minimize.cpp

        for (minstep = 0, i = 0; i < n; i++)
        {
            tmp = fabs(xx[i]);
            if (tmp < 1.0)
            {
                tmp = 1.0;
            }
            tmp      = s[i]/tmp;
            minstep += tmp*tmp;
        }
        minstep = GMX_REAL_EPS/sqrt(minstep/n);

        if (stepsize < minstep)
        {
            converged = TRUE;
            break;
        }

        // Before taking any steps along the line, store the old position
        *last       = ems;
        real *lastx = static_cast<real *>(as_rvec_array(last->s.x.data())[0]);
        real *lastf = static_cast<real *>(as_rvec_array(last->f.data())[0]);
        Epot0       = ems.epot;

        *sa         = ems;

        /* Take a step downhill.
         * In theory, we should find the actual minimum of the function in this
         * direction, somewhere along the line.
         * That is quite possible, but it turns out to take 5-10 function evaluations
         * for each line. However, we dont really need to find the exact minimum -
         * it is much better to start a new BFGS step in a modified direction as soon
         * as we are close to it. This will save a lot of energy evaluations.
         *
         * In practice, we just try to take a single step.
         * If it worked (i.e. lowered the energy), we increase the stepsize but
         * continue straight to the next BFGS step without trying to find any minimum,
         * i.e. we change the search direction too. If the line was smooth, it is
         * likely we are in a smooth region, and then it makes sense to take longer
         * steps in the modified search direction too.
         *
         * If it didn't work (higher energy), there must be a minimum somewhere between
         * the old position and the new one. Then we need to start by finding a lower
         * value before we change search direction. Since the energy was apparently
         * quite rough, we need to decrease the step size.
         *
         * Due to the finite numerical accuracy, it turns out that it is a good idea
         * to accept a SMALL increase in energy, if the derivative is still downhill.
         * This leads to lower final energies in the tests I've done. / Erik
         */

        // State "A" is the first position along the line.
        // reference position along line is initially zero
        a          = 0.0;

        // Check stepsize first. We do not allow displacements
        // larger than emstep.
        //
        do
        {
            // Pick a new position C by adding stepsize to A.
            c        = a + stepsize;

            // Calculate what the largest change in any individual coordinate
            // would be (translation along line * gradient along line)
            maxdelta = 0;
            for (i = 0; i < n; i++)
            {
                delta = c*s[i];
                if (delta > maxdelta)
                {
                    maxdelta = delta;
                }
            }
            // If any displacement is larger than the stepsize limit, reduce the step
            if (maxdelta > inputrec->em_stepsize)
            {
                stepsize *= 0.1;
            }
        }
        while (maxdelta > inputrec->em_stepsize);

        // Take a trial step and move the coordinate array xc[] to position C
        real *xc = static_cast<real *>(as_rvec_array(sc->s.x.data())[0]);
        for (i = 0; i < n; i++)
        {
            xc[i] = lastx[i] + c*s[i];
        }

        neval++;
        // Calculate energy for the trial step in position C
        evaluate_energy(fplog, cr,
                        top_global, sc, top,
                        inputrec, nrnb, wcycle, gstat,
                        vsite, constr, fcd, graph, mdAtoms, fr,
                        mu_tot, enerd, vir, pres, step, FALSE);

        // Calc line gradient in position C
        real *fc = static_cast<real *>(as_rvec_array(sc->f.data())[0]);
        for (gpc = 0, i = 0; i < n; i++)
        {
            gpc -= s[i]*fc[i]; /* f is negative gradient, thus the sign */
        }
        /* Sum the gradient along the line across CPUs */
        if (PAR(cr))
        {
            gmx_sumd(1, &gpc, cr);
        }

        // This is the max amount of increase in energy we tolerate.
        // By allowing VERY small changes (close to numerical precision) we
        // frequently find even better (lower) final energies.
        tmp = sqrt(GMX_REAL_EPS)*fabs(sa->epot);

