# include <stdlib.h>
# include <stdio.h>
# include <math.h>
# include <time.h>

# include "circle_rule.h"

/******************************************************************************/

void circle_rule ( int nt, double w[], double t[] )

/******************************************************************************/
/*
  Purpose:

    CIRCLE_RULE computes a quadrature rule for the unit circle.

  Discussion:

    The unit circle is the region:

      x * x + y * y = 1.

    The integral I(f) is then approximated by

      Q(f) = 2 * pi * sum ( 1 <= i <= NT ) W(i) * F ( cos(T(i)), sin(T(i)) ).

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    06 April 2014

  Author:

    John Burkardt

  Parameters:

    Input, int NT, the number of angles to use.

    Output, double W[NT], the weights for the rule.

    Output, double T[NT], the angles for the rule.
*/
{
  int it;
  double r8_pi = 3.141592653589793;

  for ( it = 0; it < nt; it++ )
  {
    w[it] = 1.0 / ( double ) ( nt );
    t[it] = 2.0 * r8_pi * ( double ) ( it ) / ( double ) ( nt );
  }
  return;
}
/******************************************************************************/

double circle01_monomial_integral ( int e[2] )

/******************************************************************************/
/*
  Purpose:

    CIRCLE01_MONOMIAL_INTEGRAL: integrals on circumference of unit circle in 2D.

  Discussion:

    The integration region is 

      X^2 + Y^2 = 1.

    The monomial is F(X,Y,Z) = X^E(1) * Y^E(2).

  Licensing:

    This code is distributed under the GNU LGPL license. 

  Modified:

    11 January 2014

  Author:

    John Burkardt

  Reference:

    Philip Davis, Philip Rabinowitz,
    Methods of Numerical Integration,
    Second Edition,
    Academic Press, 1984, page 263.

  Parameters:

    Input, int E[2], the exponents of X and Y in the 
    monomial.  Each exponent must be nonnegative.

    Output, double CIRCLE01_MONOMIAL_INTEGRAL, the integral.
*/
{
  int i;
  double integral;

  if ( e[0] < 0 || e[1] < 0 )
  {
    fprintf ( stderr, "\n" );
    fprintf ( stderr, "CIRCLE01_MONOMIAL_INTEGRAL - Fatal error!\n" );
    fprintf ( stderr, "  All exponents must be nonnegative.\n" );
    fprintf ( stderr, "  E[0] = %d\n", e[0] );
    fprintf ( stderr, "  E[1] = %d\n", e[1] );
    exit ( 1 );
  }

  if ( ( e[0] % 2 ) == 1 || ( e[1] % 2 ) == 1 )
  {
    integral = 0.0;
  }
  else
  {
    integral = 2.0;

    for ( i = 0; i < 2; i++ )
    {
      integral = integral * r8_gamma ( 0.5 * ( double ) ( e[i] + 1 ) );
    }

    integral = integral 
      / r8_gamma ( 0.5 * ( double ) ( e[0] + e[1] + 2 ) );

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

double r8_gamma ( double x )

/******************************************************************************/
/*
  Purpose:

    R8_GAMMA evaluates Gamma(X) for a real argument.

  Discussion:

    The C math library includes the GAMMA ( X ) function which should generally
    be used instead of this function.

    This routine calculates the gamma function for a real argument X.

    Computation is based on an algorithm outlined in reference 1.
    The program uses rational functions that approximate the gamma
    function to at least 20 significant decimal digits.  Coefficients
    for the approximation over the interval (1,2) are unpublished.
    Those for the approximation for 12 <= X are from reference 2.

  Licensing:

    This code is distributed under the GNU LGPL license. 

  Modified:

    11 January 2010

  Author:

    Original FORTRAN77 version by William Cody, Laura Stoltz.
    C version by John Burkardt.

  Reference:

    William Cody,
    An Overview of Software Development for Special Functions,
    in Numerical Analysis Dundee, 1975,
    edited by GA Watson,
    Lecture Notes in Mathematics 506,
    Springer, 1976.

    John Hart, Ward Cheney, Charles Lawson, Hans Maehly,
    Charles Mesztenyi, John Rice, Henry Thatcher,
    Christoph Witzgall,
    Computer Approximations,
    Wiley, 1968,
    LC: QA297.C64.

  Parameters:

    Input, double X, the argument of the function.

