# Documentation

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# erfc

Complementary error function

## Description

example

erfc(x) returns the Complementary Error Function evaluated for each element of x. Use the erfc function to replace 1 - erf(x) for greater accuracy when erf(x) is close to 1.

## Examples

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Find the complementary error function of a value.

erfc(0.35)
ans =

0.6206

Find the complementary error function of the elements of a vector.

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

1.5205    1.0000    0.1573    0.3086

Find the complementary error function of the elements of a matrix.

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

0.6817    1.1236
0.0000    2.0000

The bit error rate (BER) of binary phase-shift keying (BPSK), assuming additive white gaussian noise (AWGN), is

Plot the BER for BPSK for values of from 0dB to 10dB.

EbN0_dB = 0:0.1:10;
EbN0 = 10.^(EbN0_dB/10);
BER = 1/2.*erfc(sqrt(EbN0));
semilogy(EbN0_dB,BER)
grid on
ylabel('BER')
xlabel('E_b/N_0 (dB)')
title('Bit Error Rate for Binary Phase-Shift Keying')

You can use the complementary error function erfc in place of 1 - erf(x) to avoid roundoff errors when erf(x) is close to 1.

Show how to avoid roundoff errors by calculating 1 - erf(10) using erfc(10). The original calculation returns 0 while erfc(10) returns the correct result.

1 - erf(10)
erfc(10)
ans =

0

ans =

2.0885e-45

## 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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### Complementary Error Function

The complementary error function of x is defined as

$\begin{array}{c}\text{erfc}\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_{x}^{\infty }{e}^{-{t}^{2}}dt\\ =1-\text{erf}\left(x\right).\end{array}$

It is related to the error function as

$\text{erfc}\left(x\right)=1-\text{erf}\left(x\right).$

### 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 erfc and normcdf is

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

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

• For expressions of the form exp(x^2)*erfc(x), use the scaled complementary error function erfcx instead. This substitution maintains accuracy by avoiding roundoff errors for large values of x.