LALAdaptiveRungeKuttaIntegrator.c

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/*
*  Copyright (C) 2010 Michele Vallisneri, Will Farr, Evan Ochsner, 2014 A. Klein
*
*  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 2 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 with program; see the file COPYING. If not, write to the
*  Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
*  MA  02110-1301  USA
*/
 
#include <lal/LALAdaptiveRungeKuttaIntegrator.h>
 
#define XLAL_BEGINGSL \
        { \
          gsl_error_handler_t *saveGSLErrorHandler_; \
          saveGSLErrorHandler_ = gsl_set_error_handler_off();
 
#define XLAL_ENDGSL \
          gsl_set_error_handler( saveGSLErrorHandler_ ); \
        }
 
LALAdaptiveRungeKuttaIntegrator *XLALAdaptiveRungeKutta4Init(int dim, int (*dydt) (double t, const double y[], double dydt[], void *params),   /* These are XLAL functions! */
    int (*stop) (double t, const double y[], double dydt[], void *params), double eps_abs, double eps_rel)
{
    LALAdaptiveRungeKuttaIntegrator *integrator;
 
    /* allocate our custom integrator structure */
    if (!(integrator = (LALAdaptiveRungeKuttaIntegrator *) LALCalloc(1, sizeof(LALAdaptiveRungeKuttaIntegrator)))) {
        XLAL_ERROR_NULL(XLAL_ENOMEM);
    }
 
    /* allocate the GSL ODE components */
        XLAL_CALLGSL(integrator->step = gsl_odeiv_step_alloc(gsl_odeiv_step_rkf45, dim));
    //XLAL_CALLGSL(integrator->step = gsl_odeiv_step_alloc(gsl_odeiv_step_rk8pd, dim));
    XLAL_CALLGSL(integrator->control = gsl_odeiv_control_y_new(eps_abs, eps_rel));
    XLAL_CALLGSL(integrator->evolve = gsl_odeiv_evolve_alloc(dim));
 
    /* allocate the GSL system (functions, etc.) */
    integrator->sys = (gsl_odeiv_system *) LALCalloc(1, sizeof(gsl_odeiv_system));
 
    /* if something failed to be allocated, bail out */
    if (!(integrator->step) || !(integrator->control) || !(integrator->evolve) || !(integrator->sys)) {
        XLALAdaptiveRungeKuttaFree(integrator);
        XLAL_ERROR_NULL(XLAL_ENOMEM);
    }
 
    integrator->dydt = dydt;
    integrator->stop = stop;
 
    integrator->sys->function = dydt;
    integrator->sys->jacobian = NULL;
    integrator->sys->dimension = dim;
    integrator->sys->params = NULL;
 
    integrator->retries = 6;
    integrator->stopontestonly = 0;
 
    return integrator;
}
 
LALAdaptiveRungeKuttaIntegrator *XLALAdaptiveRungeKutta4InitEighthOrderInstead(int dim, int (*dydt) (double t, const double y[], double dydt[], void *params),   /* These are XLAL functions! */
    int (*stop) (double t, const double y[], double dydt[], void *params), double eps_abs, double eps_rel)
{
    LALAdaptiveRungeKuttaIntegrator *integrator;
 
    /* allocate our custom integrator structure */
    if (!(integrator = (LALAdaptiveRungeKuttaIntegrator *) LALCalloc(1, sizeof(LALAdaptiveRungeKuttaIntegrator)))) {
        XLAL_ERROR_NULL(XLAL_ENOMEM);
    }
 
    /* allocate the GSL ODE components */
    XLAL_CALLGSL(integrator->step = gsl_odeiv_step_alloc(gsl_odeiv_step_rk8pd, dim));
    XLAL_CALLGSL(integrator->control = gsl_odeiv_control_y_new(eps_abs, eps_rel));
    XLAL_CALLGSL(integrator->evolve = gsl_odeiv_evolve_alloc(dim));
 
    /* allocate the GSL system (functions, etc.) */
    integrator->sys = (gsl_odeiv_system *) LALCalloc(1, sizeof(gsl_odeiv_system));
 
    /* if something failed to be allocated, bail out */
    if (!(integrator->step) || !(integrator->control) || !(integrator->evolve) || !(integrator->sys)) {
        XLALAdaptiveRungeKuttaFree(integrator);
        XLAL_ERROR_NULL(XLAL_ENOMEM);
    }
 
    integrator->dydt = dydt;
    integrator->stop = stop;
 
    integrator->sys->function = dydt;
    integrator->sys->jacobian = NULL;
    integrator->sys->dimension = dim;
    integrator->sys->params = NULL;
 
    integrator->retries = 6;
    integrator->stopontestonly = 0;
 
    return integrator;
}
 
void XLALAdaptiveRungeKuttaFree(LALAdaptiveRungeKuttaIntegrator * integrator)
{
    if (!integrator)
        return;
 
    if (integrator->evolve)
        XLAL_CALLGSL(gsl_odeiv_evolve_free(integrator->evolve));
    if (integrator->control)
        XLAL_CALLGSL(gsl_odeiv_control_free(integrator->control));
    if (integrator->step)
        XLAL_CALLGSL(gsl_odeiv_step_free(integrator->step));
 
    LALFree(integrator->sys);
    LALFree(integrator);
 
    return;
}
 
/* Local function to store interpolated step in output array */
static int storeStateInOutput(REAL8Array ** output, REAL8 t, REAL8 * y, size_t dim, int *outputlen, int count)
{
    REAL8Array *out = *output;
    int len = *outputlen;
    size_t i;
 
    if (count > len) {
        /* Resize array! Have to make a new array, and copy over. */
        REAL8Array *new = XLALCreateREAL8ArrayL(2, dim + 1, 2 * len);
 
        if (!new) {
            return XLAL_ENOMEM;
        }
 
        for (i = 0; i < dim + 1; i++) {
            memcpy(&(new->data[2 * i * len]), &(out->data[i * len]), len * sizeof(REAL8));
        }
 
        XLALDestroyREAL8Array(out);
        out = new;
        len *= 2;
    }
 
    /* Store the current step. */
    out->data[count - 1] = t;
    for (i = 1; i < dim + 1; i++) {
        out->data[i * len + count - 1] = y[i - 1];
    }
 
    *output = out;
    *outputlen = len;
    return GSL_SUCCESS;
}
 
/* Local function to shrink output array to proper size before it's returned */
static int shrinkOutput(REAL8Array ** output, int *outputlen, int count, size_t dim)
{
    REAL8Array *out = *output;
    int len = *outputlen;
 
    REAL8Array *new = XLALCreateREAL8ArrayL(2, dim + 1, count);
 
    if (!new) {
        return XLAL_ENOMEM;
    }
 
    size_t i;
    for (i = 0; i < dim + 1; i++) {
        memcpy(&(new->data[i * count]), &(out->data[i * len]), count * sizeof(REAL8));
    }
 
    *output = new;
    *outputlen = count;
 
    XLALDestroyREAL8Array(out);
 
    return GSL_SUCCESS;
}
 
/* Copied from GSL rkf45.c */
typedef struct {
    double *k1;
    double *k2;
    double *k3;
    double *k4;
    double *k5;
    double *k6;
    double *y0;
    double *ytmp;
} rkf45_state_t;
 
