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

  This source file is part of the Avogadro project.

  Copyright (C) 2010 Eric C. Brown

  This source code is released under the New BSD License, (the "License").

  Unless required by applicable law or agreed to in writing, software
  distributed under the License is distributed on an "AS IS" BASIS,
  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  See the License for the specific language governing permissions and
  limitations under the License.

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

/*
   Based on codes written by Herman Watts, Lawrence Shampine, and John Burkardt.
*/

#include "qtaimodeintegrator.h"

#include "qtaimmathutilities.h"

namespace Avogadro::QtPlugins {

QTAIMODEIntegrator::QTAIMODEIntegrator(QTAIMWavefunctionEvaluator& eval,
                                       const qint64 mode)
  : m_eval(&eval), m_mode(mode), m_associatedSphere(0)
{
  m_betaSpheres.clear();
}

QVector3D QTAIMODEIntegrator::integrate(QVector3D x0y0z0)
{
  qreal x0 = x0y0z0.x();
  qreal y0 = x0y0z0.y();
  qreal z0 = x0y0z0.z();

  const qint64 NEQN = 3;

  abserr_save = -1.0;
  flag_save = -1000;
  h = -1.0;
  init = -1000;
  kflag = -1000;
  kop = -1;
  nfe = -1;
  relerr_save = -1.0;
  remin = 1.0e-12;

  qreal abserr;
  qint64 flag;
  qint64 i_step;
  qint64 n_step = 0;
  qreal relerr;
  qreal t;
  qreal t_out;
  qreal t_start;
  qreal t_stop;
  qreal y[3];
  qreal yp[3];

  switch (m_mode) {
    case SteepestAscentPathInElectronDensity:
      abserr = 1.e-5; //  sqrt ( r8_epsilon ( ) );
      relerr = 1.e10; //  sqrt ( r8_epsilon ( ) );
      t_start = 0.0;
      t_stop = 10.0;
      n_step = t_stop * 20;
      break;
    case CMBPMinusThreeGradientInElectronDensity:
      abserr = sqrt(r8_epsilon());
      relerr = sqrt(r8_epsilon());
      t_start = 0.0;
      t_stop = 1.0;
      n_step = t_stop * 1000;
      break;
    case CMBPMinusOneGradientInElectronDensity:
      abserr = sqrt(r8_epsilon());
      relerr = sqrt(r8_epsilon());
      t_start = 0.0;
      t_stop = 1.0;
      n_step = t_stop * 100;
      break;
    case CMBPPlusOneGradientInElectronDensity:
      abserr = sqrt(r8_epsilon());
      relerr = sqrt(r8_epsilon());
      t_start = 0.0;
      t_stop = 1.0;
      n_step = t_stop * 200;
      break;
    case CMBPPlusThreeGradientInElectronDensity:
      abserr = sqrt(r8_epsilon());
      relerr = sqrt(r8_epsilon());
      t_start = 0.0;
      t_stop = 1.0;
      n_step = t_stop * 200;
      break;
    case CMBPMinusThreeGradientInElectronDensityLaplacian:
      abserr = sqrt(r8_epsilon());
      relerr = sqrt(r8_epsilon());
      t_start = 0.0;
      t_stop = 1.0;
      n_step = t_stop * 200;
      break;
    case CMBPMinusOneGradientInElectronDensityLaplacian:
      abserr = sqrt(r8_epsilon());
      relerr = sqrt(r8_epsilon());
      t_start = 0.0;
      t_stop = 1.0;
      n_step = t_stop * 200;
      break;
    case CMBPPlusOneGradientInElectronDensityLaplacian:
      abserr = sqrt(r8_epsilon());
      relerr = sqrt(r8_epsilon());
      t_start = 0.0;
      t_stop = 1.0;
      n_step = t_stop * 200;
      break;
    case CMBPPlusThreeGradientInElectronDensityLaplacian:
      abserr = sqrt(r8_epsilon());
      relerr = sqrt(r8_epsilon());
      t_start = 0.0;
      t_stop = 1.0;
      n_step = t_stop * 200;
      break;
    default:
      qDebug() << "Catastrophic: No ODE parameters for this property.";
      break;
  }

  y[0] = x0;
  y[1] = y0;
  y[2] = z0;

