Error Functions

Error Function and its relating functions

The Error Function/Erf:

Uses the Maclaurin Series Expansion of the Error Function. The Error Function is the integral of the Gaussian Function. The Error Function is used in statistics to calculate the probability of a given value of a normal distribution.

Parameters:

z: The number for which the Error Function will be calculated.

Returns:

The Error Function of the given number.

Equation:

\[ \text{erf}\left( z \right)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt \] \[ \text{erf}\left( z \right)=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty }\frac{\left( -1 \right)^{n}z^{2x+1}}{n!\left( 2n+1 \right)} \]

Example:

use ferrate::special::Error;        
                         
fn main() {                
    let z = 2_f64;        
                          
    let erf = Error::erf(z);
                          
    println!("The Error Function of {} is: {}", z, erf);
}

Executes as:

>>> The Error Function of 2 is: 0.9999999845946841

The Complementary Error Function/Erfc:

Uses the definition of the Complementary Error Function to compute. The Complementary Error Function is the integral of the Gaussian Function.

Parameters:

z: The number for which the Complementary Error Function will be calculated.

use ferrate::special::Error;                     

fn main() {                                        
    let z = 2_f64;                                 
                                                     
    let erfc = Error::erfc(z);                      
                                                     
    println!("The Complementary Error Function of {} is: {}", z, erfc);
}

Executes as:

>>> The Complementary Error Function of 2 is: 0.000000015405315911820594

The Inverse Error Function

Approximates the Inverse of the Error Function.

Parameters:

z: The number for which the Inverse Error Function will be calculated.

use ferrate::special::Error;           

fn main() {                              
    let z = 0.975;                       
                                         
    let inverf = Error::inverf(z);       
                                         
    println!("The Inverse Error Function of {} is: {}", z, inverf);
}

Executes as:

>>> The Inverse Error Function of 0.975 is: 1.5849110680594818

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