/**
 * Fourth-order Runge-Kutta ODE integrator using Runge-Kutta-Fehlberg (RKF45)
 * steps with adaptive step size control.  Intended for use in various
 * waveform generation routines such as SpinTaylorT4 and various EOB models.
 *
 * The method is described in
 *
 * Abramowitz & Stegun, Handbook of Mathematical Functions, Tenth Printing,
 * National Bureau of Standards, Washington, DC, 1972
 * (available online at http://people.math.sfu.ca/~cbm/aands/ )
 *
 * This function also includes "on-the-fly" interpolation of the
 * differential equations at regular intervals in-between integration
 * steps. This "on-the-fly" interpolation method is derived and
 * described in the Mathematica notebook "RKF_with_interpolation.nb";
 * see
 * https://www.lsc-group.phys.uwm.edu/ligovirgo/cbcnote/InspiralPipelineDevelopment/120312111836InspiralPipelineDevelopmentImproved%20Adaptive%20Runge-Kutta%20integrator
 *
 * This method is functionally equivalent to XLALAdaptiveRungeKutta4,
 * but is nearly always faster due to the improved interpolation.
 */
int XLALAdaptiveRungeKutta4Hermite(LALAdaptiveRungeKuttaIntegrator * integrator,       /**< struct holding dydt, stopping test, stepper, etc. */
    void *params,                                                       /**< params struct used to compute dydt and stopping test */
    REAL8 * yinit,                                                      /**< pass in initial values of all variables - overwritten to final values */
    REAL8 tinit,                                                        /**< integration start time */
    REAL8 tend_in,                                                      /**< maximum integration time */
    REAL8 deltat,                                                       /**< step size for evenly sampled output */
    REAL8Array ** yout                                                  /**< array holding the evenly sampled output */
    )
{
    int errnum = 0;
    int status;
    size_t dim, retries, i;
    int outputlen = 0, count = 0;
 
    REAL8Array *output = NULL;
 
    REAL8 t, tintp, h;
 
    REAL8 *ytemp = NULL;
 
    REAL8 tend = tend_in;
 
    XLAL_BEGINGSL;
 
    /* If want to stop only on test, then tend = +/-infinity; otherwise
     * tend_in */
    if (integrator->stopontestonly) {
        if (tend < tinit)
            tend = -1.0 / 0.0;
        else
            tend = 1.0 / 0.0;
    }
 
    dim = integrator->sys->dimension;
 
    outputlen = ((int)(tend_in - tinit) / deltat);
    //if (outputlen < 0) outputlen = -outputlen;
    if (outputlen < 0) {
        XLALPrintError
            ("XLAL Error - %s: (tend_in - tinit) and deltat must have the same sign\ntend_in: %f, tinit: %f, deltat: %f\n",
            __func__, tend_in, tinit, deltat);
        errnum = XLAL_EINVAL;
        goto bail_out;
    }
    outputlen += 2;
 
    output = XLALCreateREAL8ArrayL(2, (dim + 1), outputlen);
 
    if (!output) {
        errnum = XLAL_ENOMEM;
        goto bail_out;
    }
 
    ytemp = XLALCalloc(dim, sizeof(REAL8));
 
    if (!ytemp) {
        errnum = XLAL_ENOMEM;
        goto bail_out;
    }
 
    /* Setup. */
    integrator->sys->params = params;
    integrator->returncode = 0;
    retries = integrator->retries;
    t = tinit;
    tintp = tinit;
    h = deltat;
 
    /* Copy over first step. */
    output->data[0] = tinit;
    for (i = 1; i <= dim; i++)
        output->data[i * outputlen] = yinit[i - 1];
    count = 1;
 
    /* We are starting a fresh integration; clear GSL step and evolve
     * objects. */
    gsl_odeiv_step_reset(integrator->step);
    gsl_odeiv_evolve_reset(integrator->evolve);
 
    /* Enter evolution loop.  NOTE: we *always* take at least one
     * step. */
    while (1) {
        REAL8 told = t;
 
        status =
            gsl_odeiv_evolve_apply(integrator->evolve, integrator->control, integrator->step, integrator->sys, &t, tend, &h,
            yinit);
 
        /* Check for failure, retry if haven't retried too many times
         * already. */
        if (status != GSL_SUCCESS) {
            if (retries--) {
                /* Retries to spare; reduce h, try again. */
                h /= 10.0;
                continue;
            } else {
                /* Out of retries, bail with status code. */
                integrator->returncode = status;
                break;
            }
        } else {
            /* Successful step, reset retry counter. */
            retries = integrator->retries;
        }
 
        /* Now interpolate until we would go past the current integrator time, t.
         * Note we square to get an absolute value, because we may be
         * integrating t in the positive or negative direction */
        while ((tintp + deltat) * (tintp + deltat) < t * t) {
            tintp += deltat;
 
            /* tintp = told + (t-told)*theta, 0 <= theta <= 1.  We have to
             * compute h = (t-told) because the integrator returns a
             * suggested next h, not the actual stepsize taken. */
            REAL8 hUsed = t - told;
            REAL8 theta = (tintp - told) / hUsed;
 
            /* These are the interpolating coefficients for y(t + h*theta) =
             * ynew + i1*h*k1 + i5*h*k5 + i6*h*k6 + O(h^4). */
            REAL8 i0 = 1.0 + theta * theta * (3.0 - 4.0 * theta);
            REAL8 i1 = -theta * (theta - 1.0);
            REAL8 i6 = -4.0 * theta * theta * (theta - 1.0);
            REAL8 iend = theta * theta * (4.0 * theta - 3.0);
 
            /* Grab the k's from the integrator state. */
            rkf45_state_t *rkfState = integrator->step->state;
            REAL8 *k1 = rkfState->k1;
            REAL8 *k6 = rkfState->k6;
            REAL8 *y0 = rkfState->y0;
 
            for (i = 0; i < dim; i++) {
                ytemp[i] = i0 * y0[i] + iend * yinit[i] + hUsed * i1 * k1[i] + hUsed * i6 * k6[i];
            }
 
            /* Store the interpolated value in the output array. */
            count++;
            if ((status = storeStateInOutput(&output, tintp, ytemp, dim, &outputlen, count)) == XLAL_ENOMEM) {
                errnum = XLAL_ENOMEM;
                goto bail_out;
            }
        }
 
        /* Now that we have recorded the last interpolated step that we
         * could, check for termination criteria. */
        if (!integrator->stopontestonly && t >= tend)
            break;
 
        /* If there is a stopping function in integrator, call it with the
         * last value of y and dydt from the integrator. */
        if (integrator->stop) {
            if ((status = integrator->stop(t, yinit, integrator->evolve->dydt_out, params)) != GSL_SUCCESS) {
                integrator->returncode = status;
                break;
            }
        }
    }
 
    /* Now that the interpolation is done, shrink the output array down
     * to exactly count samples. */
    shrinkOutput(&output, &outputlen, count, dim);
 
    /* Store the final *interpolated* sample in yinit. */
    for (i = 0; i < dim; i++) {
        yinit[i] = output->data[(i + 2) * outputlen - 1];
    }
 
  bail_out:
 
    XLAL_ENDGSL;
 
    /* If we have an error, then we should free allocated memory, and
     * then return. */
    XLALFree(ytemp);
 
    if (errnum) {
        if (output)
            XLALDestroyREAL8Array(output);
        *yout = NULL;
        XLAL_ERROR(errnum);
    }
 