  m_path.clear();
  m_path.append(QVector3D(y[0], y[1], y[2]));

  flag = 1;

  for (i_step = 1; i_step <= n_step; i_step++) {
    t =
      ((qreal)(n_step - i_step + 1) * t_start + (qreal)(i_step - 1) * t_stop) /
      (qreal)(n_step);

    t_out = ((qreal)(n_step - i_step) * t_start + (qreal)(i_step)*t_stop) /
            (qreal)(n_step);

    flag = QTAIMODEIntegrator::r8_rkf45(NEQN, y, yp, &t, t_out, &relerr, abserr,
                                        flag);

    m_status = flag;
    m_path.append(QVector3D(y[0], y[1], y[2]));

    if (flag == 7) {
      flag = 2;
      m_status = flag;
    }

    if (flag != 2) {
      m_status = flag;
      return QVector3D(y[0], y[1], y[2]);
    }

    if (m_betaSpheres.length() > 0) {
      for (qint64 n = 0; n < m_betaSpheres.length(); ++n) {
        Matrix<qreal, 3, 1> a(y[0], y[1], y[2]);
        Matrix<qreal, 3, 1> b(m_betaSpheres.at(n).first.x(),
                              m_betaSpheres.at(n).first.y(),
                              m_betaSpheres.at(n).first.z());

        qreal distance = QTAIMMathUtilities::distance(a, b);

        if (distance < m_betaSpheres.at(n).second) {
          m_status = 0;
          m_associatedSphere = n;
          return QVector3D(m_betaSpheres.at(n).first.x(),
                           m_betaSpheres.at(n).first.y(),
                           m_betaSpheres.at(n).first.z());
        }
      }
    } // beta spheres

  } // ODE step

  return QVector3D(y[0], y[1], y[2]);
}

void QTAIMODEIntegrator::r8_f(qreal t, qreal y[], qreal yp[])
{

  t = t; // suppress warning

  Matrix<qreal, 3, 1> gradient;

  Matrix<qreal, 3, 4> gH;
  Matrix<qreal, 3, 1> g;
  Matrix<qreal, 3, 3> H;

  Matrix<qreal, 3, 1> xyz;
  xyz << y[0], y[1], y[2];

  if (m_mode == SteepestAscentPathInElectronDensity) {
    g = m_eval->gradientOfElectronDensity(xyz);
  } else {
    if (m_mode == 1 || m_mode == 2 || m_mode == 3 || m_mode == 4) {
      gH = m_eval->gradientAndHessianOfElectronDensity(xyz);
    } else {
      gH = m_eval->gradientAndHessianOfElectronDensityLaplacian(xyz);
    }

    g(0) = gH(0, 0);
    g(1) = gH(1, 0);
    g(2) = gH(2, 0);
    H(0, 0) = gH(0, 1);
    H(1, 0) = gH(1, 1);
    H(2, 0) = gH(2, 1);
    H(0, 1) = gH(0, 2);
    H(1, 1) = gH(1, 2);
    H(2, 1) = gH(2, 2);
    H(0, 2) = gH(0, 3);
    H(1, 2) = gH(1, 3);
    H(2, 2) = gH(2, 3);
  }

  switch (m_mode) {
    case SteepestAscentPathInElectronDensity:
      gradient = g;
      break;
    case CMBPMinusThreeGradientInElectronDensity:
      gradient = QTAIMMathUtilities::minusThreeSignatureLocatorGradient(g, H);
      break;
    case CMBPMinusOneGradientInElectronDensity:
      gradient = QTAIMMathUtilities::minusOneSignatureLocatorGradient(g, H);
      break;
    case CMBPPlusOneGradientInElectronDensity:
      gradient = QTAIMMathUtilities::plusOneSignatureLocatorGradient(g, H);
      break;
    case CMBPPlusThreeGradientInElectronDensity:
      gradient = QTAIMMathUtilities::plusThreeSignatureLocatorGradient(g, H);
      break;
    case CMBPMinusThreeGradientInElectronDensityLaplacian:
      gradient = QTAIMMathUtilities::minusThreeSignatureLocatorGradient(g, H);
      break;
    case CMBPMinusOneGradientInElectronDensityLaplacian:
      gradient = QTAIMMathUtilities::minusOneSignatureLocatorGradient(g, H);
      break;
    case CMBPPlusOneGradientInElectronDensityLaplacian:
      gradient = QTAIMMathUtilities::plusOneSignatureLocatorGradient(g, H);
      break;
    case CMBPPlusThreeGradientInElectronDensityLaplacian:
      gradient = QTAIMMathUtilities::plusThreeSignatureLocatorGradient(g, H);
      break;
    default:
      qDebug() << "Catastrophic: No ODE parameters for this property.";
      break;
  }