        // Accept the step if the energy is lower in the new position C (compared to A),
        // or if it is not significantly higher and the line derivative is still negative.
        if (sc->epot < sa->epot || (gpc < 0 && sc->epot < (sa->epot + tmp)))
        {
            // Great, we found a better energy. We no longer try to alter the
            // stepsize, but simply accept this new better position. The we select a new
            // search direction instead, which will be much more efficient than continuing
            // to take smaller steps along a line. Set fnorm based on the new C position,
            // which will be used to update the stepsize to 1/fnorm further down.
            foundlower = TRUE;
        }
        else
        {
            // If we got here, the energy is NOT lower in point C, i.e. it will be the same
            // or higher than in point A. In this case it is pointless to move to point C,
            // so we will have to do more iterations along the same line to find a smaller
            // value in the interval [A=0.0,C].
            // Here, A is still 0.0, but that will change when we do a search in the interval
            // [0.0,C] below. That search we will do by interpolation or bisection rather
            // than with the stepsize, so no need to modify it. For the next search direction
            // it will be reset to 1/fnorm anyway.
            foundlower = FALSE;
        }

        if (!foundlower)
        {
            // OK, if we didn't find a lower value we will have to locate one now - there must
            // be one in the interval [a,c].
            // The same thing is valid here, though: Don't spend dozens of iterations to find
            // the line minimum. We try to interpolate based on the derivative at the endpoints,
            // and only continue until we find a lower value. In most cases this means 1-2 iterations.
            // I also have a safeguard for potentially really pathological functions so we never
            // take more than 20 steps before we give up.
            // If we already found a lower value we just skip this step and continue to the update.
            real fnorm = 0;
            nminstep   = 0;
            do
            {
                // Select a new trial point B in the interval [A,C].
                // If the derivatives at points a & c have different sign we interpolate to zero,
                // otherwise just do a bisection since there might be multiple minima/maxima
                // inside the interval.
                if (gpa < 0 && gpc > 0)
                {
                    b = a + gpa*(a-c)/(gpc-gpa);
                }
                else
                {
                    b = 0.5*(a+c);
                }

                /* safeguard if interpolation close to machine accuracy causes errors:
                 * never go outside the interval
                 */
                if (b <= a || b >= c)
                {
                    b = 0.5*(a+c);
                }

                // Take a trial step to point B
                real *xb = static_cast<real *>(as_rvec_array(sb->s.x.data())[0]);
                for (i = 0; i < n; i++)
                {
                    xb[i] = lastx[i] + b*s[i];
                }

                neval++;
                // Calculate energy for the trial step in point B
                evaluate_energy(fplog, cr,
                                top_global, sb, top,
                                inputrec, nrnb, wcycle, gstat,
                                vsite, constr, fcd, graph, mdAtoms, fr,
                                mu_tot, enerd, vir, pres, step, FALSE);
                fnorm = sb->fnorm;

                // Calculate gradient in point B
                real *fb = static_cast<real *>(as_rvec_array(sb->f.data())[0]);
                for (gpb = 0, i = 0; i < n; i++)
                {
                    gpb -= s[i]*fb[i]; /* f is negative gradient, thus the sign */

                }
                /* Sum the gradient along the line across CPUs */
                if (PAR(cr))
                {
                    gmx_sumd(1, &gpb, cr);
                }

                // Keep one of the intervals [A,B] or [B,C] based on the value of the derivative
                // at the new point B, and rename the endpoints of this new interval A and C.
                if (gpb > 0)
                {
                    /* Replace c endpoint with b */
                    c   = b;
                    /* swap states b and c */
                    swap_em_state(&sb, &sc);
                }
                else
                {
                    /* Replace a endpoint with b */
                    a   = b;
                    /* swap states a and b */
                    swap_em_state(&sa, &sb);
                }

                /*
                 * Stop search as soon as we find a value smaller than the endpoints,
                 * or if the tolerance is below machine precision.
                 * Never run more than 20 steps, no matter what.
                 */
                nminstep++;
            }
            while ((sb->epot > sa->epot || sb->epot > sc->epot) && (nminstep < 20));

            if (fabs(sb->epot - Epot0) < GMX_REAL_EPS || nminstep >= 20)
            {
                /* OK. We couldn't find a significantly lower energy.
                 * If ncorr==0 this was steepest descent, and then we give up.
                 * If not, reset memory to restart as steepest descent before quitting.
                 */
                if (ncorr == 0)
                {
                    /* Converged */
                    converged = TRUE;
                    break;
                }
                else
                {
                    /* Reset memory */
                    ncorr = 0;
                    /* Search in gradient direction */
                    for (i = 0; i < n; i++)
                    {
                        dx[point][i] = ff[i];
                    }
                    /* Reset stepsize */
                    stepsize = 1.0/fnorm;
                    continue;
                }
            }