    *yout = output;
    return outputlen;
}
 
/**
 * Fourth-order Runge-Kutta ODE integrator using Runge-Kutta-Fehlberg (RKF45)
 * steps with adaptive step size control.  Version that only outputs the final
 * state of the integration, intended for use for spin evolution in
 * lalsimulation/lib/LALSimInspiralSpinTaylor.c, based on
 * XLALAdaptiveRungeKutta4Hermite. Specifically, this assumes that the criterion
 * used to set the end of the evolution (and included in the integrator's stopping
 * condition) is given by deltat*y[1] < deltat*y1_final, where y is the vector of
 * variables.
 *
 * The method is described in
 *
 * Abramowitz & Stegun, Handbook of Mathematical Functions, Tenth Printing,
 * National Bureau of Standards, Washington, DC, 1972
 * (available online at http://people.math.sfu.ca/~cbm/aands/ )
 *
 * This function also includes "on-the-fly" interpolation of the
 * differential equations at regular intervals in-between integration
 * steps. This "on-the-fly" interpolation method is derived and
 * described in the Mathematica notebook "RKF_with_interpolation.nb";
 * see
 * https://www.lsc-group.phys.uwm.edu/ligovirgo/cbcnote/InspiralPipelineDevelopment/120312111836InspiralPipelineDevelopmentImproved%20Adaptive%20Runge-Kutta%20integrator
 */
int XLALAdaptiveRungeKutta4HermiteOnlyFinal(LALAdaptiveRungeKuttaIntegrator * integrator,       /**< struct holding dydt, stopping test, stepper, etc. */
    void *params,                                                       /**< params struct used to compute dydt and stopping test */
    REAL8 * yinit,                                                      /**< pass in initial values of all variables - overwritten to final values */
    REAL8 tinit,                                                        /**< integration start time */
    REAL8 tend_in,                                                      /**< maximum integration time */
    REAL8 y1_final,                                                     /**< final value of y[1] */
    REAL8 deltat                                                        /**< step size for integration */
    )
{
    int errnum = 0;
    int status;
    size_t dim, retries, i;
 
    REAL8 t, tintp, h;
 
    REAL8 *ytemp = NULL;
 
    REAL8 tend = tend_in;
 
    XLAL_BEGINGSL;
 
    /* If want to stop only on test, then tend = +/-infinity; otherwise
     * tend_in */
    if (integrator->stopontestonly) {
        if (tend < tinit)
            tend = -1.0 / 0.0;
        else
            tend = 1.0 / 0.0;
    }
 
    dim = integrator->sys->dimension;
 
    if ((tend < tinit && deltat > 0) || (tend > tinit && deltat < 0)) {
        XLALPrintError
            ("XLAL Error - %s: (tend_in - tinit) and deltat must have the same sign\ntend_in: %f, tinit: %f, deltat: %f\n",
            __func__, tend_in, tinit, deltat);
        errnum = XLAL_EINVAL;
        goto bail_out;
    }
 
    ytemp = XLALCalloc(dim, sizeof(REAL8));
 
    if (!ytemp) {
        errnum = XLAL_ENOMEM;
        goto bail_out;
    }
 
    /* Initialize ytemp[1] with the initial value of yinit[1] so that the initial check below is satisfied even if we are integrating backwards */
    ytemp[1] = yinit[1];
 
    /* Setup. */
    integrator->sys->params = params;
    integrator->returncode = 0;
    retries = integrator->retries;
    t = tinit;
    tintp = tinit;
    h = deltat;
 
    /* We are starting a fresh integration; clear GSL step and evolve
     * objects. */
    gsl_odeiv_step_reset(integrator->step);
    gsl_odeiv_evolve_reset(integrator->evolve);
 
    /* Enter evolution loop.  NOTE: we *always* take at least one
     * step. */
    while (1) {
        REAL8 told = t;
 
        status =
            gsl_odeiv_evolve_apply(integrator->evolve, integrator->control, integrator->step, integrator->sys, &t, tend, &h,
            yinit);
 
        /* Check for failure, retry if haven't retried too many times
         * already. */
        if (status != GSL_SUCCESS) {
            if (retries--) {
                /* Retries to spare; reduce h, try again. */
                h /= 10.0;
                continue;
            } else {
                /* Out of retries, bail with status code. */
                integrator->returncode = status;
                break;
            }
        } else {
            /* Successful step, reset retry counter. */
            retries = integrator->retries;
        }
 
        /* If we're at the final timestep, interpolate to get the output at the desired final value, y1_final.
         * Note we square to get an absolute value, because we may be
         * integrating t in the positive or negative direction
         * We multiply yinit, ytemp[1], and y1_final by deltat so that this check works when integrating forward or backward */
        if (deltat*yinit[1] >= deltat*y1_final) {
                tintp = told;
 
                REAL8 hUsed = t - told;
 
                while (deltat*ytemp[1] < deltat*y1_final) {
                  tintp += deltat;
 
                  /* tintp = told + (t-told)*theta, 0 <= theta <= 1.  We have to
                   * compute h = (t-told) because the integrator returns a
                   * suggested next h, not the actual stepsize taken. */
                  REAL8 theta = (tintp - told) / hUsed;
 
                  /* These are the interpolating coefficients for y(t + h*theta) =
                   * ynew + i1*h*k1 + i5*h*k5 + i6*h*k6 + O(h^4). */
                  REAL8 i0 = 1.0 + theta * theta * (3.0 - 4.0 * theta);
                  REAL8 i1 = -theta * (theta - 1.0);
                  REAL8 i6 = -4.0 * theta * theta * (theta - 1.0);
                  REAL8 iend = theta * theta * (4.0 * theta - 3.0);
 
                  /* Grab the k's from the integrator state. */
                  rkf45_state_t *rkfState = integrator->step->state;
                  REAL8 *k1 = rkfState->k1;
                  REAL8 *k6 = rkfState->k6;
                  REAL8 *y0 = rkfState->y0;
 
                  for (i = 0; i < dim; i++) {
                    ytemp[i] = i0 * y0[i] + iend * yinit[i] + hUsed * i1 * k1[i] + hUsed * i6 * k6[i];
                  }
                }
        }
 
        /* Now check for termination criteria. */
        if (!integrator->stopontestonly && t >= tend)
            break;
 
        /* If there is a stopping function in integrator, call it with the
         * last value of y and dydt from the integrator. */
        if (integrator->stop) {
            if ((status = integrator->stop(t, yinit, integrator->evolve->dydt_out, params)) != GSL_SUCCESS) {
                integrator->returncode = status;
                break;
            }
        }
    }
 
    /* Store the final *interpolated* sample before the stopping criterion in yinit. */
    for (i = 0; i < dim; i++) {
      yinit[i] = ytemp[i];
    }
 
 
  bail_out:
 
    XLAL_ENDGSL;
 
    /* If we have an error, then we should free allocated memory, and
     * then return. */
    XLALFree(ytemp);
 
    if (errnum) {
        XLAL_ERROR(errnum);
    }
 
    return 1;
}
 
int XLALAdaptiveRungeKutta4NoInterpolate(LALAdaptiveRungeKuttaIntegrator * integrator,
         void * params, REAL8 * yinit, REAL8 tinit, REAL8 tend, REAL8 deltat_or_h0, REAL8 min_deltat_or_h0,
                                         REAL8Array ** t_and_y_out, INT4 EOBversion)
{
 
    int errnum = 0;
    int status; /* used throughout */
 
    /* needed for the integration */
    size_t dim, outputlength=0, bufferlength, retries;
    REAL8 t, tnew, h0, h0old;
    REAL8Array *buffers = NULL;
    REAL8 *temp = NULL, *y, *y0, *dydt_in, *dydt_in0, *dydt_out, *yerr; /* aliases */
 