  qreal normGradient = std::hypot(gradient(0), gradient(1), gradient(2));

  yp[0] = gradient(0) / normGradient;
  yp[1] = gradient(1) / normGradient;
  yp[2] = gradient(2) / normGradient;
}

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

qreal QTAIMODEIntegrator::r8_abs(qreal x)

//****************************************************************************80
//
//  Purpose:
//
//    R8_ABS returns the absolute value of an R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 April 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, qreal X, the quantity whose absolute value is desired.
//
//    Output, qreal R8_ABS, the absolute value of X.
//
{
  if (0.0 <= x) {
    return x;
  } else {
    return (-x);
  }
}
//****************************************************************************80

qreal QTAIMODEIntegrator::r8_epsilon()

//****************************************************************************80
//
//  Purpose:
//
//    R8_EPSILON returns the R8 round off unit.
//
//  Discussion:
//
//    R8_EPSILON is a number R which is a power of 2 with the property that,
//    to the precision of the computer's arithmetic,
//      1 < 1 + R
//    but
//      1 = ( 1 + R / 2 )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    06 May 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, qreal R8_EPSILON, the R8 round-off unit.
//
{
  qreal r = 1.0;

  while (1.0 < (qreal)(1.0 + r)) {
    r = r / 2.0;
  }

  return (2.0 * r);
}
//****************************************************************************80

void QTAIMODEIntegrator::r8_fehl(qint64 neqn, qreal y[], qreal t, qreal h_,
                                 qreal yp[], qreal f1[], qreal f2[], qreal f3[],
                                 qreal f4[], qreal f5[], qreal s[])

//****************************************************************************80
//
//  Purpose:
//
//    R8_FEHL takes one Fehlberg fourth-fifth order step.
//
//  Discussion:
//
//    This version of the routine uses qreal real arithmetic.
//
//    This routine integrates a system of NEQN first order ordinary differential
//    equations of the form
//      dY(i)/dT = F(T,Y(1:NEQN))
//    where the initial values Y and the initial derivatives
//    YP are specified at the starting point T.
//
//    The routine advances the solution over the fixed step H and returns
//    the fifth order (sixth order accurate locally) solution
//    approximation at T+H in array S.
//
//    The formulas have been grouped to control loss of significance.
//    The routine should be called with an H not smaller than 13 units of
//    roundoff in T so that the various independent arguments can be
//    distinguished.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    27 March 2004
//
//  Author:
//
//    Original FORTRAN77 version by Herman Watts, Lawrence Shampine.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Erwin Fehlberg,
//    Low-order Classical Runge-Kutta Formulas with Stepsize Control,
//    NASA Technical Report R-315, 1969.
//
//    Lawrence Shampine, Herman Watts, S Davenport,
//    Solving Non-stiff Ordinary Differential Equations - The State of the Art,
//    SIAM Review,
//    Volume 18, pages 376-411, 1976.
//
//  Parameters:
//
//    Input, external F, a user-supplied subroutine to evaluate the
//    derivatives Y'(T), of the form:
//
//      void f ( qreal t, qreal y[], qreal yp[] )
//
//    Input, qint64 NEQN, the number of equations to be integrated.
//
//    Input, qreal Y[NEQN], the current value of the dependent variable.
//
//    Input, qreal T, the current value of the independent variable.
//
//    Input, qreal H, the step size to take.
//
//    Input, qreal YP[NEQN], the current value of the derivative of the
//    dependent variable.
//
//    Output, qreal F1[NEQN], F2[NEQN], F3[NEQN], F4[NEQN], F5[NEQN], derivative
//    values needed for the computation.
//
//    Output, qreal S[NEQN], the estimate of the solution at T+H.
//
{
  qreal ch = h_ / 4.0;

  for (qint64 i = 0; i < neqn; i++) {
    f5[i] = y[i] + ch * yp[i];
  }

  QTAIMODEIntegrator::r8_f(t + ch, f5, f1);

  ch = 3.0 * h_ / 32.0;

  for (qint64 i = 0; i < neqn; i++) {
    f5[i] = y[i] + ch * (yp[i] + 3.0 * f1[i]);
  }

