/* ode-initval/rk4imp.c

 * 

 * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman

 * 

 * This program is free software; you can redistribute it and/or modify

 * it under the terms of the GNU General Public License as published by

 * the Free Software Foundation; either version 3 of the License, or (at

 * your option) any later version.

 * 

 * This program is distributed in the hope that it will be useful, but

 * WITHOUT ANY WARRANTY; without even the implied warranty of

 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU

 * General Public License for more details.

 * 

 * You should have received a copy of the GNU General Public License

 * along with this program; if not, write to the Free Software

 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.

 */

/* Runge-Kutta 4, Gaussian implicit */

/* Author:  G. Jungman

*/

/* Error estimation by step doubling, see eg. Ascher, U.M., Petzold,

   L.R., Computer methods for ordinary differential and

   differential-algebraic equations, SIAM, Philadelphia, 1998.

   Method coefficients can also be found in it.

*/

#include <config.h>

#include <stdlib.h>

#include <string.h>

#include <gsl/gsl_math.h>

#include <gsl/gsl_errno.h>

#include <gsl/gsl_odeiv.h>

#include "odeiv_util.h"

typedef struct

{

  double *k1nu;

  double *k2nu;

  double *ytmp1;

  double *ytmp2;

  double *y0;

  double *y0_orig;

  double *y_onestep;

}

rk4imp_state_t;

static void *

rk4imp_alloc (size_t dim)

{

  rk4imp_state_t *state = (rk4imp_state_t *) malloc (sizeof (rk4imp_state_t));

  if (state == 0)

    {

      GSL_ERROR_NULL ("failed to allocate space for rk4imp_state",

                      GSL_ENOMEM);

    }

  state->k1nu = (double *) malloc (dim * sizeof (double));

  if (state->k1nu == 0)

    {

      free (state);

      GSL_ERROR_NULL ("failed to allocate space for k1nu", GSL_ENOMEM);

    }

  state->k2nu = (double *) malloc (dim * sizeof (double));

  if (state->k2nu == 0)

    {

      free (state->k1nu);

      free (state);

      GSL_ERROR_NULL ("failed to allocate space for k2nu", GSL_ENOMEM);

    }

  state->ytmp1 = (double *) malloc (dim * sizeof (double));

  if (state->ytmp1 == 0)

    {

      free (state->k2nu);

      free (state->k1nu);

      free (state);

      GSL_ERROR_NULL ("failed to allocate space for ytmp1", GSL_ENOMEM);

    }

  state->ytmp2 = (double *) malloc (dim * sizeof (double));

  if (state->ytmp2 == 0)

    {

      free (state->ytmp1);

      free (state->k2nu);

      free (state->k1nu);

      free (state);

      GSL_ERROR_NULL ("failed to allocate space for ytmp2", GSL_ENOMEM);

    }

  state->y0 = (double *) malloc (dim * sizeof (double));

  if (state->y0 == 0)

    {

      free (state->ytmp2);

      free (state->ytmp1);

      free (state->k2nu);

      free (state->k1nu);

      free (state);

      GSL_ERROR_NULL ("failed to allocate space for y0", GSL_ENOMEM);

    }

  state->y0_orig = (double *) malloc (dim * sizeof (double));

  if (state->y0_orig == 0)

    {

      free (state->y0);

      free (state->ytmp2);

      free (state->ytmp1);

      free (state->k2nu);

      free (state->k1nu);

      free (state);

      GSL_ERROR_NULL ("failed to allocate space for y0_orig", GSL_ENOMEM);

    }

  state->y_onestep = (double *) malloc (dim * sizeof (double));

  if (state->y_onestep == 0)

    {

      free (state->y0_orig);

      free (state->y0);

      free (state->ytmp2);

      free (state->ytmp1);

      free (state->k2nu);

      free (state->k1nu);

      free (state);

      GSL_ERROR_NULL ("failed to allocate space for y_onestep", GSL_ENOMEM);

    }

  return state;

}

static int

rk4imp_step (double *y, rk4imp_state_t *state, 

	     const double h, const double t, 

	     const size_t dim, const gsl_odeiv_system *sys)

{

  /* Makes a Runge-Kutta 4th order implicit advance with step size h.

     y0 is initial values of variables y. 

     The implicit matrix equations to solve are:

     Y1 = y0 + h * a11 * f(t + h * c1, Y1) + h * a12 * f(t + h * c2, Y2) 

     Y2 = y0 + h * a21 * f(t + h * c1, Y1) + h * a22 * f(t + h * c2, Y2) 

     y = y0 + h * b1 * f(t + h * c1, Y1) + h * b2 * f(t + h * c2, Y2)

     with constant coefficients a, b and c. For this method

     they are: b=[0.5 0.5] c=[(3-sqrt(3))/6 (3+sqrt(3))/6]

     a11=1/4, a12=(3-2*sqrt(3))/12, a21=(3+2*sqrt(3))/12 and a22=1/4

  */

  const double ir3 = 1.0 / M_SQRT3;

  const int iter_steps = 3;

  int nu;

  size_t i;

  double *const k1nu = state->k1nu;

  double *const k2nu = state->k2nu;

  double *const ytmp1 = state->ytmp1;

  double *const ytmp2 = state->ytmp2;

  /* iterative solution of Y1 and Y2.

     Note: This method does not check for convergence of the

     iterative solution! 

