Stability Level

Functions

Types and Values

Object Hierarchy

    GEnum
    ╰── NcmSplineFuncType

Includes

#include <numcosmo/math/ncm_spline_func.h>

Description

This function implements 4 different methods to automatically determine the NcmVector of knots, $\mathbf{x}$, of a NcmSpline given a relative error between the function $f$ to be interpolated and the spline result $\hat{f}$.

All available methods start with $n_0$ knots, $\mathbf{x}_0$, distributed across [xi , xf ], including both limiting points. The value of $n_0$ depends on the chosen interpolation method given by s , e.g., NcmSplineCubicNotaknot has $n_0 = 6$.

The function $f$ is first interpolated at the $\mathbf{x}_0$ knots, producing the interpolated function $\hat{f}_0$. Next, the existing $n_0 - 1$ bins, $\Delta \mathbf{x}_0 = \mathbf{x}_0^{i+1} - \mathbf{x}_0^{i}$, are divided in half and test points are placed at those positions, $\overline{\mathbf{x}}_0 = \frac{\mathbf{x}_0^{i+1} + \mathbf{x}_0^{i}}{2}$. The following tests are done for each one of the bins $\Delta \mathbf{x}_0$ separately: \begin{equation*} \left| \frac{ \hat{f}_0(\overline{\mathbf{x}}_0) - f(\overline{\mathbf{x}}_0)}{f(\overline{\mathbf{x}}_0)} \right| < \mathrm{rel \_ error} \end{equation*} and \begin{equation*} \left| \frac{ \int_{\Delta \mathbf{x}_0} \hat{f}_0 - \int_{\Delta \mathbf{x}_0} f }{ \int_{\Delta \mathbf{x}_0} f } \right| < \mathrm{rel \_ error}. \end{equation*} Where $\int_{\Delta \mathbf{x}_0} f$ is the integral of the input function $f$ evaluated using Simpson's rule \begin{equation*} \int_{\Delta \mathbf{x}_0} f = \frac{\Delta \mathbf{x}_0}{6} \left[ f \left( \mathbf{x}_0^{i} \right) + 4 f \left( \overline{\mathbf{x}}_0 \right) + f \left( \mathbf{x}_0^{i+1} \right) \right] \end{equation*}

and the interpolated function $\hat{f}_0$ is integrated by applying ncm_spline_eval_integ(). If any bin passes those relations then, its associated test point $\overline{\mathbf{x}}_0$ together with both knots, are defined as a good representation of the function $f$ and this specific bin does not need to be refined anymore. If not, then $\mathbf{x}_0 \cup \overline{\mathbf{x}}_0$ is splitted once again into two more symmetric test points around $\overline{\mathbf{x}}_0$. A new set of knots is defined, $\mathbf{x}_1 = \mathbf{x}_0 \cup \overline{\mathbf{x}}_0$. The ones which did not pass the tests define another set of test points $\overline{\mathbf{x}}_1$ that lie between the $\mathbf{x}_0\cup \overline{\mathbf{x}}_0$. The new set of knots $\mathbf{x}_1$ are used to create a new interpolated function $\hat{f}_1$, always in the full range [xi , xf ]. The same tests are performed as before, but now with $\hat{f}_1(\overline{\mathbf{x}}_1)$, $f(\overline{\mathbf{x}}_1)$ and the integral now has limits $\Delta \mathbf{x}_1 = \Delta \mathbf{x}_0/2$, but only for those bins that did not pass the previous test. Therefore, the tests are always applied on three knots at a time. This procedure is repeated until the desired accuracy is met across the whole range [xi , xf ]. Note that it will most probably create a inhomogeneous set of knots.

Functions

gdouble
(*NcmSplineFuncF) (gdouble x,
                   GObject *obj);

void
ncm_spline_set_func (NcmSpline *s,
                     NcmSplineFuncType ftype,
                     gsl_function *F,
                     const gdouble xi,
                     const gdouble xf,
                     gsize max_nodes,
                     const gdouble rel_error);

void
ncm_spline_set_func_scale (NcmSpline *s,
                           NcmSplineFuncType ftype,
                           gsl_function *F,
                           const gdouble xi,
                           const gdouble xf,
                           gsize max_nodes,
                           const gdouble rel_error,
                           const gdouble scale,
                           gint refine,
                           gdouble refine_ns);

void
ncm_spline_set_func1 (NcmSpline *s,
                      NcmSplineFuncType ftype,
                      NcmSplineFuncF F,
                      GObject *obj,
                      gdouble xi,
                      gdouble xf,
                      gsize max_nodes,
                      gdouble rel_error);

void
ncm_spline_set_func_grid1 (NcmSpline *s,
                           NcmSplineFuncType ftype,
                           NcmSplineFuncF F,
                           GObject *obj,
                           gdouble xi,
                           gdouble xf,
                           gsize nnodes);

void
ncm_spline_set_func_grid (NcmSpline *s,
                          NcmSplineFuncType ftype,
                          gsl_function *F,
                          const gdouble xi,
                          const gdouble xf,
                          gsize nnodes);

Types and Values

#define NCM_SPLINE_FUNC_DEFAULT_MAX_NODES 10000000

#define NCM_SPLINE_KNOT_DIFF_TOL (GSL_DBL_EPSILON * 1.0e2)