    /* note: for speed, this replaces the single CALLGSL wrapper applied before each GSL call */
    XLAL_BEGINGSL;
 
    /* allocate the buffers!
     * note: REAL8Array has a field dimLength (UINT4Vector) with dimensions, and a field data that points to a single memory block;
     * dimLength itself has fields length and data */
    dim = integrator->sys->dimension;
    bufferlength = (int)((tend - tinit) / deltat_or_h0) + 2;   /* allow for the initial value and possibly a final semi-step */
 
    UINT4 dimn;/* Variable for different loop indices below */
    if(EOBversion==2) dimn = dim + 1;
    else dimn = dim + 4;//v3opt: Include three derivatives
 
    buffers = XLALCreateREAL8ArrayL(2, dimn/*dim + 1*/, bufferlength); /* 2-dimensional array, ((dim+1)) x bufferlength */
 
    temp = LALCalloc(6 * dim, sizeof(REAL8));
 
    if (!buffers || !temp) {
        errnum = XLAL_ENOMEM;
        goto bail_out;
    }
 
    y = temp;
    y0 = temp + dim;
    dydt_in = temp + 2 * dim;
    dydt_in0 = temp + 3 * dim;
    dydt_out = temp + 4 * dim;
    yerr = temp + 5 * dim;      /* aliases */
 
    /* set up to get started */
    integrator->sys->params = params;
 
    integrator->returncode = 0;
 
    retries = integrator->retries;
 
    t = tinit;
    h0 = deltat_or_h0;
    h0old = h0; /* initialized so that it will not trigger the check h0<h0old at the first step */
    memcpy(y, yinit, dim * sizeof(REAL8));
 
    /* store the first data point */
    buffers->data[0] = t;
    for (unsigned int i = 1; i <= dim; i++)
        buffers->data[i * bufferlength] = y[i - 1];
 
    /* compute derivatives at the initial time (dydt_in), bail out if impossible */
    if ((status = integrator->dydt(t, y, dydt_in, params)) != GSL_SUCCESS) {
        integrator->returncode = status;
        errnum = XLAL_EFAILED;
        goto bail_out;
    }
 
    if(EOBversion==3){
      for (unsigned int i = 1; i <= 3; i++) //OPTV3: include the initial derivatives
        buffers->data[(dim+i)*bufferlength] = dydt_in[i-1];
    }
 
    UINT4 loop;/*variable for different loop indices below. */
    if(EOBversion==2) loop = dim;
    else loop = dim + 3;
 
    while (1) {
 
        if (!integrator->stopontestonly && t >= tend) {
            break;
        }
 
        if (integrator->stop) {
            if ((status = integrator->stop(t, y, dydt_in, params)) != GSL_SUCCESS) {
                integrator->returncode = status;
                break;
            }
        }
 
        /* ready to try stepping! */
      try_step:
 
        /* if we would be stepping beyond the final time, stop there instead... */
        if (!integrator->stopontestonly && t + h0 > tend)
            h0 = tend - t;
 
        memcpy(y0, y, dim * sizeof(REAL8));     /* save y to y0, dydt_in to dydt_in0 */
        memcpy(dydt_in0, dydt_in, dim * sizeof(REAL8));
 
        /* call the GSL stepper function */
        status = gsl_odeiv_step_apply(integrator->step, t, h0, y, yerr, dydt_in, dydt_out, integrator->sys);
        /* note: If the user-supplied functions defined in the system dydt return a status other than GSL_SUCCESS,
         * the step will be aborted. In this case, the elements of y will be restored to their pre-step values,
         * and the error code from the user-supplied function will be returned. */
 
        /* did the stepper report a derivative-evaluation error? */
        if (status != GSL_SUCCESS) {
            if (retries--) {
                h0 = h0 / 10.0; /* if we have singularity retries left, reduce the timestep and try again */
                goto try_step;
            } else {
                integrator->returncode = status;
                break;  /* otherwise exit the loop */
            }
        } else {
            retries = integrator->retries;      /* we stepped successfully, reset the singularity retries */
        }
 
        tnew = t + h0;
 
        /* call the GSL error-checking function */
        status = gsl_odeiv_control_hadjust(integrator->control, integrator->step, y, yerr, dydt_out, &h0);
 
        /* Enforce a minimal allowed time step */
        /* To ignore this type of constraint, set min_deltat_or_h0 = 0 */
        if (h0 < min_deltat_or_h0) h0 = min_deltat_or_h0;
 
        /* did the error-checker reduce the stepsize?
         * note: previously, was using status == GSL_ODEIV_HADJ_DEC
         * other possible return codes are GSL_ODEIV_HADJ_INC if it was increased,
         * GSL_ODEIV_HADJ_NIL if it was unchanged
         * since we introduced a minimal step size we simply compare to the saved value of h0 */
        /* if (status == GSL_ODEIV_HADJ_DEC) { */
        if (h0 < h0old) {
            h0old = h0;
            memcpy(y, y0, dim * sizeof(REAL8)); /* if so, undo the step, and try again */
            memcpy(dydt_in, dydt_in0, dim * sizeof(REAL8));
            goto try_step;
        }
 
        /* update the current time and input derivatives, save the time step */
        t = tnew;
        h0old = h0;
        memcpy(dydt_in, dydt_out, dim * sizeof(REAL8));
        outputlength++;
 
        /* check if interpolation buffers need to be extended */
        if (outputlength >= bufferlength) {
            REAL8Array *rebuffers;
 
            /* sadly, we cannot use ResizeREAL8Array, because it would only work if we extended the first array dimension,
             * so we need to copy everything over and switch the buffers. Oh well. */
            if (!(rebuffers = XLALCreateREAL8ArrayL(2, dimn, 2 * bufferlength))) {
                errnum = XLAL_ENOMEM;   /* ouch, that hurt */
                goto bail_out;
            } else {
              for (unsigned int i = 0; i <= loop /* dim */; i++)
                    memcpy(&rebuffers->data[i * 2 * bufferlength], &buffers->data[i * bufferlength], outputlength * sizeof(REAL8));
                XLALDestroyREAL8Array(buffers);
                buffers = rebuffers;
                bufferlength *= 2;
            }
        }
 
        /* copy time and state into output buffers */
        buffers->data[outputlength] = t;
        for (unsigned int i = 1; i <= loop; i++)
            buffers->data[i * bufferlength + outputlength] = y[i - 1];   /* y does not have time */
        if(EOBversion==3){
          for (unsigned int i = 1; i <= 3; i++)
            buffers->data[(dim+i) * bufferlength + outputlength] = dydt_out[i - 1];  //OPTV3: Include 3 derivatives
        }
    }
 
    /* copy the final state into yinit */
 
    memcpy(yinit, y, dim * sizeof(REAL8));
 