  QTAIMODEIntegrator::r8_f(t + 3.0 * h_ / 8.0, f5, f2);

  ch = h_ / 2197.0;

  for (qint64 i = 0; i < neqn; i++) {
    f5[i] = y[i] + ch * (1932.0 * yp[i] + (7296.0 * f2[i] - 7200.0 * f1[i]));
  }

  QTAIMODEIntegrator::r8_f(t + 12.0 * h_ / 13.0, f5, f3);

  ch = h_ / 4104.0;

  for (qint64 i = 0; i < neqn; i++) {
    f5[i] = y[i] + ch * ((8341.0 * yp[i] - 845.0 * f3[i]) +
                         (29440.0 * f2[i] - 32832.0 * f1[i]));
  }

  QTAIMODEIntegrator::r8_f(t + h_, f5, f4);

  ch = h_ / 20520.0;

  for (qint64 i = 0; i < neqn; i++) {
    f1[i] = y[i] + ch * ((-6080.0 * yp[i] + (9295.0 * f3[i] - 5643.0 * f4[i])) +
                         (41040.0 * f1[i] - 28352.0 * f2[i]));
  }

  QTAIMODEIntegrator::r8_f(t + h_ / 2.0, f1, f5);
  //
  //  Ready to compute the approximate solution at T+H.
  //
  ch = h_ / 7618050.0;

  for (qint64 i = 0; i < neqn; i++) {
    s[i] = y[i] +
           ch * ((902880.0 * yp[i] + (3855735.0 * f3[i] - 1371249.0 * f4[i])) +
                 (3953664.0 * f2[i] + 277020.0 * f5[i]));
  }
}
//****************************************************************************80

qreal QTAIMODEIntegrator::r8_max(qreal x, qreal y)

//****************************************************************************80
//
//  Purpose:
//
//    R8_MAX returns the maximum of two R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    10 January 2002
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, qreal X, Y, the quantities to compare.
//
//    Output, qreal R8_MAX, the maximum of X and Y.
//
{
  if (y < x) {
    return x;
  } else {
    return y;
  }
}
//****************************************************************************80

qreal QTAIMODEIntegrator::r8_min(qreal x, qreal y)

//****************************************************************************80
//
//  Purpose:
//
//    R8_MIN returns the minimum of two R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    09 May 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, qreal X, Y, the quantities to compare.
//
//    Output, qreal R8_MIN, the minimum of X and Y.
//
{
  if (y < x) {
    return y;
  } else {
    return x;
  }
}
//****************************************************************************80

qint64 QTAIMODEIntegrator::r8_rkf45(qint64 neqn, qreal y[], qreal yp[],
                                    qreal* t, qreal tout, qreal* relerr,
                                    qreal abserr, qint64 flag)