    Output, double R8_GAMMA, the value of the function.
*/
{
  double c[7] = {
   -1.910444077728E-03, 
    8.4171387781295E-04, 
   -5.952379913043012E-04, 
    7.93650793500350248E-04, 
   -2.777777777777681622553E-03, 
    8.333333333333333331554247E-02, 
    5.7083835261E-03 };
  double eps = 2.22E-16;
  double fact;
  int i;
  int n;
  double p[8] = {
  -1.71618513886549492533811E+00,
   2.47656508055759199108314E+01, 
  -3.79804256470945635097577E+02,
   6.29331155312818442661052E+02, 
   8.66966202790413211295064E+02,
  -3.14512729688483675254357E+04, 
  -3.61444134186911729807069E+04,
   6.64561438202405440627855E+04 };
  int parity;
  const double pi = 3.1415926535897932384626434;
  double q[8] = {
  -3.08402300119738975254353E+01,
   3.15350626979604161529144E+02, 
  -1.01515636749021914166146E+03,
  -3.10777167157231109440444E+03, 
   2.25381184209801510330112E+04,
   4.75584627752788110767815E+03, 
  -1.34659959864969306392456E+05,
  -1.15132259675553483497211E+05 };
  double res;
  const double sqrtpi = 0.9189385332046727417803297;
  double sum;
  double value;
  double xbig = 171.624;
  double xden;
  double xinf = 1.79E+308;
  double xminin = 2.23E-308;
  double xnum;
  double y;
  double y1;
  double ysq;
  double z;

  parity = 0;
  fact = 1.0;
  n = 0;
  y = x;
/*
  Argument is negative.
*/
  if ( y <= 0.0 )
  {
    y = - x;
    y1 = ( double ) ( int ) ( y );
    res = y - y1;

    if ( res != 0.0 )
    {
      if ( y1 != ( double ) ( int ) ( y1 * 0.5 ) * 2.0 )
      {
        parity = 1;
      }

      fact = - pi / sin ( pi * res );
      y = y + 1.0;
    }
    else
    {
      res = xinf;
      value = res;
      return value;
    }
  }
/*
  Argument is positive.
*/
  if ( y < eps )
  {
/*
  Argument < EPS.
*/
    if ( xminin <= y )
    {
      res = 1.0 / y;
    }
    else
    {
      res = xinf;
      value = res;
      return value;
    }
  }
  else if ( y < 12.0 )
  {
    y1 = y;
/*
  0.0 < argument < 1.0.
*/
    if ( y < 1.0 )
    {
      z = y;
      y = y + 1.0;
    }
/*
  1.0 < argument < 12.0.
  Reduce argument if necessary.
*/
    else
    {
      n = ( int ) ( y ) - 1;
      y = y - ( double ) ( n );
      z = y - 1.0;
    }
/*
  Evaluate approximation for 1.0 < argument < 2.0.
*/
    xnum = 0.0;
    xden = 1.0;
    for ( i = 0; i < 8; i++ )
    {
      xnum = ( xnum + p[i] ) * z;
      xden = xden * z + q[i];
    }
    res = xnum / xden + 1.0;
/*
  Adjust result for case  0.0 < argument < 1.0.
*/
    if ( y1 < y )
    {
      res = res / y1;
    }
/*
  Adjust result for case 2.0 < argument < 12.0.
*/
    else if ( y < y1 )
    {
      for ( i = 1; i <= n; i++ )
      {
        res = res * y;
        y = y + 1.0;
      }
    }
  }
  else
  {
/*
  Evaluate for 12.0 <= argument.
*/
    if ( y <= xbig )
    {
      ysq = y * y;
      sum = c[6];
      for ( i = 0; i < 6; i++ )
      {
        sum = sum / ysq + c[i];
      }
      sum = sum / y - y + sqrtpi;
      sum = sum + ( y - 0.5 ) * log ( y );
      res = exp ( sum );
    }
    else
    {
      res = xinf;
      value = res;
      return value;
    }
  }
/*
  Final adjustments and return.
*/
  if ( parity )
  {
    res = - res;
  }

  if ( fact != 1.0 )
  {
    res = fact / res;
  }

  value = res;

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

void timestamp ( )

/******************************************************************************/
/*
  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:

    24 September 2003

  Author:

    John Burkardt

  Parameters:

    None
*/
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct tm *tm;
  time_t now;

  now = time ( NULL );
  tm = localtime ( &now );

  strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm );

  fprintf ( stdout, "%s\n", time_buffer );

  return;
# undef TIME_SIZE
}