            /* Select min energy state of A & C, put the best in xx/ff/Epot
             */
            if (sc->epot < sa->epot)
            {
                /* Use state C */
                ems        = *sc;
                step_taken = c;
            }
            else
            {
                /* Use state A */
                ems        = *sa;
                step_taken = a;
            }

        }
        else
        {
            /* found lower */
            /* Use state C */
            ems        = *sc;
            step_taken = c;
        }

        /* Update the memory information, and calculate a new
         * approximation of the inverse hessian
         */

        /* Have new data in Epot, xx, ff */
        if (ncorr < nmaxcorr)
        {
            ncorr++;
        }

        for (i = 0; i < n; i++)
        {
            dg[point][i]  = lastf[i]-ff[i];
            dx[point][i] *= step_taken;
        }

        dgdg = 0;
        dgdx = 0;
        for (i = 0; i < n; i++)
        {
            dgdg += dg[point][i]*dg[point][i];
            dgdx += dg[point][i]*dx[point][i];
        }

        diag = dgdx/dgdg;

        rho[point] = 1.0/dgdx;
        point++;

        if (point >= nmaxcorr)
        {
            point = 0;
        }

        /* Update */
        for (i = 0; i < n; i++)
        {
            p[i] = ff[i];
        }

        cp = point;

        /* Recursive update. First go back over the memory points */
        for (k = 0; k < ncorr; k++)
        {
            cp--;
            if (cp < 0)
            {
                cp = ncorr-1;
            }

            sq = 0;
            for (i = 0; i < n; i++)
            {
                sq += dx[cp][i]*p[i];
            }

            alpha[cp] = rho[cp]*sq;

            for (i = 0; i < n; i++)
            {
                p[i] -= alpha[cp]*dg[cp][i];
            }
        }

        for (i = 0; i < n; i++)
        {
            p[i] *= diag;
        }

        /* And then go forward again */
        for (k = 0; k < ncorr; k++)
        {
            yr = 0;
            for (i = 0; i < n; i++)
            {
                yr += p[i]*dg[cp][i];
            }

            beta = rho[cp]*yr;
            beta = alpha[cp]-beta;

            for (i = 0; i < n; i++)
            {
                p[i] += beta*dx[cp][i];
            }

            cp++;
            if (cp >= ncorr)
            {
                cp = 0;
            }
        }

        for (i = 0; i < n; i++)
        {
            if (!frozen[i])
            {
                dx[point][i] = p[i];
            }
            else
            {
                dx[point][i] = 0;
            }
        }

        /* Print it if necessary */
        if (MASTER(cr))
        {
            if (mdrunOptions.verbose)
            {
                double sqrtNumAtoms = sqrt(static_cast<double>(state_global->natoms));
                fprintf(stderr, "\rStep %d, Epot=%12.6e, Fnorm=%9.3e, Fmax=%9.3e (atom %d)\n",
                        step, ems.epot, ems.fnorm/sqrtNumAtoms, ems.fmax, ems.a_fmax + 1);
                fflush(stderr);
            }
            /* Store the new (lower) energies */
            upd_mdebin(mdebin, FALSE, FALSE, (double)step,
                       mdatoms->tmass, enerd, state_global, inputrec->fepvals, inputrec->expandedvals, state_global->box,
                       nullptr, nullptr, vir, pres, nullptr, mu_tot, constr);
            do_log = do_per_step(step, inputrec->nstlog);
            do_ene = do_per_step(step, inputrec->nstenergy);
            if (do_log)
            {
                print_ebin_header(fplog, step, step);
            }
            print_ebin(mdoutf_get_fp_ene(outf), do_ene, FALSE, FALSE,
                       do_log ? fplog : nullptr, step, step, eprNORMAL,
                       mdebin, fcd, &(top_global->groups), &(inputrec->opts), nullptr);
        }

        /* Send x and E to IMD client, if bIMD is TRUE. */
        if (do_IMD(inputrec->bIMD, step, cr, TRUE, state_global->box, as_rvec_array(state_global->x.data()), inputrec, 0, wcycle) && MASTER(cr))
        {
            IMD_send_positions(inputrec->imd);
        }

        // Reset stepsize in we are doing more iterations
        stepsize = 1.0/ems.fnorm;