//****************************************************************************80
//
//  Purpose:
//
//    R8_RKF45 carries out the Runge-Kutta-Fehlberg method.
//
//  Discussion:
//
//    This version of the routine uses qreal real arithmetic.
//
//    This routine is primarily designed to solve non-stiff and mildly stiff
//    differential equations when derivative evaluations are inexpensive.
//    It should generally not be used when the user is demanding
//    high accuracy.
//
//    This routine integrates a system of NEQN first-order ordinary differential
//    equations of the form:
//
//      dY(i)/dT = F(T,Y(1),Y(2),...,Y(NEQN))
//
//    where the Y(1:NEQN) are given at T.
//
//    Typically the subroutine is used to integrate from T to TOUT but it
//    can be used as a one-step integrator to advance the solution a
//    single step in the direction of TOUT.  On return, the parameters in
//    the call list are set for continuing the integration.  The user has
//    only to call again (and perhaps define a new value for TOUT).
//
//    Before the first call, the user must
//
//    * supply the subroutine F(T,Y,YP) to evaluate the right hand side;
//      and declare F in an EXTERNAL statement;
//
//    * initialize the parameters:
//      NEQN, Y(1:NEQN), T, TOUT, RELERR, ABSERR, FLAG.
//      In particular, T should initially be the starting point for integration,
//      Y should be the value of the initial conditions, and FLAG should
//      normally be +1.
//
//    Normally, the user only sets the value of FLAG before the first call, and
//    thereafter, the program manages the value.  On the first call, FLAG should
//    normally be +1 (or -1 for single step mode.)  On normal return, FLAG will
//    have been reset by the program to the value of 2 (or -2 in single
//    step mode), and the user can continue to call the routine with that
//    value of FLAG.
//
//    (When the input magnitude of FLAG is 1, this indicates to the program
//    that it is necessary to do some initialization work.  An input magnitude
//    of 2 lets the program know that initialization can be skipped,
//    and that useful information was computed earlier.)
//
//    The routine returns with all the information needed to continue
//    the integration.  If the integration reached TOUT, the user need only
//    define a new TOUT and call again.  In the one-step integrator
//    mode, returning with FLAG = -2, the user must keep in mind that
//    each step taken is in the direction of the current TOUT.  Upon
//    reaching TOUT, indicated by the output value of FLAG switching to 2,
//    the user must define a new TOUT and reset FLAG to -2 to continue
//    in the one-step integrator mode.
//
//    In some cases, an error or difficulty occurs during a call.  In that case,
//    the output value of FLAG is used to indicate that there is a problem
//    that the user must address.  These values include:
//
//    * 3, integration was not completed because the input value of RELERR, the
//      relative error tolerance, was too small.  RELERR has been increased
//      appropriately for continuing.  If the user accepts the output value of
//      RELERR, then simply reset FLAG to 2 and continue.
//
//    * 4, integration was not completed because more than MAXNFE derivative
//      evaluations were needed.  This is approximately (MAXNFE/6) steps.
//      The user may continue by simply calling again.  The function counter
//      will be reset to 0, and another MAXNFE function evaluations are allowed.
//
//    * 5, integration was not completed because the solution vanished,
//      making a pure relative error test impossible.  The user must use
//      a non-zero ABSERR to continue.  Using the one-step integration mode
//      for one step is a good way to proceed.
//
//    * 6, integration was not completed because the requested accuracy
//      could not be achieved, even using the smallest allowable stepsize.
//      The user must increase the error tolerances ABSERR or RELERR before
//      continuing.  It is also necessary to reset FLAG to 2 (or -2 when
//      the one-step integration mode is being used).  The occurrence of
//      FLAG = 6 indicates a trouble spot.  The solution is changing
//      rapidly, or a singularity may be present.  It often is inadvisable
//      to continue.
//
//    * 7, it is likely that this routine is inefficient for solving
//      this problem.  Too much output is restricting the natural stepsize
//      choice.  The user should use the one-step integration mode with
//      the stepsize determined by the code.  If the user insists upon
//      continuing the integration, reset FLAG to 2 before calling
//      again.  Otherwise, execution will be terminated.
//
//    * 8, invalid input parameters, indicates one of the following:
//      NEQN <= 0;
//      T = TOUT and |FLAG| /= 1;
//      RELERR < 0 or ABSERR < 0;
//      FLAG == 0  or FLAG < -2 or 8 < FLAG.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    27 March 2004
//
//  Author:
//
//    Original FORTRAN77 version by Herman Watts, Lawrence Shampine.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Erwin Fehlberg,
//    Low-order Classical Runge-Kutta Formulas with Stepsize Control,
//    NASA Technical Report R-315, 1969.
//
//    Lawrence Shampine, Herman Watts, S Davenport,
//    Solving Non-stiff Ordinary Differential Equations - The State of the Art,
//    SIAM Review,
//    Volume 18, pages 376-411, 1976.
//
//  Parameters:
//
//    Input, external F, a user-supplied subroutine to evaluate the
//    derivatives Y'(T), of the form:
//
//      void f ( qreal t, qreal y[], qreal yp[] )
//
//    Input, qint64 NEQN, the number of equations to be integrated.
//
//    Input/output, qreal Y[NEQN], the current solution vector at T.
//
//    Input/output, qreal YP[NEQN], the derivative of the current solution
//    vector at T.  The user should not set or alter this information!
//
//    Input/output, qreal *T, the current value of the independent variable.
//
//    Input, qreal TOUT, the output point at which solution is desired.
//    TOUT = T is allowed on the first call only, in which case the routine
//    returns with FLAG = 2 if continuation is possible.
//
//    Input, qreal *RELERR, ABSERR, the relative and absolute error tolerances
//    for the local error test.  At each step the code requires:
//      abs ( local error ) <= RELERR * abs ( Y ) + ABSERR
//    for each component of the local error and the solution vector Y.
//    RELERR cannot be "too small".  If the routine believes RELERR has been
//    set too small, it will reset RELERR to an acceptable value and return
//    immediately for user action.
//
//    Input, qint64 FLAG, indicator for status of integration. On the first
//    call,
//    set FLAG to +1 for normal use, or to -1 for single step mode.  On
//    subsequent continuation steps, FLAG should be +2, or -2 for single
//    step mode.
//
//    Output, qint64 RKF45_D, indicator for status of integration.  A value of 2
//    or -2 indicates normal progress, while any other value indicates a
//    problem that should be addressed.
//
{
// ECB: originally 3000
#define MAXNFE 1000

