# Documentation

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# erf

Error function

## Description

example

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

## 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 =

-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

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

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 and .

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 represents the temperature at position and time , the heat equation is

where is a constant.

For a material with heat coefficient , and for the initial condition for and elsewhere, the solution to the heat equation is

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

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

### Tall Array Support

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

### 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

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

• 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).