    /* if we have completed at least one step, allocate the GSL interpolation object and the output array */
    if (outputlength == 0)
        goto bail_out;
 
    if(EOBversion==3){
      outputlength++;
      (*t_and_y_out) = XLALCreateREAL8ArrayL(2,(dim + 4),outputlength);//OPTV3: Include derivatives
    }else{
      (*t_and_y_out) = XLALCreateREAL8ArrayL(2,(dim + 3),outputlength); //OPTV3: Include derivatives
    }
 
      if (!(*t_and_y_out)) {
        errnum = XLAL_ENOMEM;   /* ouch again, ran out of memory */
        if (*t_and_y_out)
          XLALDestroyREAL8Array(*t_and_y_out);
        outputlength = 0;
        goto bail_out;
    }
 
    for(UINT8 j=0;j<outputlength;j++) {
      (*t_and_y_out)->data[j] = buffers->data[j];
      for(UINT8 i=1;i<=loop;i++) {
        (*t_and_y_out)->data[i*outputlength + j] = buffers->data[i*bufferlength + j];
      }
    }
    /* deallocate stuff and return */
  bail_out:
 
    XLAL_ENDGSL;
 
    if (buffers)
        XLALDestroyREAL8Array(buffers); /* let's be careful, although all these checks may not be needed */
    if (temp)
        LALFree(temp);
 
    if (errnum)
        XLAL_ERROR(errnum);
 
    return outputlength;
}
 
int XLALAdaptiveRungeKuttaDenseandSparseOutput(LALAdaptiveRungeKuttaIntegrator * integrator,
         void * params, REAL8 * yinit, REAL8 tinit, REAL8 tend, REAL8 deltat,
         REAL8Array ** sparse_output, REAL8Array ** dense_output)
{
    /* Error-checking variables used throughout */
    int errnum = 0;
    int status;
 
    /* Integration and interpolation variables */
    size_t dim = integrator->sys->dimension;
    UINT4 sparse_outputlength = 0;
    UINT4 dense_outputlength = 1;
    size_t sparse_bufferlength, dense_bufferlength, retries;
    REAL8 t = tinit;
    REAL8 tnew;
    REAL8 h0 = deltat;
    REAL8Array *sparse_buffers = NULL;
    REAL8Array *dense_buffers = NULL;
    REAL8 *temp = NULL, *y, *y0, *dydt_in, *dydt_in0, *dydt_out, *yerr;
    REAL8 interp_t = tinit + h0;
 
    /* For speed, this replaces the single CALLGSL wrapper applied before each GSL call */
    XLAL_BEGINGSL;
 
    /* Allocate buffers!
     * Note: REAL8Array has a field dimLength (UINT4Vector) with dimensions, and a field data that points to a single memory block;
     * dimLength itself has fields length and data */
    sparse_bufferlength = (int)((tend - tinit) / h0) + 2;   /* allow for the initial value and possibly a final semi-step */
    dense_bufferlength = (int)((tend - tinit) / h0) + 2;
 
    const UINT4 dimn = dim + 1;/* Time is not included in input dimesions, but is included in ouput arrays. */
 
    sparse_buffers = XLALCreateREAL8ArrayL(2, dimn, sparse_bufferlength);
    dense_buffers = XLALCreateREAL8ArrayL(2, dimn, dense_bufferlength);
    temp = LALCalloc(6 * dim, sizeof(REAL8));
 
    if (!sparse_buffers || !dense_buffers || !temp) {
        errnum = XLAL_ENOMEM;
        goto bail_out;
    }
 
    /* Aliases */
    y = temp;
    y0 = temp + dim;
    dydt_in = temp + 2 * dim;
    dydt_in0 = temp + 3 * dim;
    dydt_out = temp + 4 * dim;
    yerr = temp + 5 * dim;
 
    /* Integrator set up */
    integrator->sys->params = params;
    integrator->returncode = 0;
    retries = integrator->retries;
 
    memcpy(y, yinit, dim * sizeof(REAL8));
 
    /* Store the first data point. */
    sparse_buffers->data[0] = t;
    dense_buffers->data[0] = t;
    for (UINT4 i = 1; i <= dim; i++){
      UINT4 iminus1 = i - 1;
      sparse_buffers->data[i * sparse_bufferlength] = y[iminus1];
      dense_buffers->data[i * dense_bufferlength] = y[iminus1];
    }
 
    /* Compute derivatives at the initial time (dydt_in); bail out if impossible. */
    if ((status = integrator->dydt(t, y, dydt_in, params)) != GSL_SUCCESS) {
        integrator->returncode = status;
        errnum = XLAL_EFAILED;
        goto bail_out;
    }
 
    while (1) {
 
        if (!integrator->stopontestonly && t >= tend) {
            break;
        }
 
        if (integrator->stop) {
            if ((status = integrator->stop(t, y, dydt_in, params)) != GSL_SUCCESS) {
              integrator->returncode = status;
                break;
            }
        }
 
        /* Try stepping! */
      try_step:
 
        /* If we would be stepping beyond the final time, stop there instead. */
        if (!integrator->stopontestonly && t + h0 > tend)
            h0 = tend - t;
 
        /* Save y to y0 and dydt_in to dydt_in0. */
        memcpy(y0, y, dim * sizeof(REAL8));
        memcpy(dydt_in0, dydt_in, dim * sizeof(REAL8));
 
        /* Call the GSL stepper function. */
        status = gsl_odeiv_step_apply(integrator->step, t, h0, y, yerr, dydt_in, dydt_out, integrator->sys);
        /* Note: If the user-supplied functions defined in the system dydt return a status other than GSL_SUCCESS,
         * the step will be aborted. In this case, the elements of y will be restored to their pre-step values,
         * and the error code from the user-supplied function will be returned. */
 
        /* Did the stepper report a derivative-evaluation error? */
        if (status != GSL_SUCCESS) {
          if (retries--) {
            /* If we have singularity retries left, reduce the timestep and try again... */
            h0 = h0 / 10.0;
            goto try_step;
          } else {
            integrator->returncode = status;
            /* ...otherwise exit the loop. */
            break;
          }
        } else {
          /* We stepped successfully; reset the singularity retries. */
          retries = integrator->retries;
        }
 
        tnew = t + h0;
 
        /* Call the GSL error-checking function. */
        status = gsl_odeiv_control_hadjust(integrator->control, integrator->step, y, yerr, dydt_out, &h0);
 
        /* Did the error-checker reduce the stepsize?
         * Note: other possible return codes are GSL_ODEIV_HADJ_INC if it was increased;
         * GSL_ODEIV_HADJ_NIL if it was unchanged. */
        if (status == GSL_ODEIV_HADJ_DEC) {
          /* If so, undo the step, and try again */
          memcpy(y, y0, dim * sizeof(REAL8));
          memcpy(dydt_in, dydt_in0, dim * sizeof(REAL8));
          goto try_step;
        }
 
        if (2*dense_outputlength >= dense_bufferlength) {
          REAL8Array *rebuffers;
 
          /* Sadly, we cannot use ResizeREAL8Array because it would only work if we extended the first array dimension;
           * thus we copy everything and switch the buffers. */
          if (!(rebuffers = XLALCreateREAL8ArrayL(2, dimn, 2 * dense_bufferlength))) {
            errnum = XLAL_ENOMEM;
            goto bail_out;
          } else {
            for (UINT4 k = 0; k <= dim; k++)
              memcpy(&rebuffers->data[k * 2 * dense_bufferlength], &dense_buffers->data[k * dense_bufferlength], dense_outputlength * sizeof(REAL8));
            XLALDestroyREAL8Array(dense_buffers);
            dense_buffers = rebuffers;
            dense_bufferlength *= 2;
          }
        }
 
