Documentation

erf

Error function

Syntax

Description

example

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

Examples

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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 =

   -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 =

    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

$$\phi(x) = \frac{1}{2} \biggl(1+\rm erf\Bigl(
\frac{x-\mu}{\sigma\sqrt{2}}\Bigr)\biggr).$$

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

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

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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x — Inputreal 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

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Error Function

The error function erf of x is

erf(x)=2π0xet2dt.

Tips

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

    normcdf(x)=12(1erf(x2)).

  • 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).

See Also

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Introduced before R2006a

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