erfcx

Scaled complementary error function

Syntax

erfcx(x)

Description

erfcx(x) returns the value of the Scaled Complementary Error Function for each element of x. Use the erfcx function to replace expressions containing exp(x^2)*erfc(x) to avoid underflow or overflow errors.

Examples

Find Scaled Complementary Error Function

erfcx(5)

ans = 
0.1107

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

V = [-Inf -1 0 1 10 Inf];
erfcx(V)

ans = 1×6

       Inf    5.0090    1.0000    0.4276    0.0561         0

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

M = [-0.5 15; 3.2 1];
erfcx(M)

ans = 2×2

    1.9524    0.0375
    0.1687    0.4276

Avoid Round-off Errors Using Scaled Complementary Error Function

You can use the scaled complementary error function erfcx in place of exp(x^2)*erfc(x) to avoid underflow or overflow errors.

Show how to avoid round-off errors by calculating exp(35^2)*erfc(35) using erfcx(35). The original calculation returns NaN while erfcx(35) returns the correct result.

x = 35;
exp(x^2)*erfc(x)

ans = 
NaN

erfcx(x)

ans = 
0.0161

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

Scaled Complementary Error Function

The scaled complementary error function erfcx(x) is defined as

$$\text{erfcx}(x)=e^{x^2}\text{erfc}(x).$$

For large X, erfcx(X) is approximately $$\left(\frac{1}{\sqrt{\pi }}\right)\frac{1}{x}.$$

Tips

• For expressions of the form exp(-x^2)*erfcx(x), use the complementary error function erfc 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 erfcx 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 erfcx 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