# qfuncinv

Inverse Q function

## 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

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