        {
          const REAL8 interp_t_old = interp_t;
          const UINT4 dense_outputlength_old = dense_outputlength;
          while (interp_t < tnew) {
            dense_buffers->data[dense_outputlength] = interp_t;
            dense_outputlength++;
            interp_t = tinit + dense_outputlength*deltat;
          }
          interp_t = interp_t_old;
          dense_outputlength = dense_outputlength_old;
        }
 
        {
          const REAL8 h = tnew - t;
          const REAL8 h_inv = 1.0/h;
 
          for (UINT4 i = 0; i < dim; i++) {
            REAL8 interp_t_old = interp_t;
            UINT4 dense_outputlength_old = dense_outputlength;
            REAL8 y0i = y0[i];
            REAL8 yi = y[i];
 
            while (interp_t < tnew) {
              const REAL8 theta = (interp_t - t)*h_inv;
              dense_buffers->data[(i+1)*dense_bufferlength + dense_outputlength] =
                (1.0 - theta)*y0i + theta*yi + theta*(theta-1.0)*( (1.0 - 2.0*theta)*(yi - y0i) + h*( (theta-1.0)*dydt_in[i] + theta*dydt_out[i]));
              dense_outputlength++;
              interp_t = tinit + dense_outputlength*deltat;
            }
 
            if(i<(dim-1)){
              interp_t = interp_t_old;
              dense_outputlength = dense_outputlength_old;
            }
          }
        }
 
        /* Update the current time and input derivatives. */
        t = tnew;
        memcpy(dydt_in, dydt_out, dim * sizeof(REAL8));
        sparse_outputlength++;
 
        /* Check if interpolation buffers need to be extended. */
        if (sparse_outputlength >= sparse_bufferlength) {
          REAL8Array *rebuffers;
 
          /* Sadly, we cannot use ResizeREAL8Array because it would only work if we extended the first array dimension;
           * thus we copy everything and switch the buffers. */
          if (!(rebuffers = XLALCreateREAL8ArrayL(2, dimn, 2 * sparse_bufferlength))) {
            errnum = XLAL_ENOMEM;
            goto bail_out;
          } else {
            for (UINT4 i = 0; i <= dim; i++)
              memcpy(&rebuffers->data[i * 2 * sparse_bufferlength], &sparse_buffers->data[i * sparse_bufferlength], sparse_outputlength * sizeof(REAL8));
            XLALDestroyREAL8Array(sparse_buffers);
            sparse_buffers = rebuffers;
            sparse_bufferlength *= 2;
          }
        }
 
        /* Copy time and state into buffers. */
        sparse_buffers->data[sparse_outputlength] = t;
        for (UINT4 i = 1; i <= dim; i++)
            sparse_buffers->data[i * sparse_bufferlength + sparse_outputlength] = y[i - 1];
    }
 
    if (sparse_outputlength == 0 || dense_outputlength == 1)
        goto bail_out;
 
    sparse_outputlength++;
 
    (*sparse_output) = XLALCreateREAL8ArrayL(2, dim+1, sparse_outputlength);
    (*dense_output) = XLALCreateREAL8ArrayL(2, dim+1, dense_outputlength);
 
    if (!(*sparse_output) || !(*dense_output)) {
      errnum = XLAL_ENOMEM;   /* ouch again, ran out of memory */
      if (*sparse_output){
        XLALDestroyREAL8Array(*sparse_output);
        sparse_outputlength = 0;
      }
      if (*dense_output){
        XLALDestroyREAL8Array(*dense_output);
        dense_outputlength = 0;
      }
      goto bail_out;
    }
 
    for(UINT4 j = 0; j < sparse_outputlength; j++) {
      (*sparse_output)->data[j] = sparse_buffers->data[j];
      for(UINT4 i = 1; i <= dim; i++) {
        (*sparse_output)->data[i * sparse_outputlength + j] = sparse_buffers->data[i * sparse_bufferlength + j];
      }
    }
 
    for(UINT4 j = 0; j < dense_outputlength; j++) {
      (*dense_output)->data[j] = dense_buffers->data[j];
      for(UINT4 i = 1; i <= dim; i++) {
        (*dense_output)->data[i * dense_outputlength + j] = dense_buffers->data[i * dense_bufferlength + j];
      }
    }
 
    /* Deallocate memory and return sparse_outputlength. */
  bail_out:
 
    XLAL_ENDGSL;
 
    if (sparse_buffers)
      XLALDestroyREAL8Array(sparse_buffers);
    if (dense_buffers)
      XLALDestroyREAL8Array(dense_buffers);
    if (temp)
      LALFree(temp);
    if (errnum)
      XLAL_ERROR(errnum);
 
    return sparse_outputlength;
}
 
int XLALAdaptiveRungeKutta4(LALAdaptiveRungeKuttaIntegrator * integrator,
    void *params, REAL8 * yinit, REAL8 tinit, REAL8 tend, REAL8 deltat, REAL8Array ** yout)
{
    int errnum = 0;
    int status; /* used throughout */
 
    /* needed for the integration */
    size_t dim, bufferlength, cnt, retries;
    REAL8 t, tnew, h0;
    REAL8Array *buffers = NULL;
    REAL8 *temp = NULL, *y, *y0, *dydt_in, *dydt_in0, *dydt_out, *yerr; /* aliases */
 
    /* needed for the final interpolation */
    gsl_spline *interp = NULL;
    gsl_interp_accel *accel = NULL;
    int outputlen = 0;
    REAL8Array *output = NULL;
    REAL8 *times, *vector;      /* aliases */
 
    /* note: for speed, this replaces the single CALLGSL wrapper applied before each GSL call */
    XLAL_BEGINGSL;
 
    /* allocate the buffers!
     * note: REAL8Array has a field dimLength (UINT4Vector) with dimensions, and a field data that points to a single memory block;
     * dimLength itself has fields length and data */
    dim = integrator->sys->dimension;
    bufferlength = (int)((tend - tinit) / deltat) + 2;  /* allow for the initial value and possibly a final semi-step */
    buffers = XLALCreateREAL8ArrayL(2, dim + 1, bufferlength);  /* 2-dimensional array, (dim+1) x bufferlength */
    temp = LALCalloc(6 * dim, sizeof(REAL8));
 
    if (!buffers || !temp) {
        errnum = XLAL_ENOMEM;
        goto bail_out;
    }
 
    y = temp;
    y0 = temp + dim;
    dydt_in = temp + 2 * dim;
    dydt_in0 = temp + 3 * dim;
    dydt_out = temp + 4 * dim;
    yerr = temp + 5 * dim;      /* aliases */
 
    /* set up to get started */
    integrator->sys->params = params;
 
    integrator->returncode = 0;
 
    cnt = 0;
    retries = integrator->retries;
 
    t = tinit;
    h0 = deltat;
    memcpy(y, yinit, dim * sizeof(REAL8));
 
    /* store the first data point */
    buffers->data[0] = t;
    for (unsigned int i = 1; i <= dim; i++)
        buffers->data[i * bufferlength] = y[i - 1];
 
