erfc

Complementary error function

Syntax

erfc(x)

Description

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

Find Complementary Error Function

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 = 1×4

    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 = 2×2

    0.6817    1.1236
    0.0000    2.0000

Find Bit Error Rate of Binary Phase-Shift Keying

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

$$P_b=\frac{1}{2}erfc\left(\sqrt{\frac{E_b}{N_0}}\right).$$

Plot the BER for BPSK for values of $$E_b/N_0$$ 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')

Avoid Round-off Errors Using Complementary Error Function

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

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

1 - erf(10)

ans = 
0

erfc(10)

ans = 
2.0885e-45

Input Arguments

x — Input

real 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

Complementary Error Function

The complementary error function of x is defined as

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

It is related to the error function as

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

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 erfc and normcdf is

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

• 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.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

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

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Strict single-precision calculations are not supported. In the generated code, single-precision inputs produce single-precision outputs. However, variables inside the function might be double-precision.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The erfc function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a