erfinv

Inverse error function

Syntax

erfinv(x)

Description

erfinv(x) returns the Inverse Error Function evaluated for each element of x. For inputs outside the interval [-1 1], erfinv returns NaN.

Examples

Find Inverse Error Function of Value

erfinv(0.25)

ans = 
0.2253

For inputs outside [-1,1], erfinv returns NaN. For -1 and 1, erfinv returns -Inf and Inf, respectively.

erfinv([-2 -1 1 2])

ans = 1×4

   NaN  -Inf   Inf   NaN

Find the inverse error function of the elements of a matrix.

M = [0 -0.5; 0.9 -0.2];
erfinv(M)

ans = 2×2

         0   -0.4769
    1.1631   -0.1791

Plot the Inverse Error Function

Plot the inverse error function for -1 < x < 1.

x = -1:0.01:1;
y = erfinv(x);
plot(x,y)
grid on
xlabel('x')
ylabel('erfinv(x)')
title('Inverse Error Function for -1 < x < 1')

Generate Gaussian Distributed Random Numbers

Generate Gaussian distributed random numbers using uniformly distributed random numbers. To convert a uniformly distributed random number $$x$$ to a Gaussian distributed random number $$y$$, use the transform

$$y=\sqrt{2}er{f}^{-1}(x).$$

Note that because x has the form -1 + 2*rand(1,10000), you can improve accuracy by using erfcinv instead of erfinv. For details, see Tips.

Generate 10,000 uniformly distributed random numbers on the interval [-1,1]. Transform them into Gaussian distributed random numbers. Show that the numbers follow the form of the Gaussian distribution using a histogram plot.

rng('default')
x = -1 + 2*rand(1,10000);
y = sqrt(2)*erfinv(x);
h = histogram(y);

Input Arguments

x — Input

real number | vector of real numbers | matrix of real numbers | multidimensional array of real numbers

Input, specified as a real number, or a vector, matrix, or multidimensional array of real numbers. x cannot be sparse.

Data Types: single | double

More About

Inverse Error Function

The inverse error function erfinv is defined as the inverse of the error function, such that

$$\text{erfinv}\left(\text{erf}\left(x\right)\right)=x.$$

Tips

• For expressions of the form erfinv(1-x), use the complementary inverse error function erfcinv instead. This substitution maintains accuracy. When x is close to 1, then 1 - x is a small number and may be rounded down to 0. Instead, replace erfinv(1-x) with erfcinv(x).

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The erfinv function fully supports tall arrays. For more information, see Tall Arrays.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Strict single-precision calculations are not supported. In the generated code, single-precision inputs produce single-precision outputs. However, variables inside the function might be double-precision.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The erfinv function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a