  */

  for (nu = 0; nu < iter_steps; nu++)

    {

      for (i = 0; i < dim; i++)

        {

          ytmp1[i] =

            y[i] + h * (0.25 * k1nu[i] + 0.5 * (0.5 - ir3) * k2nu[i]);

          ytmp2[i] =

            y[i] + h * (0.25 * k2nu[i] + 0.5 * (0.5 + ir3) * k1nu[i]);

        }

      {

        int s =

	  GSL_ODEIV_FN_EVAL (sys, t + 0.5 * h * (1.0 - ir3), ytmp1, k1nu);

	if (s != GSL_SUCCESS)

	  {

	    return s;

	  }    

      }

      {

        int s =

	  GSL_ODEIV_FN_EVAL (sys, t + 0.5 * h * (1.0 + ir3), ytmp2, k2nu);

	if (s != GSL_SUCCESS)

	  {

	    return s;

	  }    

      }

    }

  /* assignment */

  for (i = 0; i < dim; i++)

    {

      const double d_i = 0.5 * (k1nu[i] + k2nu[i]);

      y[i] += h * d_i;

    }

  return GSL_SUCCESS;

}

static int

rk4imp_apply (void *vstate,

              size_t dim,

              double t,

              double h,

              double y[],

              double yerr[],

              const double dydt_in[],

              double dydt_out[], 

              const gsl_odeiv_system * sys)

{

  rk4imp_state_t *state = (rk4imp_state_t *) vstate;

  size_t i;

  double *y0 = state->y0;

  double *y0_orig = state->y0_orig;

  double *y_onestep = state->y_onestep;

  double *k1nu = state->k1nu;

  double *k2nu = state->k2nu;

  /* Initialization step */

  DBL_MEMCPY (y0, y, dim);

  /* Save initial values in case of failure */

  DBL_MEMCPY (y0_orig, y, dim);

  if (dydt_in != 0)

    {

      DBL_MEMCPY (k1nu, dydt_in, dim);

    }

  else

    {

      int s = GSL_ODEIV_FN_EVAL (sys, t, y, k1nu);

      if (s != GSL_SUCCESS)

        {

          return s;

        }

    }

  DBL_MEMCPY (k2nu, k1nu, dim);

  /* First traverse h with one step (save to y_onestep) */

  DBL_MEMCPY (y_onestep, y, dim);

  {

    int s = rk4imp_step (y_onestep, state, h, t, dim, sys);

    if (s != GSL_SUCCESS) 

      {

	return s;

      }

  }

 /* Then with two steps with half step length (save to y) */ 

  {

    int s = rk4imp_step (y, state, h/2.0, t, dim, sys);

    if (s != GSL_SUCCESS) 

      {

	/* Restore original y vector */

	DBL_MEMCPY (y, y0_orig, dim);

	return s;

      }

  }

  DBL_MEMCPY (y0, y, dim);

  {

    int s = GSL_ODEIV_FN_EVAL (sys, t + h/2.0, y, k1nu);

    if (s != GSL_SUCCESS)

      {

	/* Restore original y vector */

	DBL_MEMCPY (y, y0_orig, dim);

	return s;

      }

  }

  DBL_MEMCPY (k2nu, k1nu, dim);

  {

    int s = rk4imp_step (y, state, h/2.0, t + h/2.0, dim, sys);

    if (s != GSL_SUCCESS) 

      {

	/* Restore original y vector */

	DBL_MEMCPY (y, y0_orig, dim);

	return s;

      }

  }

  /* Derivatives at output */

  if (dydt_out != NULL) 

    {

      int s = GSL_ODEIV_FN_EVAL (sys, t + h, y, dydt_out);

      if (s != GSL_SUCCESS) {

	/* Restore original y vector */

	DBL_MEMCPY (y, y0_orig, dim);

	return s;

      } 

    }

  /* Error estimation */

  /* Denominator in step doubling error equation 

   *  yerr = 0.5 * | y(onestep) - y(twosteps) | / (2^order - 1)  

   */

  for (i = 0; i < dim; i++) 

    {

      yerr[i] = 8.0 * 0.5 * (y[i] - y_onestep[i]) / 15.0;

    }

  return GSL_SUCCESS;

}

static int

rk4imp_reset (void *vstate, size_t dim)

{

  rk4imp_state_t *state = (rk4imp_state_t *) vstate;

  DBL_ZERO_MEMSET (state->y_onestep, dim);

  DBL_ZERO_MEMSET (state->y0_orig, dim);

  DBL_ZERO_MEMSET (state->y0, dim);

  DBL_ZERO_MEMSET (state->k1nu, dim);

  DBL_ZERO_MEMSET (state->k2nu, dim);

  DBL_ZERO_MEMSET (state->ytmp1, dim);

  DBL_ZERO_MEMSET (state->ytmp2, dim);

  return GSL_SUCCESS;

}

static unsigned int

rk4imp_order (void *vstate)

{

  rk4imp_state_t *state = (rk4imp_state_t *) vstate;

  state = 0; /* prevent warnings about unused parameters */

  return 4;

}

static void

rk4imp_free (void *vstate)

{

  rk4imp_state_t *state = (rk4imp_state_t *) vstate;

  free (state->y_onestep);

  free (state->y0_orig);

  free (state->y0);

  free (state->k1nu);

  free (state->k2nu);

  free (state->ytmp1);

  free (state->ytmp2);

  free (state);

}

static const gsl_odeiv_step_type rk4imp_type = { "rk4imp",      /* name */

  1,                             /* can use dydt_in? */

  1,                             /* gives exact dydt_out? */

  &rk4imp_alloc,

  &rk4imp_apply,

  &rk4imp_reset,

  &rk4imp_order,

  &rk4imp_free

};

const gsl_odeiv_step_type *gsl_odeiv_step_rk4imp = &rk4imp_type;