        /* Stop when the maximum force lies below tolerance.
         * If we have reached machine precision, converged is already set to true.
         */
        converged = converged || (ems.fmax < inputrec->em_tol);

    }   /* End of the loop */

    /* IMD cleanup, if bIMD is TRUE. */
    IMD_finalize(inputrec->bIMD, inputrec->imd);

    if (converged)
    {
        step--; /* we never took that last step in this case */

    }
    if (ems.fmax > inputrec->em_tol)
    {
        if (MASTER(cr))
        {
            warn_step(stderr, inputrec->em_tol, step-1 == number_steps, FALSE);
            warn_step(fplog, inputrec->em_tol, step-1 == number_steps, FALSE);
        }
        converged = FALSE;
    }

    /* If we printed energy and/or logfile last step (which was the last step)
     * we don't have to do it again, but otherwise print the final values.
     */
    if (!do_log) /* Write final value to log since we didn't do anythin last step */
    {
        print_ebin_header(fplog, step, step);
    }
    if (!do_ene || !do_log) /* Write final energy file entries */
    {
        print_ebin(mdoutf_get_fp_ene(outf), !do_ene, FALSE, FALSE,
                   !do_log ? fplog : nullptr, step, step, eprNORMAL,
                   mdebin, fcd, &(top_global->groups), &(inputrec->opts), nullptr);
    }

    /* Print some stuff... */
    if (MASTER(cr))
    {
        fprintf(stderr, "\nwriting lowest energy coordinates.\n");
    }

    /* IMPORTANT!
     * For accurate normal mode calculation it is imperative that we
     * store the last conformation into the full precision binary trajectory.
     *
     * However, we should only do it if we did NOT already write this step
     * above (which we did if do_x or do_f was true).
     */
    do_x = !do_per_step(step, inputrec->nstxout);
    do_f = !do_per_step(step, inputrec->nstfout);
    write_em_traj(fplog, cr, outf, do_x, do_f, ftp2fn(efSTO, nfile, fnm),
                  top_global, inputrec, step,
                  &ems, state_global, observablesHistory);

    if (MASTER(cr))
    {
        double sqrtNumAtoms = sqrt(static_cast<double>(state_global->natoms));
        print_converged(stderr, LBFGS, inputrec->em_tol, step, converged,
                        number_steps, &ems, sqrtNumAtoms);
        print_converged(fplog, LBFGS, inputrec->em_tol, step, converged,
                        number_steps, &ems, sqrtNumAtoms);

        fprintf(fplog, "\nPerformed %d energy evaluations in total.\n", neval);
    }

    finish_em(cr, outf, walltime_accounting, wcycle);

    /* To print the actual number of steps we needed somewhere */
    walltime_accounting_set_nsteps_done(walltime_accounting, step);

    return 0;
}   /* That's all folks */

/*! \brief Do steepest descents minimization
    \copydoc integrator_t(FILE *fplog, t_commrec *cr, const gmx::MDLogger &mdlog,
                          int nfile, const t_filenm fnm[],
                          const gmx_output_env_t *oenv,
                          const MdrunOptions &mdrunOptions,
                          gmx_vsite_t *vsite, gmx_constr_t constr,
                          gmx::IMDOutputProvider *outputProvider,
                          t_inputrec *inputrec,
                          gmx_mtop_t *top_global, t_fcdata *fcd,
                          t_state *state_global,
                          gmx::MDAtoms *mdAtoms,
                          t_nrnb *nrnb, gmx_wallcycle_t wcycle,
                          gmx_edsam_t ed,
                          t_forcerec *fr,
                          const ReplicaExchangeParameters &replExParams,
                          gmx_walltime_accounting_t walltime_accounting)
 */
double do_steep(FILE *fplog, t_commrec *cr, const gmx::MDLogger gmx_unused &mdlog,
                int nfile, const t_filenm fnm[],
                const gmx_output_env_t gmx_unused *oenv,
                const MdrunOptions &mdrunOptions,
                gmx_vsite_t *vsite, gmx_constr_t constr,
                gmx::IMDOutputProvider *outputProvider,
                t_inputrec *inputrec,
                gmx_mtop_t *top_global, t_fcdata *fcd,
                t_state *state_global,
                ObservablesHistory *observablesHistory,
                gmx::MDAtoms *mdAtoms,
                t_nrnb *nrnb, gmx_wallcycle_t wcycle,
                t_forcerec *fr,
                const ReplicaExchangeParameters gmx_unused &replExParams,
                gmx_membed_t gmx_unused *membed,
                gmx_walltime_accounting_t walltime_accounting)
{
    const char       *SD = "Steepest Descents";
    gmx_localtop_t   *top;
    gmx_enerdata_t   *enerd;
    gmx_global_stat_t gstat;
    t_graph          *graph;
    real              stepsize;
    real              ustep;
    gmx_mdoutf_t      outf;
    t_mdebin         *mdebin;
    gmx_bool          bDone, bAbort, do_x, do_f;
    tensor            vir, pres;
    rvec              mu_tot;
    int               nsteps;
    int               count          = 0;
    int               steps_accepted = 0;
    auto              mdatoms        = mdAtoms->mdatoms();

