simplex_integrals.cpp

# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <cmath>
# include <ctime>
# include <cstring>

using namespace std;

# include "simplex_integrals.hpp"

//****************************************************************************80

int *i4vec_uniform_ab_new ( int n, int a, int b, int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    I4VEC_UNIFORM_AB_NEW returns a scaled pseudorandom I4VEC.
//
//  Discussion:
//
//    An I4VEC is a vector of I4's.
//
//    The pseudorandom numbers should be uniformly distributed
//    between A and B.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    24 May 2012
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Second Edition,
//    Springer, 1987,
//    ISBN: 0387964673,
//    LC: QA76.9.C65.B73.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, December 1986, pages 362-376.
//
//    Pierre L'Ecuyer,
//    Random Number Generation,
//    in Handbook of Simulation,
//    edited by Jerry Banks,
//    Wiley, 1998,
//    ISBN: 0471134031,
//    LC: T57.62.H37.
//
//    Peter Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, Number 2, 1969, pages 136-143.
//
//  Parameters:
//
//    Input, int N, the dimension of the vector.
//
//    Input, int A, B, the limits of the interval.
//
//    Input/output, int &SEED, the "seed" value, which should NOT be 0.
//    On output, SEED has been updated.
//
//    Output, int IVEC_UNIFORM_AB_NEW[N], a vector of random values 
//    between A and B.
//
{
  int c;
  int i;
  const int i4_huge = 2147483647;
  int k;
  float r;
  int value;
  int *x;
  
  if ( seed == 0 )
  {
    cerr << "\n";
    cerr << "I4VEC_UNIFORM_AB_NEW - Fatal error!\n";
    cerr << "  Input value of SEED = 0.\n";
    exit ( 1 );
  }
//
//  Guarantee A <= B.
//
  if ( b < a )
  {
    c = a;
    a = b;
    b = c;
  }

  x = new int[n];

  for ( i = 0; i < n; i++ )
  {
    k = seed / 127773;

    seed = 16807 * ( seed - k * 127773 ) - k * 2836;

    if ( seed < 0 )
    {
      seed = seed + i4_huge;
    }

    r = ( float ) ( seed ) * 4.656612875E-10;
//
//  Scale R to lie between A-0.5 and B+0.5.
//
    r = ( 1.0 - r ) * ( ( float ) a - 0.5 ) 
      +         r   * ( ( float ) b + 0.5 );
//
//  Use rounding to convert R to an integer between A and B.
//
    value = round ( r );
//
//  Guarantee A <= VALUE <= B.
//
    if ( value < a )
    {
      value = a;
    }
    if ( b < value )
    {
      value = b;
    }

    x[i] = value;
  }

  return x;
}
//    Input, int EXPON, the exponents.
//
//****************************************************************************80

double r8vec_sum ( int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_SUM returns the sum of an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    15 October 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, double A[N], the vector.
//
//    Output, double R8VEC_SUM, the sum of the vector.
//
{
  int i;
  double value;

  value = 0.0;
  for ( i = 0; i < n; i++ )
  {
    value = value + a[i];
  }

  return value;
}
//****************************************************************************80

double *r8vec_uniform_01_new ( int n, int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_UNIFORM_01_NEW returns a new unit pseudorandom R8VEC.
//
//  Discussion:
//
//    This routine implements the recursion
//
//      seed = ( 16807 * seed ) mod ( 2^31 - 1 )
//      u = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    19 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Second Edition,
//    Springer, 1987,
//    ISBN: 0387964673,
//    LC: QA76.9.C65.B73.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, December 1986, pages 362-376.
//
//    Pierre L'Ecuyer,
//    Random Number Generation,
//    in Handbook of Simulation,
//    edited by Jerry Banks,
//    Wiley, 1998,
//    ISBN: 0471134031,
//    LC: T57.62.H37.
//
//    Peter Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, Number 2, 1969, pages 136-143.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input/output, int &SEED, a seed for the random number generator.
//
//    Output, double R8VEC_UNIFORM_01_NEW[N], the vector of pseudorandom values.
//
{
  int i;
  int i4_huge = 2147483647;
  int k;
  double *r;