    /* compute derivatives at the initial time (dydt_in), bail out if impossible */
    if ((status = integrator->dydt(t, y, dydt_in, params)) != GSL_SUCCESS) {
        integrator->returncode = status;
        errnum = XLAL_EFAILED;
        goto bail_out;
    }
 
    while (1) {
 
        if (!integrator->stopontestonly && t >= tend) {
            break;
        }
 
        if (integrator->stop) {
            if ((status = integrator->stop(t, y, dydt_in, params)) != GSL_SUCCESS) {
                integrator->returncode = status;
                break;
            }
        }
 
        /* ready to try stepping! */
      try_step:
 
        /* if we would be stepping beyond the final time, stop there instead... */
        if (!integrator->stopontestonly && t + h0 > tend)
            h0 = tend - t;
 
        memcpy(y0, y, dim * sizeof(REAL8));     /* save y to y0, dydt_in to dydt_in0 */
        memcpy(dydt_in0, dydt_in, dim * sizeof(REAL8));
 
        /* call the GSL stepper function */
        status = gsl_odeiv_step_apply(integrator->step, t, h0, y, yerr, dydt_in, dydt_out, integrator->sys);
        /* note: If the user-supplied functions defined in the system dydt return a status other than GSL_SUCCESS,
         * the step will be aborted. In this case, the elements of y will be restored to their pre-step values,
         * and the error code from the user-supplied function will be returned. */
 
        /* did the stepper report a derivative-evaluation error? */
        if (status != GSL_SUCCESS) {
            if (retries--) {
                h0 = h0 / 10.0; /* if we have singularity retries left, reduce the timestep and try again */
                goto try_step;
            } else {
                integrator->returncode = status;
                break;  /* otherwise exit the loop */
            }
        } else {
            retries = integrator->retries;      /* we stepped successfully, reset the singularity retries */
        }
 
        tnew = t + h0;
 
        /* call the GSL error-checking function */
        status = gsl_odeiv_control_hadjust(integrator->control, integrator->step, y, yerr, dydt_out, &h0);
 
        /* did the error-checker reduce the stepsize?
         * note: other possible return codes are GSL_ODEIV_HADJ_INC if it was increased,
         * GSL_ODEIV_HADJ_NIL if it was unchanged */
        if (status == GSL_ODEIV_HADJ_DEC) {
            memcpy(y, y0, dim * sizeof(REAL8)); /* if so, undo the step, and try again */
            memcpy(dydt_in, dydt_in0, dim * sizeof(REAL8));
            goto try_step;
        }
 
        /* update the current time and input derivatives */
        t = tnew;
        memcpy(dydt_in, dydt_out, dim * sizeof(REAL8));
        cnt++;
 
        /* check if interpolation buffers need to be extended */
        if (cnt >= bufferlength) {
            REAL8Array *rebuffers;
 
            /* sadly, we cannot use ResizeREAL8Array, because it would only work if we extended the first array dimension,
             * so we need to copy everything over and switch the buffers. Oh well. */
            if (!(rebuffers = XLALCreateREAL8ArrayL(2, dim + 1, 2 * bufferlength))) {
                errnum = XLAL_ENOMEM;   /* ouch, that hurt */
                goto bail_out;
            } else {
                for (unsigned int i = 0; i <= dim; i++)
                    memcpy(&rebuffers->data[i * 2 * bufferlength], &buffers->data[i * bufferlength], cnt * sizeof(REAL8));
                XLALDestroyREAL8Array(buffers);
                buffers = rebuffers;
                bufferlength *= 2;
            }
        }
 
        /* copy time and state into interpolation buffers */
        buffers->data[cnt] = t;
        for (unsigned int i = 1; i <= dim; i++)
            buffers->data[i * bufferlength + cnt] = y[i - 1];   /* y does not have time */
    }
 
    /* copy the final state into yinit */
 
    memcpy(yinit, y, dim * sizeof(REAL8));
 
    /* if we have completed at least one step, allocate the GSL interpolation object and the output array */
    if (cnt == 0)
        goto bail_out;
 
    interp = gsl_spline_alloc(gsl_interp_cspline, cnt + 1);
    accel = gsl_interp_accel_alloc();
 
    outputlen = (int)(t / deltat) + 1;
    output = XLALCreateREAL8ArrayL(2, dim + 1, outputlen);
 
    if (!interp || !accel || !output) {
        errnum = XLAL_ENOMEM;   /* ouch again, ran out of memory */
        if (output)
            XLALDestroyREAL8Array(output);
        outputlen = 0;
        goto bail_out;
    }
 
    /* make an array of times */
    times = output->data;
    for (int j = 0; j < outputlen; j++)
        times[j] = tinit + deltat * j;
 
    /* interpolate! */
    for (unsigned int i = 1; i <= dim; i++) {
        gsl_spline_init(interp, &buffers->data[0], &buffers->data[bufferlength * i], cnt + 1);
 
        vector = output->data + outputlen * i;
        for (int j = 0; j < outputlen; j++) {
            gsl_spline_eval_e(interp, times[j], accel, &(vector[j]));
        }
    }
 
    /* deallocate stuff and return */
  bail_out:
 
    XLAL_ENDGSL;
 
    if (buffers)
        XLALDestroyREAL8Array(buffers); /* let's be careful, although all these checks may not be needed */
    if (temp)
        LALFree(temp);
 
    if (interp)
        XLAL_CALLGSL(gsl_spline_free(interp));
    if (accel)
        XLAL_CALLGSL(gsl_interp_accel_free(accel));
 
    if (errnum)
        XLAL_ERROR(errnum);
 
    *yout = output;
    return outputlen;
}
 
/**
 * Fourth-order Runge-Kutta ODE integrator using Runge-Kutta-Fehlberg (RKF45)
 * steps with adaptive step size control.  Intended for use in Fourier domain
 * waveform generation routines based on SpinTaylorTN models.
 *
 * The method is described in
 *
 * Abramowitz & Stegun, Handbook of Mathematical Functions, Tenth Printing,
 * National Bureau of Standards, Washington, DC, 1972
 * (available online at http://people.math.sfu.ca/~cbm/aands/ )
 *
 * This method is equivalent to XLALAdaptiveRungeKutta4 and
 * XLALAdaptiveRungeKutta4Hermite, but does not includes any interpolation.
 *
 */
int XLALAdaptiveRungeKutta4IrregularIntervals(LALAdaptiveRungeKuttaIntegrator * integrator,      /**< struct holding dydt, stopping test, stepper, etc. */
    void *params,                                                       /**< params struct used to compute dydt and stopping test */
    REAL8 * yinit,                                                      /**< pass in initial values of all variables - overwritten to final values */
    REAL8 tinit,                                                        /**< integration start time */
    REAL8 tend_in,                                                      /**< maximum integration time */
    REAL8Array ** yout                                                  /**< array holding the unevenly sampled output */
    )
{
    UINT4 MaxRK4Steps = 1000000;
    int errnum = 0;
    int status; /* used throughout */
    unsigned int i;
 
    // if we integrate forward in time, we start at i=0, and if we integrate backwards, we start at i=MaxRK4Steps-1
    unsigned int iStart, iCurrent;
    int di;
 
    REAL8 tend = tend_in;
 
    /* needed for the integration */
    size_t dim, bufferlength, cnt, retries;
    REAL8 t, tnew, h0;
    REAL8Array *buffers = NULL;
    REAL8 *temp = NULL, *y, *y0, *dydt_in, *dydt_in0, *dydt_out, *yerr; /* aliases */
 
    int outputlen = 0;
    REAL8Array *output = NULL;
 