    /* Create 2 states on the stack and extract pointers that we will swap */
    em_state_t  s0 {}, s1 {};
    em_state_t *s_min = &s0;
    em_state_t *s_try = &s1;

    /* Init em and store the local state in s_try */
    init_em(fplog, SD, cr, outputProvider, inputrec, mdrunOptions,
            state_global, top_global, s_try, &top,
            nrnb, mu_tot, fr, &enerd, &graph, mdAtoms, &gstat,
            vsite, constr, nullptr,
            nfile, fnm, &outf, &mdebin, wcycle);

    /* Print to log file  */
    print_em_start(fplog, cr, walltime_accounting, wcycle, SD);

    /* Set variables for stepsize (in nm). This is the largest
     * step that we are going to make in any direction.
     */
    ustep    = inputrec->em_stepsize;
    stepsize = 0;

    /* Max number of steps  */
    nsteps = inputrec->nsteps;

    if (MASTER(cr))
    {
        /* Print to the screen  */
        sp_header(stderr, SD, inputrec->em_tol, nsteps);
    }
    if (fplog)
    {
        sp_header(fplog, SD, inputrec->em_tol, nsteps);
    }

    /**** HERE STARTS THE LOOP ****
     * count is the counter for the number of steps
     * bDone will be TRUE when the minimization has converged
     * bAbort will be TRUE when nsteps steps have been performed or when
     * the stepsize becomes smaller than is reasonable for machine precision
     */
    count  = 0;
    bDone  = FALSE;
    bAbort = FALSE;
    while (!bDone && !bAbort)
    {
        bAbort = (nsteps >= 0) && (count == nsteps);

        /* set new coordinates, except for first step */
        bool validStep = true;
        if (count > 0)
        {
            validStep =
                do_em_step(cr, inputrec, mdatoms, fr->bMolPBC,
                           s_min, stepsize, &s_min->f, s_try,
                           constr, top, nrnb, wcycle, count);
        }

        if (validStep)
        {
            evaluate_energy(fplog, cr,
                            top_global, s_try, top,
                            inputrec, nrnb, wcycle, gstat,
                            vsite, constr, fcd, graph, mdAtoms, fr,
                            mu_tot, enerd, vir, pres, count, count == 0);
        }
        else
        {
            // Signal constraint error during stepping with energy=inf
            s_try->epot = std::numeric_limits<real>::infinity();
        }

        if (MASTER(cr))
        {
            print_ebin_header(fplog, count, count);
        }

        if (count == 0)
        {
            s_min->epot = s_try->epot;
        }

        /* Print it if necessary  */
        if (MASTER(cr))
        {
            if (mdrunOptions.verbose)
            {
                fprintf(stderr, "Step=%5d, Dmax= %6.1e nm, Epot= %12.5e Fmax= %11.5e, atom= %d%c",
                        count, ustep, s_try->epot, s_try->fmax, s_try->a_fmax+1,
                        ( (count == 0) || (s_try->epot < s_min->epot) ) ? '\n' : '\r');
                fflush(stderr);
            }

            if ( (count == 0) || (s_try->epot < s_min->epot) )
            {
                /* Store the new (lower) energies  */
                upd_mdebin(mdebin, FALSE, FALSE, (double)count,
                           mdatoms->tmass, enerd, &s_try->s, inputrec->fepvals, inputrec->expandedvals,
                           s_try->s.box, nullptr, nullptr, vir, pres, nullptr, mu_tot, constr);