  if ( seed == 0 )
  {
    cerr << "\n";
    cerr << "R8VEC_UNIFORM_01_NEW - Fatal error!\n";
    cerr << "  Input value of SEED = 0.\n";
    exit ( 1 );
  }

  r = new double[n];

  for ( i = 0; i < n; i++ )
  {
    k = seed / 127773;

    seed = 16807 * ( seed - k * 127773 ) - k * 2836;

    if ( seed < 0 )
    {
      seed = seed + i4_huge;
    }

    r[i] = ( double ) ( seed ) * 4.656612875E-10;
  }

  return r;
}
//****************************************************************************80

double simplex01_monomial_integral ( int m, int e[] )

//****************************************************************************80
//
//  Purpose:
//
//    SIMPLEX01_MONOMIAL_INTEGRAL: integrals in the unit simplex in M dimensions.
//
//  Discussion:
//
//    The monomial is F(X) = product ( 1 <= I <= M ) X(I)^E(I).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    16 January 2014
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the spatial dimension.
//
//    Input, int E[M], the exponents.  
//    Each exponent must be nonnegative.
//
//    Output, double SIMPLEX01_MONOMIAL_INTEGRAL, the integral.
//
{
  int i;
  double integral;
  int j;
  int k;

  for ( i = 0; i < m; i++ )
  {
    if ( e[i] < 0 )
    {
      cerr << "\n";
      cerr << "SIMPLEX01_MONOMIAL_INTEGRAL - Fatal error!\n";
      cerr << "  All exponents must be nonnegative.\n";
      exit ( 1 );
    }
  }

  k = 0;
  integral = 1.0;

  for ( i = 0; i < m; i++ )
  {
    for ( j = 1; j <= e[i]; j++ )
    {
      k = k + 1;
      integral = integral * ( double ) ( j ) / ( double ) ( k );
    }
  }

  for ( i = 0; i < m; i++ )
  {
    k = k + 1;
    integral = integral / ( double ) ( k );
  }

  return integral;
}
//****************************************************************************80

double *simplex01_sample ( int m, int n, int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    SIMPLEX01_SAMPLE samples the unit simplex in M dimensions.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    16 January 2015
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Reuven Rubinstein,
//    Monte Carlo Optimization, Simulation, and Sensitivity 
//    of Queueing Networks,
//    Krieger, 1992,
//    ISBN: 0894647644,
//    LC: QA298.R79.
//
//  Parameters:
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of points.
//
//    Input/output, int &SEED, a seed for the random 
//    number generator.
//
//    Output, double SIMPLEX01_SAMPLE_01[M*N], the points.
//
{
  double *e;
  double e_sum;
  int i;
  int j;
  double *x;

  x = new double[m*n];

  for ( j = 0; j < n; j++ )
  {
    e = r8vec_uniform_01_new ( m + 1, seed );

    for ( i = 0; i < m + 1; i++ )
    {
      e[i] = - log ( e[i] );
    }
    e_sum = r8vec_sum ( m + 1, e );

    for ( i = 0; i < m; i++ )
    {
      x[i+j*m] = e[i] / e_sum;
    }
    delete [] e;
  }

  return x;
}
//****************************************************************************80

double simplex01_volume ( int m )

//****************************************************************************80
//
//  Purpose:
//
//    SIMPLEX01_VOLUME returns the volume of the unit simplex in M dimensions.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    16 January 2014
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the spatial dimension.
//
//    Output, double SIMPLEX01_VOLUME, the volume.
//
{
  int i;
  double volume;

  volume = 1.0;
  for ( i = 1; i <= m; i++ )
  {
    volume = volume / ( double ) ( i );
  }

  return volume;
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct std::tm *tm_ptr;
  size_t len;
  std::time_t now;

  now = std::time ( NULL );
  tm_ptr = std::localtime ( &now );

  len = std::strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm_ptr );

  std::cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}