  qreal ae;
  qreal dt;
  qreal ee;
  qreal eeoet;
  qreal eps;
  qreal esttol;
  qreal et;
  qreal* f1;
  qreal* f2;
  qreal* f3;
  qreal* f4;
  qreal* f5;
  bool hfaild;
  qreal hmin;
  qint64 i;
  qint64 k;
  qint64 mflag;
  bool output;
  qreal relerr_min;
  qreal s;
  qreal scale;
  qreal tol;
  qreal toln;
  qreal ypk;
  //
  //  Check the input parameters.
  //
  eps = r8_epsilon();

  if (neqn < 1) {
    return 8;
  }

  if ((*relerr) < 0.0) {
    return 8;
  }

  if (abserr < 0.0) {
    return 8;
  }

  if (flag == 0 || 8 < flag || flag < -2) {
    return 8;
  }

  mflag = (qint64)abs((long)flag);
  //
  //  Is this a continuation call?
  //
  if (mflag != 1) {
    if (*t == tout && kflag != 3) {
      return 8;
    }
    //
    //  FLAG = -2 or +2:
    //
    if (mflag == 2) {
      if (kflag == 3) {
        flag = flag_save;
        mflag = (qint64)abs((long)flag);
      } else if (init == 0) {
        flag = flag_save;
      } else if (kflag == 4) {
        nfe = 0;
      } else if (kflag == 5 && abserr == 0.0) {
        // error
      } else if (kflag == 6 && (*relerr) <= relerr_save &&
                 abserr <= abserr_save) {
        // error
      }
    }
    //
    //  FLAG = 3, 4, 5, 6, 7 or 8.
    //
    else {
      if (flag == 3) {
        flag = flag_save;
        if (kflag == 3) {
          mflag = (qint64)abs((long)flag);
        }
      } else if (flag == 4) {
        nfe = 0;
        flag = flag_save;
        if (kflag == 3) {
          mflag = (qint64)abs((long)flag);
        }
      } else if (flag == 5 && 0.0 < abserr) {
        flag = flag_save;
        if (kflag == 3) {
          mflag = (qint64)abs((long)flag);
        }
      }
      //
      //  Integration cannot be continued because the user did not respond to
      //  the instructions pertaining to FLAG = 5, 6, 7 or 8.
      //
      else {
        // error
      }
    }
  }
  //
  //  Save the input value of FLAG.
  //  Set the continuation flag KFLAG for subsequent input checking.
  //
  flag_save = flag;
  kflag = 0;
  //
  //  Save RELERR and ABSERR for checking input on subsequent calls.
  //
  relerr_save = (*relerr);
  abserr_save = abserr;
  //
  //  Restrict the relative error tolerance to be at least
  //
  //    2*EPS+REMIN
  //
  //  to avoid limiting precision difficulties arising from impossible
  //  accuracy requests.
  //
  relerr_min = 2.0 * r8_epsilon() + remin;
  //
  //  Is the relative error tolerance too small?
  //
  if ((*relerr) < relerr_min) {
    (*relerr) = relerr_min;
    kflag = 3;
    return 3;
  }