                /* Prepare IMD energy record, if bIMD is TRUE. */
                IMD_fill_energy_record(inputrec->bIMD, inputrec->imd, enerd, count, TRUE);

                print_ebin(mdoutf_get_fp_ene(outf), TRUE,
                           do_per_step(steps_accepted, inputrec->nstdisreout),
                           do_per_step(steps_accepted, inputrec->nstorireout),
                           fplog, count, count, eprNORMAL,
                           mdebin, fcd, &(top_global->groups), &(inputrec->opts), nullptr);
                fflush(fplog);
            }
        }

        /* Now if the new energy is smaller than the previous...
         * or if this is the first step!
         * or if we did random steps!
         */

        if ( (count == 0) || (s_try->epot < s_min->epot) )
        {
            steps_accepted++;

            /* Test whether the convergence criterion is met...  */
            bDone = (s_try->fmax < inputrec->em_tol);

            /* Copy the arrays for force, positions and energy  */
            /* The 'Min' array always holds the coords and forces of the minimal
               sampled energy  */
            swap_em_state(&s_min, &s_try);
            if (count > 0)
            {
                ustep *= 1.2;
            }

            /* Write to trn, if necessary */
            do_x = do_per_step(steps_accepted, inputrec->nstxout);
            do_f = do_per_step(steps_accepted, inputrec->nstfout);
            write_em_traj(fplog, cr, outf, do_x, do_f, nullptr,
                          top_global, inputrec, count,
                          s_min, state_global, observablesHistory);
        }
        else
        {
            /* If energy is not smaller make the step smaller...  */
            ustep *= 0.5;

            if (DOMAINDECOMP(cr) && s_min->s.ddp_count != cr->dd->ddp_count)
            {
                /* Reload the old state */
                em_dd_partition_system(fplog, count, cr, top_global, inputrec,
                                       s_min, top, mdAtoms, fr, vsite, constr,
                                       nrnb, wcycle);
            }
        }

        /* Determine new step  */
        stepsize = ustep/s_min->fmax;

        /* Check if stepsize is too small, with 1 nm as a characteristic length */
#if GMX_DOUBLE
        if (count == nsteps || ustep < 1e-12)
#else
        if (count == nsteps || ustep < 1e-6)
#endif
        {
            if (MASTER(cr))
            {
                warn_step(stderr, inputrec->em_tol, count == nsteps, constr != nullptr);
                warn_step(fplog, inputrec->em_tol, count == nsteps, constr != nullptr);
            }
            bAbort = TRUE;
        }

        /* Send IMD energies and positions, if bIMD is TRUE. */
        if (do_IMD(inputrec->bIMD, count, cr, TRUE, state_global->box,
                   MASTER(cr) ? as_rvec_array(state_global->x.data()) : nullptr,
                   inputrec, 0, wcycle) &&
            MASTER(cr))
        {
            IMD_send_positions(inputrec->imd);
        }

        count++;
    }   /* End of the loop  */

    /* IMD cleanup, if bIMD is TRUE. */
    IMD_finalize(inputrec->bIMD, inputrec->imd);

    /* Print some data...  */
    if (MASTER(cr))
    {
        fprintf(stderr, "\nwriting lowest energy coordinates.\n");
    }
    write_em_traj(fplog, cr, outf, TRUE, inputrec->nstfout, ftp2fn(efSTO, nfile, fnm),
                  top_global, inputrec, count,
                  s_min, state_global, observablesHistory);

    if (MASTER(cr))
    {
        double sqrtNumAtoms = sqrt(static_cast<double>(state_global->natoms));

        print_converged(stderr, SD, inputrec->em_tol, count, bDone, nsteps,
                        s_min, sqrtNumAtoms);
        print_converged(fplog, SD, inputrec->em_tol, count, bDone, nsteps,
                        s_min, sqrtNumAtoms);
    }

    finish_em(cr, outf, walltime_accounting, wcycle);

