36.9969
26.9549
28.3914
36.3664
27.7799
30.8197
36.4897
33.9317
31.4805
33.8794



Question #4:
#include <iostream>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <math.h>

int
main (void)
{
  const gsl_rng_type * T;
  gsl_rng * r;
     
  int i, j, n = 10000, drunks=100;
  double a=0,b=2, direction[n], distance=0,abs=0, Average=0;
	
     
  T = gsl_rng_default;
  r = gsl_rng_alloc (T);

     
  srand((time(0)));                
  long int seed = random(); 
  gsl_rng_env_setup();
  gsl_rng_set(r,seed); 


  for( i=0; i < drunks; i++){
    for (j = 0; j < n; j++) 
    {
      unsigned int k = gsl_ran_flat (r, a, b);
  
      if(k==0){	
        direction[i] =-1;
      }
      else direction[i] = 1;	
      distance+=direction[i];
		
    }
	
	abs=fabs(distance);
	Average+=abs;
	
  }

  Average /= drunks;
	
     
  std::cout << " the average distance D a drunk is from his/her starting position at the end of the journey of 10, 000 steps is: \t" << Average << "l"<<std::endl;
  gsl_rng_free (r);
  return 0;
}

//output
the average distance D a drunk is from his/her starting position at the end of the journey of 10, 000 steps is:        861.3l ;


Question #5:

(a);

f(x) = 1/sqrt(pi)*exp(-x**2)*x**2;
plot f(x)



(b)


#include <cmath>
#include <math.h>
#include <gsl/gsl_integration.h>
#include <iostream>
#include <iomanip>

  using std::setprecision;

double f (double x, void * params) {
  double f = 1/sqrt(3.14159)*exp(-pow(x,2))*pow(x,2);
  return f;
}
     
int main (void)
{
  gsl_integration_workspace * w 
    = gsl_integration_workspace_alloc (1000);
       
  double result, error, total;
  double alpha = 1.0;
     
  gsl_function F;
  F.function = &f;
  F.params = &alpha;
     
  gsl_integration_qags (&F, 0.995, 1.005, 0, 1e-8, 1000,
                        w, &result, &error); 

  gsl_integration_qags (&F, 0, 100, 0, 1e-8, 1000,
                        w, &total, &error); 
	
     
  std::cout<<setprecision(22)<<"the answer is: \t"<<result/total<<std::endl;
     
  gsl_integration_workspace_free (w);
     
  return 0;
}

//output
the answer is:  0.008302011580802648796351