  dt = tout - *t;
  //
  //  Initialization:
  //
  //  Set the initialization completion indicator, INIT;
  //  set the indicator for too many output points, KOP;
  //  evaluate the initial derivatives
  //  set the counter for function evaluations, NFE;
  //  estimate the starting stepsize.
  //
  f1 = new qreal[neqn];
  f2 = new qreal[neqn];
  f3 = new qreal[neqn];
  f4 = new qreal[neqn];
  f5 = new qreal[neqn];

  if (mflag == 1) {
    init = 0;
    kop = 0;
    QTAIMODEIntegrator::r8_f(*t, y, yp);
    nfe = 1;

    if (*t == tout) {
      return 2;
    }
  }

  if (init == 0) {
    init = 1;
    h = r8_abs(dt);
    toln = 0.0;

    for (k = 0; k < neqn; k++) {
      tol = (*relerr) * r8_abs(y[k]) + abserr;
      if (0.0 < tol) {
        toln = tol;
        ypk = r8_abs(yp[k]);
        if (tol < ypk * pow(h, 5)) {
          h = pow((tol / ypk), 0.2);
        }
      }
    }

    if (toln <= 0.0) {
      h = 0.0;
    }

    h = r8_max(h, 26.0 * eps * r8_max(r8_abs(*t), r8_abs(dt)));

    if (flag < 0) {
      flag_save = -2;
    } else {
      flag_save = 2;
    }
  }
  //
  //  Set stepsize for integration in the direction from T to TOUT.
  //
  h = r8_sign(dt) * r8_abs(h);
  //
  //  Test to see if too may output points are being requested.
  //
  if (2.0 * r8_abs(dt) <= r8_abs(h)) {
    kop = kop + 1;
  }
  //
  //  Unnecessary frequency of output.
  //
  if (kop == 100) {
    kop = 0;
    delete[] f1;
    delete[] f2;
    delete[] f3;
    delete[] f4;
    delete[] f5;
    return 7;
  }
  //
  //  If we are too close to the output point, then simply extrapolate and
  //  return.
  //
  if (r8_abs(dt) <= 26.0 * eps * r8_abs(*t)) {
    *t = tout;
    for (i = 0; i < neqn; i++) {
      y[i] = y[i] + dt * yp[i];
    }
    QTAIMODEIntegrator::r8_f(*t, y, yp);
    nfe = nfe + 1;

    delete[] f1;
    delete[] f2;
    delete[] f3;
    delete[] f4;
    delete[] f5;
    return 2;
  }
  //
  //  Initialize the output point indicator.
  //
  output = false;
  //
  //  To avoid premature underflow in the error tolerance function,
  //  scale the error tolerances.
  //
  scale = 2.0 / (*relerr);
  ae = scale * abserr;
  //
  //  Step by step integration.
  //
  for (;;) {
    hfaild = false;
    //
    //  Set the smallest allowable stepsize.
    //
    hmin = 26.0 * eps * r8_abs(*t);
    //
    //  Adjust the stepsize if necessary to hit the output point.
    //
    //  Look ahead two steps to avoid drastic changes in the stepsize and
    //  thus lessen the impact of output points on the code.
    //
    dt = tout - *t;