    /* To print the actual number of steps we needed somewhere */
    inputrec->nsteps = count;

    walltime_accounting_set_nsteps_done(walltime_accounting, count);

    return 0;
}   /* That's all folks */

/*! \brief Do normal modes analysis
    \copydoc integrator_t(FILE *fplog, t_commrec *cr, const gmx::MDLogger &mdlog,
                          int nfile, const t_filenm fnm[],
                          const gmx_output_env_t *oenv,
                          const MdrunOptions &mdrunOptions,
                          gmx_vsite_t *vsite, gmx_constr_t constr,
                          gmx::IMDOutputProvider *outputProvider,
                          t_inputrec *inputrec,
                          gmx_mtop_t *top_global, t_fcdata *fcd,
                          t_state *state_global,
                          gmx::MDAtoms *mdAtoms,
                          t_nrnb *nrnb, gmx_wallcycle_t wcycle,
                          gmx_edsam_t ed,
                          t_forcerec *fr,
                          const ReplicaExchangeParameters &replExParams,
                          gmx_walltime_accounting_t walltime_accounting)
 */
double do_nm(FILE *fplog, t_commrec *cr, const gmx::MDLogger &mdlog,
             int nfile, const t_filenm fnm[],
             const gmx_output_env_t gmx_unused *oenv,
             const MdrunOptions &mdrunOptions,
             gmx_vsite_t *vsite, gmx_constr_t constr,
             gmx::IMDOutputProvider *outputProvider,
             t_inputrec *inputrec,
             gmx_mtop_t *top_global, t_fcdata *fcd,
             t_state *state_global,
             ObservablesHistory gmx_unused *observablesHistory,
             gmx::MDAtoms *mdAtoms,
             t_nrnb *nrnb, gmx_wallcycle_t wcycle,
             t_forcerec *fr,
             const ReplicaExchangeParameters gmx_unused &replExParams,
             gmx_membed_t gmx_unused *membed,
             gmx_walltime_accounting_t walltime_accounting)
{
    const char          *NM = "Normal Mode Analysis";
    gmx_mdoutf_t         outf;
    int                  nnodes, node;
    gmx_localtop_t      *top;
    gmx_enerdata_t      *enerd;
    gmx_global_stat_t    gstat;
    t_graph             *graph;
    tensor               vir, pres;
    rvec                 mu_tot;
    rvec                *fneg, *dfdx;
    gmx_bool             bSparse; /* use sparse matrix storage format */
    size_t               sz;
    gmx_sparsematrix_t * sparse_matrix           = nullptr;
    real           *     full_matrix             = nullptr;

    /* added with respect to mdrun */
    int                       row, col;
    real                      der_range = 10.0*sqrt(GMX_REAL_EPS);
    real                      x_min;
    bool                      bIsMaster = MASTER(cr);
    auto                      mdatoms   = mdAtoms->mdatoms();

    if (constr != nullptr)
    {
        gmx_fatal(FARGS, "Constraints present with Normal Mode Analysis, this combination is not supported");
    }

    gmx_shellfc_t *shellfc;

    em_state_t     state_work {};

    /* Init em and store the local state in state_minimum */
    init_em(fplog, NM, cr, outputProvider, inputrec, mdrunOptions,
            state_global, top_global, &state_work, &top,
            nrnb, mu_tot, fr, &enerd, &graph, mdAtoms, &gstat,
            vsite, constr, &shellfc,
            nfile, fnm, &outf, nullptr, wcycle);

    std::vector<size_t> atom_index = get_atom_index(top_global);
    snew(fneg, atom_index.size());
    snew(dfdx, atom_index.size());

#if !GMX_DOUBLE
    if (bIsMaster)
    {
        fprintf(stderr,
                "NOTE: This version of GROMACS has been compiled in single precision,\n"
                "      which MIGHT not be accurate enough for normal mode analysis.\n"
                "      GROMACS now uses sparse matrix storage, so the memory requirements\n"
                "      are fairly modest even if you recompile in double precision.\n\n");
    }
#endif

    /* Check if we can/should use sparse storage format.
     *
     * Sparse format is only useful when the Hessian itself is sparse, which it
     * will be when we use a cutoff.
     * For small systems (n<1000) it is easier to always use full matrix format, though.
     */
    if (EEL_FULL(fr->ic->eeltype) || fr->rlist == 0.0)
    {
        GMX_LOG(mdlog.warning).appendText("Non-cutoff electrostatics used, forcing full Hessian format.");
        bSparse = FALSE;
    }
    else if (atom_index.size() < 1000)
    {
        GMX_LOG(mdlog.warning).appendTextFormatted("Small system size (N=%d), using full Hessian format.",
                                                   atom_index.size());
        bSparse = FALSE;
    }
    else
    {
        GMX_LOG(mdlog.warning).appendText("Using compressed symmetric sparse Hessian format.");
        bSparse = TRUE;
    }

