(a)If $| q | \leq 0.3$ , $x_{p}$ is computed by a rational Chebyshev approximation

$x_{p} = s \frac{A (s^{2})}{B (s^{2})},$

where $s = \sqrt{2 π} q$ and $A$ , $B$ are polynomials of degree $7$ .
(b)If $0.3 < | q | \leq 0.42$ , $x_{p}$ is computed by a rational Chebyshev approximation

$x_{p} = sign q (\frac{C (t)}{D (t)}),$

where $t = | q | - 0.3$ and $C$ , $D$ are polynomials of degree $5$ .
(c)If $| q | > 0.42$ , $x_{p}$ is computed as

$x_{p} = sign q [(\frac{E (u)}{F (u)}) + u],$

where $u = \sqrt{−2 \times \log (\min (p, 1 - p))}$ and $E$ , $F$ are polynomials of degree $6$ .

For the upper tail probability

- x_{p}

is returned, while for the two tail probabilities the value

x_{p^{*}}

is returned, where

p^{*}

is the required tail probability computed from the input value of

p

4 References

NIST Digital Library of Mathematical Functions

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Wichura (1988) Algorithm AS 241: the percentage points of the Normal distribution Appl. Statist. 37 477–484

5 Arguments

1: $tail$ – Character(1) Input

On entry: indicates which tail the supplied probability represents.

$tail ='L'$: The lower probability, i.e., $P (X \leq x_{p})$ .
$tail ='U'$: The upper probability, i.e., $P (X \geq x_{p})$ .
$tail ='S'$: The two tail (significance level) probability, i.e., $P (X \geq | x_{p} |) + P (X \leq - | x_{p} |)$ .
$tail ='C'$: The two tail (confidence interval) probability, i.e., $P (X \leq | x_{p} |) - P (X \leq - | x_{p} |)$ .

Constraint:

tail ='L'

'U'

'S'

'C'

2: $p$ – Real (Kind=nag_wp) Input

On entry:

p

, the probability from the standard Normal distribution as defined by tail.

Constraint:

0.0 < p < 1.0

3: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

If on exit

ifail \neq 0

, then g01faf returns

0.0

$ifail = 1$: On entry, $tail = ⟨ value ⟩$ .
Constraint: $tail ='L'$ , $'U'$ , $'S'$ or $'C'$ .

$ifail = 2$: On entry, $p = ⟨ value ⟩$ .
Constraint: $p < 1.0$ .

On entry, $p = ⟨ value ⟩$ .
Constraint: $p > 0.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The accuracy is mainly limited by the machine precision.

8 Parallelism and Performance

g01faf is not threaded in any implementation.

9 Further Comments

None.

10 Example

Four values of tail and p are input and the deviates calculated and printed.

g01fa: FL CL CPP AD

NAG FL Interfaceg01faf (inv_​cdf_​normal)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g01faf (inv_cdf_normal)