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

# include "runge.h"

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

double runge_antideriv ( double x )

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

    runge_antideriv() evaluates the antiderivative of the Runge function.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    24 March 2025

  Author:

    John Burkardt

  Input:

    double x: the argument of the function.

  Output:

    double runge_antideriv: the value of the antiderivative.
*/
{
  double value;

  value = atan ( 5.0 * x ) / 5.0;

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

double runge_deriv ( double x )

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

    runge_deriv() evaluates the derivative of the Runge function.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    24 March 2025

  Author:

    John Burkardt

  Input:

    double x: the argument of the function.

  Output:

    double runge_deriv: the value of the derivative.
*/
{
  double value;

  value = - 50.0 * x / pow ( 1.0 + 25.0 * x * x, 2 );

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

double runge_deriv2 ( double x )

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

    runge_deriv2() evaluates the second derivative of the Runge function.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    24 March 2025

  Author:

    John Burkardt

  Input:

    double x: the argument of the function.

  Output:

    double runge_deriv2, the value of the second derivative.
*/
{
  double u;
  double up;
  double v;
  double value;
  double vp;
 
  u = - 50.0 * x;
  up = - 50.0;
  v = pow ( 1.0 + 25.0 * x * x, 2 );
  vp = 2.0 * ( 1.0 + 25.0 * x * x ) * ( 50.0 * x );

  value = ( up * v - u * vp ) / pow ( v, 2 );

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

double runge_fun ( double x )

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

    runge_fun() evaluates the Runge function.

  Discussion:

    This function causes a breakdown for
    polynomial interpolation over equally spaced nodes in [-1,+1].

    Runge originally considered the function 1/(1+x^2) over the 
    interval [-5,+5].  For convenience, a rescaled version is considered
    over the interval [-1,+1].

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    24 March 2025

  Author:

    John Burkardt

  Input:

    double x: the argument of the function.

  Output:

    double runge_fun: the value of the function.
*/
{
  double value;

  value = 1.0 / ( 1.0 + 25.0 * x * x );

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

double runge_power_series ( double x, int n )

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

    runge_power_series() evaluates a power series for the Runge function.

  Discussion:

    The Runge function considered here has the form
      f(x) = 1 / ( 1 + 25x^2 )

    The power series is 1 - (5x)^2 + (5x)^4 - (5x)^6 + (5x)^8 - (5x)^10 ...

    The power series is only well defined in the open interval 
      -1/5 < x < 1/5
    where successive terms gradually go to zero.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    25 March 2025

  Author:

    John Burkardt

  Input:

    real x: the argument of the function.

    integer n: the number of terms to be computed.

  Output:

    real value: the value of the function.
*/
{
  int i;
  double term;
  double value;

  value = 0.0;
  term = 1.0;
  for ( i = 1; i <= n; i++ )
  {
    value = value + term;
    term = - term * pow ( 5.0 * x, 2 );
  }

  return value;
}