    /* Number of dimensions, based on real atoms, that is not vsites or shell */
    sz = DIM*atom_index.size();

    fprintf(stderr, "Allocating Hessian memory...\n\n");

    if (bSparse)
    {
        sparse_matrix = gmx_sparsematrix_init(sz);
        sparse_matrix->compressed_symmetric = TRUE;
    }
    else
    {
        snew(full_matrix, sz*sz);
    }

    init_nrnb(nrnb);

    where();

    /* Write start time and temperature */
    print_em_start(fplog, cr, walltime_accounting, wcycle, NM);

    /* fudge nr of steps to nr of atoms */
    inputrec->nsteps = atom_index.size()*2;

    if (bIsMaster)
    {
        fprintf(stderr, "starting normal mode calculation '%s'\n%d steps.\n\n",
                *(top_global->name), (int)inputrec->nsteps);
    }

    nnodes = cr->nnodes;

    /* Make evaluate_energy do a single node force calculation */
    cr->nnodes = 1;
    evaluate_energy(fplog, cr,
                    top_global, &state_work, top,
                    inputrec, nrnb, wcycle, gstat,
                    vsite, constr, fcd, graph, mdAtoms, fr,
                    mu_tot, enerd, vir, pres, -1, TRUE);
    cr->nnodes = nnodes;

    /* if forces are not small, warn user */
    get_state_f_norm_max(cr, &(inputrec->opts), mdatoms, &state_work);

    GMX_LOG(mdlog.warning).appendTextFormatted("Maximum force:%12.5e", state_work.fmax);
    if (state_work.fmax > 1.0e-3)
    {
        GMX_LOG(mdlog.warning).appendText(
                "The force is probably not small enough to "
                "ensure that you are at a minimum.\n"
                "Be aware that negative eigenvalues may occur\n"
                "when the resulting matrix is diagonalized.");
    }

    /***********************************************************
     *
     *      Loop over all pairs in matrix
     *
     *      do_force called twice. Once with positive and
     *      once with negative displacement
     *
     ************************************************************/

    /* Steps are divided one by one over the nodes */
    bool bNS = true;
    for (unsigned int aid = cr->nodeid; aid < atom_index.size(); aid += nnodes)
    {
        size_t atom = atom_index[aid];
        for (size_t d = 0; d < DIM; d++)
        {
            gmx_bool    bBornRadii  = FALSE;
            gmx_int64_t step        = 0;
            int         force_flags = GMX_FORCE_STATECHANGED | GMX_FORCE_ALLFORCES;
            double      t           = 0;

            x_min = state_work.s.x[atom][d];

            for (unsigned int dx = 0; (dx < 2); dx++)
            {
                if (dx == 0)
                {
                    state_work.s.x[atom][d] = x_min - der_range;
                }
                else
                {
                    state_work.s.x[atom][d] = x_min + der_range;
                }

                /* Make evaluate_energy do a single node force calculation */
                cr->nnodes = 1;
                if (shellfc)
                {
                    /* Now is the time to relax the shells */
                    (void) relax_shell_flexcon(fplog, cr, mdrunOptions.verbose, step,
                                               inputrec, bNS, force_flags,
                                               top,
                                               constr, enerd, fcd,
                                               &state_work.s, &state_work.f, vir, mdatoms,
                                               nrnb, wcycle, graph, &top_global->groups,
                                               shellfc, fr, bBornRadii, t, mu_tot,
                                               vsite,
                                               DdOpenBalanceRegionBeforeForceComputation::no,
                                               DdCloseBalanceRegionAfterForceComputation::no);
                    bNS = false;
                    step++;
                }
                else
                {
                    evaluate_energy(fplog, cr,
                                    top_global, &state_work, top,
                                    inputrec, nrnb, wcycle, gstat,
                                    vsite, constr, fcd, graph, mdAtoms, fr,
                                    mu_tot, enerd, vir, pres, aid*2+dx, FALSE);
                }

                cr->nnodes = nnodes;

                if (dx == 0)
                {
                    for (size_t i = 0; i < atom_index.size(); i++)
                    {
                        copy_rvec(state_work.f[atom_index[i]], fneg[i]);
                    }
                }
            }

            /* x is restored to original */