Maths › Special ›

error Fn C inv

Evaluates the inverse of the complementary error function.

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Interface
ErrorFnC Inv
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Private project under development, to help contact the author:

Interface

C++

ErrorFnC Inv

doubleerrorFnC_inv( double y )

The inverse of the complementary error function is defined as the function $\inline \mathrm{erfc}^{-1}:(0,2) \rightarrow \mathbb{R}$ which satisfies:

$\begin{array}{rcll} \mathrm{erfc}\left(\mathrm{erfc}^{-1}(x)\right) &=& x, &\qquad \forall x \in (0,2)\\ \mathrm{erfc}^{-1}\left(\mathrm{erfc}(x)\right) &=& x, &\qquad \forall x \in \mathbb{R} \end{array}$

(2)

where $\inline \displaystyle \mathrm{erfc}$ is the complementary error function. Some special values are:

$\begin{array}{rcl} \mathrm{erfc}^{-1}(0) &=& \infty \\ \\ \mathrm{erfc}^{-1}(1) &=& 0 \\ \\ \mathrm{erfc}^{-1}(2) &=& -\infty. \end{array}$

(3)

The graph of this function is shown below.

There is an error with your graph parameters for errorFnC_inv with options y=0:2

Error Message:Function errorFnC_inv failed. Ensure that: Invalid C++

The following property also holds:

$\mathrm{erfc}^{-1}(x) = \mathrm{erf}^{-1}(1 - x), \qquad \forall x \in (0,2)$

(4)

where $\inline \displaystyle \mathrm{erf}^{-1}$ is the inverse of the error function. Based on this last formula, you may notice how the output of the example code below is linked to the example output in the errorFn_inv module.

References:

Mathworld, http://mathworld.wolfram.com/InverseErfc.html

Example 1

#include <codecogs/maths/special/errorfnc_inv.h>
#include <stdio.h>
 
int main(  )
{
  // display the value of the function at important points
  printf("x = 0     y = %.15lf\n",   Maths::Special::errorFnC_inv(0.0));
  printf("x = 1     y = %.15lf\n",   Maths::Special::errorFnC_inv(1.0));
  printf("x = 2     y = %.15lf\n\n", Maths::Special::errorFnC_inv(2.0));
 
  // display several values of the function
  // at equally spaced abscissas with a step of 0.1
  for (double x = 1.1; x < 1.99; x += 0.1)
    printf("x = %.1lf   y = %.15lf\n", 
    x, Maths::Special::errorFnC_inv(x));
 
  return 0;
}

Output

x = 0     y = 1.#INF00000000000
x = 1     y = 0.000000000000000
x = 2     y = -1.#INF00000000000
 
x = 1.1   y = -0.088855990494258
x = 1.2   y = -0.179143454621292
x = 1.3   y = -0.272462714726755
x = 1.4   y = -0.370807158593558
x = 1.5   y = -0.476936276204471
x = 1.6   y = -0.595116081449996
x = 1.7   y = -0.732869077959218
x = 1.8   y = -0.906193802436824
x = 1.9   y = -1.163087153676677

Parameters

y	the value at which to evaluate the function ( $\inline 0 \leq y \leq 2$ )

Returns

The inverse of the complementary error function.

Authors

Lucian Bentea (September 2006)

Source Code

This module is private, for owner's use only.

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