# erf

Error function

## Description

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

example

## Examples

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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

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

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

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')

Where $u\left(x,t\right)$ 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 $u\left(x,0\right)=a$ for $x>b$ and $u\left(x,0\right)=0$ elsewhere, the solution to the heat equation is

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

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

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

Data Types: single | double

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### Error Function

The error function erf of x is

$\text{erf(}x\text{)=}\frac{2}{\sqrt{\pi }}{\int }_{0}^{x}{e}^{-{t}^{2}}dt.$

## 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).

## Version History

Introduced before R2006a