ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains classes and procedures for computing the mathematical Inverse Error Function. More...
Data Types | |
interface | getErfInv |
Generate and return the Inverse Error Function \(\ms{erf}^{-1}(x)\) for an input real value in range \((-1, +1)\) as defined in the details section of pm_mathErf. More... | |
interface | setErfInv |
Return the Inverse Error Function \(\ms{erf}^{-1}(x)\) for an input real value in range \((-1, +1)\) as defined in the details section of pm_mathErf. More... | |
Variables | |
character(*, SK), parameter | MODULE_NAME = "@pm_mathErf" |
This module contains classes and procedures for computing the mathematical Inverse Error Function.
The error function (or the Gauss error function), denoted by \(\ms{erf}(\cdot)\), is a complex function of a complex variable defined as,
\begin{equation} \ms{erf}(x) = {\frac{2}{\sqrt{\pi}}} \int_{0}^{x} e^{-t^{2}} ~ \mathrm{d}t ~. \end{equation}
This integral is a special (non-elementary) Sigmoid function that occurs in probability, statistics, and partial differential equations.
In many of these applications, the function argument is however a real number.
If the function argument is real, then the function value is also real.
For non-negative values of \(x\), the error function has the following interpretation: For a random variable \(Y\) that is normally distributed with mean \(0\) and standard deviation \(\frac{1}{\sqrt{2}}\), \(\ms{erf}(x)\) is the probability that \(Y\) falls in the range \([−x, x]\).
Two closely related functions are the complementary error function (erfc) defined as,
\begin{equation} \ms{erfc}(x) = 1 - \ms{erf}(x) ~, \end{equation}
and the imaginary error function (erfi) defined as,
\begin{equation} \ms{erfi}(x) = -i\ms{erf}(ix) ~, \end{equation}
where \(i\) is the imaginary unit.
Inverse error function
Given a complex number \(x\), there is not a unique complex number \(w\) satisfying \(\ms{erf}(w) = x\).
Therefore, a true inverse function would be multivalued.
However, for \(−1 < x < 1\), there is a unique real number denoted \(\ms{erf}^{−1}(x)\) satisfying,
\begin{equation} \ms{erf}\left(\ms{erf}^{-1}(x)\right) = x ~. \end{equation}
The inverse error function is usually defined with domain \((−1, 1)\) and it is restricted to this domain in many computer algebra systems.
However, it can be extended to the disk \(|x| < 1\) of the complex plane, using the Maclaurin series,
\begin{equation} \ms{erf}^{-1}(x) = \sum_{k=0}^{\infty} \frac{c_{k}}{2k+1} \left({\frac{\sqrt{\pi}}{2}} x \right)^{2k+1} ~, \end{equation}
where \(c_0 = 1\) and,
\begin{equation} \begin{aligned} c_k & = \sum_{m=0}^{k-1} \frac{c_{m} c_{k-1-m}}{(m+1) (2m+1)} \\ & = \left\{1, 1, \frac{7}{6}, \frac{127}{90}, \frac{4369}{2520}, \frac{34807}{16200}, \ldots \right\} ~. \end{aligned} \end{equation}
Therefore,
\begin{equation} \ms{erf}^{-1}(x) = \frac{\sqrt{\pi}}{2} \left(x + \frac{\pi}{12} x^{3} + \frac{7\pi^{2}}{480} x^{5} + \frac{127\pi^{3}}{40320} x^{7} + \frac{4369\pi^{4}}{5806080} x^{9} + \frac{34807\pi^{5}}{182476800} x^{11} + \cdots \right) ~. \end{equation}
The error function value at \(x = \pm\infty\) is equal to \(\pm1\).
For \(|x| < 1\), \(\ms{erf}\left(\ms{erf}^{−1}(x)\right) = x\).
The inverse complementary error function is defined as,
\begin{equation} \ms{erfc}^{-1}(1 - x) = \ms{erf}^{-1}(x) ~. \end{equation}
For real \(x\), there is a unique real number \(\ms{erfi}^{−1}(x)\) satisfying \(\ms{erfi}(\ms{erfi}^{−1}(x) = x\).
The inverse imaginary error function is defined as \(\ms{erfi}^{−1}(x)\).
Numerical computation
The erf()
and erfc()
intrinsic Fortran functions readily return the value of the Error function at any given input value with arbitrary real
type and kind.
Theoretically, for any real \(x\), the Newton root-finding method can be used to compute the inverse error function \(\ms{erfi}^{−1}(x)\) and for −1 ≤ x ≤ 1, the following Maclaurin series converges:
\begin{equation} \ms{erfi}^{-1}(x) = \sum_{k=0}^{\infty}{\frac{(-1)^{k}c_{k}}{2k+1}}\left({\frac{\sqrt{\pi}}{2}}(x)\right)^{2k+1} ~, \end{equation}
where \(c_k\) is defined as above.
The procedures of this module combine multiple varying-precision approaches to make a decision at compile-time about the best strategy for computing the inverse error function.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
character(*, SK), parameter pm_mathErf::MODULE_NAME = "@pm_mathErf" |
Definition at line 111 of file pm_mathErf.F90.