# erfcx

Scaled complementary error function

## Description

example

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

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### 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 =

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 =

1.9524    0.0375
0.1687    0.4276

### Avoid Roundoff 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 roundoff 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)
erfcx(x)
ans =

NaN

ans =

0.0161

## Input Arguments

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### x — Inputreal 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

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### Scaled Complementary Error Function

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

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

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.