    if (2.0 * r8_abs(h) <= r8_abs(dt)) {
    } else
    //
    //  Will the next successful step complete the integration to the output
    //  point?
    //
    {
      if (r8_abs(dt) <= r8_abs(h)) {
        output = true;
        h = dt;
      } else {
        h = 0.5 * dt;
      }
    }
    //
    //  Here begins the core integrator for taking a single step.
    //
    //  The tolerances have been scaled to avoid premature underflow in
    //  computing the error tolerance function ET.
    //  To avoid problems with zero crossings, relative error is measured
    //  using the average of the magnitudes of the solution at the
    //  beginning and end of a step.
    //  The error estimate formula has been grouped to control loss of
    //  significance.
    //
    //  To distinguish the various arguments, H is not permitted
    //  to become smaller than 26 units of roundoff in T.
    //  Practical limits on the change in the stepsize are enforced to
    //  smooth the stepsize selection process and to avoid excessive
    //  chattering on problems having discontinuities.
    //  To prevent unnecessary failures, the code uses 9/10 the stepsize
    //  it estimates will succeed.
    //
    //  After a step failure, the stepsize is not allowed to increase for
    //  the next attempted step.  This makes the code more efficient on
    //  problems having discontinuities and more effective in general
    //  since local extrapolation is being used and extra caution seems
    //  warranted.
    //
    //  Test the number of derivative function evaluations.
    //  If okay, try to advance the integration from T to T+H.
    //
    for (;;) {
      //
      //  Have we done too much work?
      //
      if (MAXNFE < nfe) {
        kflag = 4;
        delete[] f1;
        delete[] f2;
        delete[] f3;
        delete[] f4;
        delete[] f5;
        return 4;
      }
      //
      //  Advance an approximate solution over one step of length H.
      //
      r8_fehl(neqn, y, *t, h, yp, f1, f2, f3, f4, f5, f1);
      nfe = nfe + 5;
      //
      //  Compute and test allowable tolerances versus local error estimates
      //  and remove scaling of tolerances.  The relative error is
      //  measured with respect to the average of the magnitudes of the
      //  solution at the beginning and end of the step.
      //
      eeoet = 0.0;

      for (k = 0; k < neqn; k++) {
        et = r8_abs(y[k]) + r8_abs(f1[k]) + ae;

        if (et <= 0.0) {
          delete[] f1;
          delete[] f2;
          delete[] f3;
          delete[] f4;
          delete[] f5;
          return 5;
        }

        ee = r8_abs((-2090.0 * yp[k] + (21970.0 * f3[k] - 15048.0 * f4[k])) +
                    (22528.0 * f2[k] - 27360.0 * f5[k]));

        eeoet = r8_max(eeoet, ee / et);
      }

      esttol = r8_abs(h) * eeoet * scale / 752400.0;

      if (esttol <= 1.0) {
        break;
      }
      //
      //  Unsuccessful step.  Reduce the stepsize, try again.
      //  The decrease is limited to a factor of 1/10.
      //
      hfaild = true;
      output = false;

      if (esttol < 59049.0) {
        s = 0.9 / pow(esttol, 0.2);
      } else {
        s = 0.1;
      }

      h = s * h;

      if (r8_abs(h) < hmin) {
        kflag = 6;
        delete[] f1;
        delete[] f2;
        delete[] f3;
        delete[] f4;
        delete[] f5;
        return 6;
      }
    }
    //
    //  We exited the loop because we took a successful step.
    //  Store the solution for T+H, and evaluate the derivative there.
    //
    *t = *t + h;
    for (i = 0; i < neqn; i++) {
      y[i] = f1[i];
    }
    QTAIMODEIntegrator::r8_f(*t, y, yp);
    nfe = nfe + 1;
    //
    //  Choose the next stepsize.  The increase is limited to a factor of 5.
    //  If the step failed, the next stepsize is not allowed to increase.
    //
    if (0.0001889568 < esttol) {
      s = 0.9 / pow(esttol, 0.2);
    } else {
      s = 5.0;
    }

    if (hfaild) {
      s = r8_min(s, 1.0);
    }

    h = r8_sign(h) * r8_max(s * r8_abs(h), hmin);
    //
    //  End of core integrator
    //
    //  Should we take another step?
    //
    if (output) {
      *t = tout;
      delete[] f1;
      delete[] f2;
      delete[] f3;
      delete[] f4;
      delete[] f5;
      return 2;
    }

    if (flag <= 0) {
      delete[] f1;
      delete[] f2;
      delete[] f3;
      delete[] f4;
      delete[] f5;
      return (-2);
    }
  }
#undef MAXNFE
}
//****************************************************************************80

qreal QTAIMODEIntegrator::r8_sign(qreal x)

//****************************************************************************80
//
//  Purpose:
//
//    R8_SIGN returns the sign of an R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    27 March 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, qreal X, the number whose sign is desired.
//
//    Output, qreal R8_SIGN, the sign of X.
//
{
  if (x < 0.0) {
    return (-1.0);
  } else {
    return (+1.0);
  }
}
//****************************************************************************80

} // namespace Avogadro::QtPlugins
