How can i print the mu from the levenberg marquardt method using the GNU scientific library (gsl)?

Question

I have tried this example from the official gsl website : gsl website with examples.

the code of the example is the following :

#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multifit_nlinear.h>

/* parameters to model */
struct model_params
{
  double a1;
  double a2;
  double a3;
  double a4;
  double a5;
};

/* Branin function */
int
func_f (const gsl_vector * x, void *params, gsl_vector * f)
{
  struct model_params *par = (struct model_params *) params;
  double x1 = gsl_vector_get(x, 0);
  double x2 = gsl_vector_get(x, 1);
  double f1 = x2 + par->a1 * x1 * x1 + par->a2 * x1 + par->a3;
  double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));

  gsl_vector_set(f, 0, f1);
  gsl_vector_set(f, 1, f2);

  return GSL_SUCCESS;
}

int
func_df (const gsl_vector * x, void *params, gsl_matrix * J)
{
  struct model_params *par = (struct model_params *) params;
  double x1 = gsl_vector_get(x, 0);
  double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));

  gsl_matrix_set(J, 0, 0, 2.0 * par->a1 * x1 + par->a2);
  gsl_matrix_set(J, 0, 1, 1.0);

  gsl_matrix_set(J, 1, 0, -0.5 * par->a4 / f2 * (1.0 - par->a5) * sin(x1));
  gsl_matrix_set(J, 1, 1, 0.0);

  return GSL_SUCCESS;
}

int
func_fvv (const gsl_vector * x, const gsl_vector * v,
          void *params, gsl_vector * fvv)
{
  struct model_params *par = (struct model_params *) params;
  double x1 = gsl_vector_get(x, 0);
  double v1 = gsl_vector_get(v, 0);
  double c = cos(x1);
  double s = sin(x1);
  double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * c);
  double t = 0.5 * par->a4 * (1.0 - par->a5) / f2;

  gsl_vector_set(fvv, 0, 2.0 * par->a1 * v1 * v1);
  gsl_vector_set(fvv, 1, -t * (c + s*s/f2) * v1 * v1);

  return GSL_SUCCESS;
}

void callback(const size_t iter, void *params,
         const gsl_multifit_nlinear_workspace *w)
{
  gsl_vector * x = gsl_multifit_nlinear_position(w);
  double x1 = gsl_vector_get(x, 0);
  double x2 = gsl_vector_get(x, 1);

  /* print out current location */
  printf("%f %f\n", x1, x2);
}

void solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
             gsl_multifit_nlinear_parameters *params)
{
  const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
  const size_t max_iter = 200;
  const double xtol = 1.0e-8;
  const double gtol = 1.0e-8;
  const double ftol = 1.0e-8;
  const size_t n = fdf->n;
  const size_t p = fdf->p;
  gsl_multifit_nlinear_workspace *work =
    gsl_multifit_nlinear_alloc(T, params, n, p);
  gsl_vector * f = gsl_multifit_nlinear_residual(work);
  gsl_vector * x = gsl_multifit_nlinear_position(work);
  int info;
  double chisq0, chisq, rcond;

  printf("# %s/%s\n",
         gsl_multifit_nlinear_name(work),
         gsl_multifit_nlinear_trs_name(work));

  /* initialize solver */
  gsl_multifit_nlinear_init(x0, fdf, work);

  /* store initial cost */
  gsl_blas_ddot(f, f, &chisq0);

  /* iterate until convergence */
  gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
                              callback, NULL, &info, work);

  /* store final cost */
  gsl_blas_ddot(f, f, &chisq);

  /* store cond(J(x)) */
  gsl_multifit_nlinear_rcond(&rcond, work);

  /* print summary */
  fprintf(stderr, "%-25s %-6zu %-5zu %-5zu %-13.4e %-12.4e %-13.4e (%.2e, %.2e)\n",
          gsl_multifit_nlinear_trs_name(work),
          gsl_multifit_nlinear_niter(work),
          fdf->nevalf,
          fdf->nevaldf,
          chisq0,
          chisq,
          1.0 / rcond,
          gsl_vector_get(x, 0),
          gsl_vector_get(x, 1));

  printf("\n\n");

  gsl_multifit_nlinear_free(work);
}

int main (void)
{
  const size_t n = 2;
  const size_t p = 2;
  gsl_vector *f = gsl_vector_alloc(n);
  gsl_vector *x = gsl_vector_alloc(p);
  gsl_multifit_nlinear_fdf fdf;
  gsl_multifit_nlinear_parameters fdf_params =
    gsl_multifit_nlinear_default_parameters();
  struct model_params params;

  params.a1 = -5.1 / (4.0 * M_PI * M_PI);
  params.a2 = 5.0 / M_PI;
  params.a3 = -6.0;
  params.a4 = 10.0;
  params.a5 = 1.0 / (8.0 * M_PI);

  /* print map of Phi(x1, x2) */
  {
    double x1, x2, chisq;

    for (x1 = -5.0; x1 < 15.0; x1 += 0.1)
      {
        for (x2 = -5.0; x2 < 15.0; x2 += 0.1)
          {
            gsl_vector_set(x, 0, x1);
            gsl_vector_set(x, 1, x2);
            func_f(x, &params, f);

            gsl_blas_ddot(f, f, &chisq);

            printf("%f %f %f\n", x1, x2, chisq);
          }
        printf("\n");
      }
    printf("\n\n");
  }

  /* define function to be minimized */
  fdf.f = func_f;
  fdf.df = func_df;
  fdf.fvv = func_fvv;
  fdf.n = n;
  fdf.p = p;
  fdf.params = &params;

  /* starting point */
  gsl_vector_set(x, 0, 6.0);
  gsl_vector_set(x, 1, 14.5);

  fprintf(stderr, "%-25s %-6s %-5s %-5s %-13s %-12s %-13s %-15s\n",
          "Method", "NITER", "NFEV", "NJEV", "Initial Cost",
          "Final cost", "Final cond(J)", "Final x");

  fdf_params.trs = gsl_multifit_nlinear_trs_lm;
  solve_system(x, &fdf, &fdf_params);

  fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
  solve_system(x, &fdf, &fdf_params);

  fdf_params.trs = gsl_multifit_nlinear_trs_dogleg;
  solve_system(x, &fdf, &fdf_params);

  fdf_params.trs = gsl_multifit_nlinear_trs_ddogleg;
  solve_system(x, &fdf, &fdf_params);

  fdf_params.trs = gsl_multifit_nlinear_trs_subspace2D;
  solve_system(x, &fdf, &fdf_params);

  gsl_vector_free(f);
  gsl_vector_free(x);

  return 0;
}

It actually has a few methods such as the levenberg marquardt , the levenberg marquadt with geodesic acceleration, the dogleg, the double dogleg and the 2d subspace. what i am interested in here is the levenberg marquardt. I would like to specifically print out the mu in every step of the iteration but i have no idea how you can get the mu. this is how the levenberg marquadt method is defined : enter image description here

and this is how it is solved on every iteration through a least-squares method : enter image description here

i checked the source file of the leveberg marquadt method . they do calculate the mu but i dont know how to retrieve it in this case.

i tried something like the following :

 gsl_multifit_nlinear_workspace *work =
    gsl_multifit_nlinear_alloc(T, params, n, p);

    
    gsl_multifit_nlinear_trust_state *trs_st = w->state;
    const double mu = *(trs_st->mu);
    fprintf(stdout, "mu = %g \n", mu);

this would be inside the solve_system function mentioned above.