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Error function




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


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Find the error function of a value.

ans = 0.7175

Find the error function of the elements of a vector.

V = [-0.5 0 1 0.72];
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];
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 σ and mean μ 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 μ=0 and σ=1.

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

Figure contains an axes object. The axes object with title C D F blank o f blank n o r m a l blank d i s t r i b u t i o n blank w i t h blank mu blank = blank 0 blank a n d blank sigma blank = blank 1 contains an object of type line.

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


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


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;
hold on
for i = 1:3
    u(i,:) = (a/2)*(erf((x-b)/sqrt(4*k*t(i))));
grid on
legend('t = 0.1','t = 5','t = 100','Location','best')
title('Temperatures across material at t = 0.1, t = 5, and t = 100')

Figure contains an axes object. The axes object with title Temperatures across material at t = 0.1, t = 5, and t = 100 contains 3 objects of type line. These objects represent t = 0.1, t = 5, 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

More About

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Error Function

The error function erf of x is



  • 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


  • 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

See Also

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Introduced before R2006a