    /* note: for speed, this replaces the single CALLGSL wrapper applied before each GSL call */
    XLAL_BEGINGSL;
 
    /* allocate the buffers!
     * note: REAL8Array has a field dimLength (UINT4Vector) with dimensions, and a field data that points to a single memory block;
     * dimLength itself has fields length and data */
    dim = integrator->sys->dimension;
    bufferlength = MaxRK4Steps;
    buffers = XLALCreateREAL8ArrayL(2, dim + 2, bufferlength);  /* 2-dimensional array, (dim+2) x bufferlength */
    temp = LALCalloc(6 * dim, sizeof(REAL8));
 
    if (!buffers || !temp) {
        errnum = XLAL_ENOMEM;
        goto bail_out;
    }
 
    y = temp;
    y0 = temp + dim;
    dydt_in = temp + 2 * dim;
    dydt_in0 = temp + 3 * dim;
    dydt_out = temp + 4 * dim;
    yerr = temp + 5 * dim;      /* aliases */
 
    /* set up to get started */
    integrator->sys->params = params;
 
    integrator->returncode = 0;
 
    cnt = 0;
    retries = integrator->retries;
 
    t = tinit;
    if (tend > tinit) {
        h0 = 1.;
    } else {
        h0 = -1.;
    }
    memcpy(y, yinit, dim * sizeof(REAL8));
 
    // current index starts at iStart, and increases in steps of di. We've reached the memory limit if cnt == bufferlength.
    if (h0 > 0.) {
        iStart = 0;
        di = 1;
    } else {
        iStart = MaxRK4Steps - 1;
        di = -1;
    }
 
    iCurrent = iStart;
 
    /* store the first data point */
    buffers->data[iCurrent] = t;
    for (i = 1; i <= dim; i++)
        buffers->data[i * bufferlength + iCurrent] = y[i - 1];
 
    /* compute derivatives at the initial time (dydt_in), bail out if impossible */
    if ((status = integrator->dydt(t, y, dydt_in, params)) != GSL_SUCCESS) {
        integrator->returncode = status;
        errnum = XLAL_EFAILED;
        goto bail_out;
    }
 
    buffers->data[i * bufferlength + iCurrent] = dydt_in[1];    /* add domega/dt. here i=dim+1 */
 
    while (1) {
 
        if (!integrator->stopontestonly && t >= tend) {
            break;
        }
 
        if (integrator->stop) {
            if ((status = integrator->stop(t, y, dydt_in, params)) != GSL_SUCCESS) {
                integrator->returncode = status;
                break;
            }
        }
 
        /* ready to try stepping! */
      try_step:
 
        /* if we would be stepping beyond the final time, stop there instead... */
        if (!integrator->stopontestonly && t + h0 > tend)
            h0 = tend - t;
 
        memcpy(y0, y, dim * sizeof(REAL8));     /* save y to y0, dydt_in to dydt_in0 */
        memcpy(dydt_in0, dydt_in, dim * sizeof(REAL8));
 
        /* call the GSL stepper function */
        status = gsl_odeiv_step_apply(integrator->step, t, h0, y, yerr, dydt_in, dydt_out, integrator->sys);
        /* note: If the user-supplied functions defined in the system dydt return a status other than GSL_SUCCESS,
         * the step will be aborted. In this case, the elements of y will be restored to their pre-step values,
         * and the error code from the user-supplied function will be returned. */
 
        /* did the stepper report a derivative-evaluation error? */
        if (status != GSL_SUCCESS) {
            if (retries--) {
                h0 = h0 / 10.0; /* if we have singularity retries left, reduce the timestep and try again */
                goto try_step;
            } else {
                integrator->returncode = status;
                break;  /* otherwise exit the loop */
            }
        } else {
            retries = integrator->retries;      /* we stepped successfully, reset the singularity retries */
        }
 
        tnew = t + h0;
 
        /* call the GSL error-checking function */
        status = gsl_odeiv_control_hadjust(integrator->control, integrator->step, y, yerr, dydt_out, &h0);
 
        /* did the error-checker reduce the stepsize?
         * note: other possible return codes are GSL_ODEIV_HADJ_INC if it was increased, GSL_ODEIV_HADJ_NIL if it was unchanged */
        if (status == GSL_ODEIV_HADJ_DEC) {
            memcpy(y, y0, dim * sizeof(REAL8)); /* if so, undo the step, and try again */
            memcpy(dydt_in, dydt_in0, dim * sizeof(REAL8));
            goto try_step;
        }
 
        /* update the current time and input derivatives */
        t = tnew;
        memcpy(dydt_in, dydt_out, dim * sizeof(REAL8));
        cnt++;
        iCurrent += di;
 
        /* check if interpolation buffers need to be extended */
        if (cnt >= bufferlength) {
            REAL8Array *rebuffers;
 
            /* sadly, we cannot use ResizeREAL8Array, because it would only work if we extended the first array dimension,
             * so we need to copy everything over and switch the buffers. Oh well. */
            if (!(rebuffers = XLALCreateREAL8ArrayL(2, dim + 1, 2 * bufferlength))) {
                errnum = XLAL_ENOMEM;   /* ouch, that hurt */
                goto bail_out;
            } else {
                for (i = 0; i <= dim; i++)
                    memcpy(&rebuffers->data[i * 2 * bufferlength], &buffers->data[i * bufferlength], cnt * sizeof(REAL8));
                XLALDestroyREAL8Array(buffers);
                buffers = rebuffers;
                bufferlength *= 2;
                if (di < 0) {
                    iCurrent += bufferlength;
                }
            }
        }
 
        /* copy time and state into interpolation buffers */
        buffers->data[iCurrent] = t;
        for (i = 1; i <= dim; i++)
            buffers->data[i * bufferlength + iCurrent] = y[i - 1];      /* y does not have time */
 
        buffers->data[i * bufferlength + iCurrent] = dydt_in[1];        /* add domega/dt. here i=dim+1 */
    }
 
    /* copy the final state into yinit */
 
    memcpy(yinit, y, dim * sizeof(REAL8));
 
    /* if we have completed at least one step, allocate the output array */
    if (cnt == 0)
        goto bail_out;
 
    outputlen = cnt + 1;
    output = XLALCreateREAL8ArrayL(2, dim + 2, outputlen);
 
    // depending on the direction of integration, we copy starting from 0 or from the current index
    if (di > 0) {
        iStart = 0;
    } else {
        iStart = iCurrent;
    }
 
    if (!output) {
        errnum = XLAL_ENOMEM;   /* ouch again, ran out of memory */
        if (output)
            XLALDestroyREAL8Array(output);
        outputlen = 0;
        goto bail_out;
    }
 
    for (i = 0; i <= dim + 1; i++) {
        memcpy(&(output->data[i * outputlen]), &(buffers->data[i * bufferlength + iStart]), outputlen * sizeof(REAL8));
    }
 
    /* deallocate stuff and return */
  bail_out:
 
    XLAL_ENDGSL;
 
    if (buffers)
        XLALDestroyREAL8Array(buffers); /* let's be careful, although all these checks may not be needed */
    if (temp)
        LALFree(temp);
 
    if (errnum)
        XLAL_ERROR(errnum);
 
    *yout = output;
    return outputlen;
}