| @cindex error function |
| @cindex erf(x) |
| @cindex erfc(x) |
| |
| The error function is described in Abramowitz & Stegun, Chapter 7. The |
| functions in this section are declared in the header file |
| @file{gsl_sf_erf.h}. |
| |
| @menu |
| * Error Function:: |
| * Complementary Error Function:: |
| * Log Complementary Error Function:: |
| * Probability functions:: |
| @end menu |
| |
| @node Error Function |
| @subsection Error Function |
| |
| @deftypefun double gsl_sf_erf (double @var{x}) |
| @deftypefunx int gsl_sf_erf_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the error function @c{$\erf(x)$} |
| @math{erf(x)}, where |
| @c{$\erf(x) = (2/\sqrt{\pi}) \int_0^x dt \exp(-t^2)$} |
| @math{erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2)}. |
| @comment Exceptional Return Values: none |
| @end deftypefun |
| |
| @node Complementary Error Function |
| @subsection Complementary Error Function |
| |
| @deftypefun double gsl_sf_erfc (double @var{x}) |
| @deftypefunx int gsl_sf_erfc_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the complementary error function |
| @c{$\erfc(x) = 1 - \erf(x) = (2/\sqrt{\pi}) \int_x^\infty \exp(-t^2)$} |
| @math{erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2)}. |
| @comment Exceptional Return Values: none |
| @end deftypefun |
| |
| |
| @node Log Complementary Error Function |
| @subsection Log Complementary Error Function |
| |
| @deftypefun double gsl_sf_log_erfc (double @var{x}) |
| @deftypefunx int gsl_sf_log_erfc_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the logarithm of the complementary error function |
| @math{\log(\erfc(x))}. |
| @comment Exceptional Return Values: none |
| @end deftypefun |
| |
| |
| @node Probability functions |
| @subsection Probability functions |
| |
| The probability functions for the Normal or Gaussian distribution are |
| described in Abramowitz & Stegun, Section 26.2. |
| |
| @deftypefun double gsl_sf_erf_Z (double @var{x}) |
| @deftypefunx int gsl_sf_erf_Z_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the Gaussian probability density function |
| @c{$Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)$} |
| @math{Z(x) = (1/\sqrt@{2\pi@}) \exp(-x^2/2)}. |
| @end deftypefun |
| |
| @deftypefun double gsl_sf_erf_Q (double @var{x}) |
| @deftypefunx int gsl_sf_erf_Q_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the upper tail of the Gaussian probability |
| function |
| @c{$Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2)$} |
| @math{Q(x) = (1/\sqrt@{2\pi@}) \int_x^\infty dt \exp(-t^2/2)}. |
| @comment Exceptional Return Values: none |
| @end deftypefun |
| |
| @cindex hazard function, normal distribution |
| @cindex Mill's ratio, inverse |
| The @dfn{hazard function} for the normal distribution, |
| also known as the inverse Mill's ratio, is defined as, |
| @tex |
| \beforedisplay |
| $$ |
| h(x) = {Z(x)\over Q(x)} = \sqrt{2 \over \pi} {\exp(-x^2 / 2) \over \erfc(x/\sqrt 2)} |
| $$ |
| \afterdisplay |
| @end tex |
| @ifinfo |
| |
| @example |
| h(x) = Z(x)/Q(x) = \sqrt@{2/\pi@} \exp(-x^2 / 2) / \erfc(x/\sqrt 2) |
| @end example |
| |
| @end ifinfo |
| @noindent |
| It decreases rapidly as @math{x} approaches @math{-\infty} and asymptotes |
| to @math{h(x) \sim x} as @math{x} approaches @math{+\infty}. |
| |
| @deftypefun double gsl_sf_hazard (double @var{x}) |
| @deftypefunx int gsl_sf_hazard_e (double @var{x}, gsl_sf_result * @var{result}) |
| These routines compute the hazard function for the normal distribution. |
| @comment Exceptional Return Values: GSL_EUNDRFLW |
| @end deftypefun |