erf

Error function

Syntax

erf(x)

Description

erf(x) returns the Error Function evaluated for each element of x.

Examples

Find Error Function

Find the error function of a value.

erf(0.76)

ans = 
0.7175

Find the error function of the elements of a vector.

V = [-0.5 0 1 0.72];
erf(V)

ans = 1×4

   -0.5205         0    0.8427    0.6914

Find the error function of the elements of a matrix.

M = [0.29 -0.11; 3.1 -2.9];
erf(M)

ans = 2×2

    0.3183   -0.1236
    1.0000   -1.0000

Find Cumulative Distribution Function of Normal Distribution

The cumulative distribution function (CDF) of the normal, or Gaussian, distribution with standard deviation $$\sigma$$ and mean $$\mu$$ is

$$\varphi (x)=\frac{1}{2}(1+erf\left(\frac{x-\mu }{\sigma \sqrt{2}}\right)).$$

Note that for increased computational accuracy, you can rewrite the formula in terms of erfc. For details, see Tips.

Plot the CDF of the normal distribution with $$\mu =0$$ and $$\sigma =1$$.

x = -3:0.1:3;
y = (1/2)*(1+erf(x/sqrt(2)));
plot(x,y)
grid on
title('CDF of normal distribution with \mu = 0 and \sigma = 1')
xlabel('x')
ylabel('CDF')

Calculate Solution of Heat Equation with Initial Condition

Where $$u(x,t)$$ represents the temperature at position $$x$$ and time $$t$$, the heat equation is

$$\frac{\partial u}{\partial t}=c\frac{{\partial }^{2}u}{\partial x^2},$$

where $$c$$ is a constant.

For a material with heat coefficient $$k$$, and for the initial condition $$u(x,0)=a$$ for $$x>b$$ and $$u(x,0)=0$$ elsewhere, the solution to the heat equation is

$$u(x,t)=\frac{a}{2}(erf\left(\frac{x-b}{\sqrt{4kt}}\right)).$$

For k = 2, a = 5, and b = 1, plot the solution of the heat equation at times t = 0.1, 5, and 100.

x = -4:0.01:6;
t = [0.1 5 100];
a = 5;
k = 2;
b = 1;
figure(1)
hold on
for i = 1:3
    u(i,:) = (a/2)*(erf((x-b)/sqrt(4*k*t(i))));
    plot(x,u(i,:))
end
grid on
xlabel('x')
ylabel('Temperature')
legend('t = 0.1','t = 5','t = 100','Location','best')
title('Temperatures across material at t = 0.1, t = 5, and t = 100')

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

Error Function

The error function erf of x is

$$\text{erf(}x\text{)=}\frac{2}{\sqrt{\pi }}{\displaystyle {\int }_0^x{e}^{-t^2}}dt.$$

Tips

• You can also find the standard normal probability distribution using the function normcdf (Statistics and Machine Learning Toolbox). The relationship between the error function erf and normcdf is

  $$\text{normcdf}\left(x\right)=\frac{1}{2}\left(1-\text{erf}\left(\frac{-x}{\sqrt{2}}\right)\right).$$

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

Extended Capabilities

Tall Arrays

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

The erf 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 erf 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