# qfuncinv

Inverse Q function

## Syntax

``z = qfuncinv(y)``

## Description

example

````z = qfuncinv(y)` returns the input argument of the Q function for which the output value of the Q function is `y`. For more information, see Algorithms.```

## Examples

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Recover the Q function input argument by using the inverse Q function. Show the inverse relationship between Q function and its inverse.

Calculate the Q function values for a real-valued input.

```x1 = [0 1 2; 3 4 5]; y1 = qfunc(x1)```
```y1 = 2×3 0.5000 0.1587 0.0228 0.0013 0.0000 0.0000 ```

Recover the Q function input argument by calculating the inverse Q function values for `y1`.

`x1_recovered = qfuncinv(y1)`
```x1_recovered = 2×3 0 1 2 3 4 5 ```

Confirm the original and recovered Q functions arguments are the same.

`isequal (x1,x1_recovered)`
```ans = logical 1 ```

Calculate the inverse of values representing Q function output values.

```y2 = 0:0.2:1; x2 = qfuncinv(y2)```
```x2 = 1×6 Inf 0.8416 0.2533 -0.2533 -0.8416 -Inf ```

Recover the Q function output argument by calculating the Q function values for `x2`.

`y2_recovered = qfunc(x2)`
```y2_recovered = 1×6 0 0.2000 0.4000 0.6000 0.8000 1.0000 ```

Confirm the original values and recovered inverse Q functions arguments are the same.

`isequal (y2,y2_recovered)`
```ans = logical 1 ```

## Input Arguments

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Q function output, specified as a scalar, matrix, or array. Input values must be in the range [0, 1].

Data Types: `double`

## Output Arguments

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Q function input argument, returned as a real-valued scalar, matrix, or array. `z` has the same dimensions as input `y`.

## Algorithms

For a scalar x, the Q function is (1 – f), where f is the result of the cumulative distribution function of the standardized normal random variable. The Q function is defined as

`$Q\left(x\right)=\frac{1}{\sqrt{2\pi }}\underset{x}{\overset{\infty }{\int }}\mathrm{exp}\left(-{t}^{2}/2\right)dt$`

The Q function is related to the complementary error function, erfc, according to

`$Q\left(x\right)=\frac{1}{2}\text{erfc}\left(\frac{x}{\sqrt{2}}\right)$`

## Version History

Introduced before R2006a