(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: $tail$ – Nag_TailProbability Input

On entry: indicates which tail the supplied probability represents.

$tail = Nag_LowerTail$: The lower probability, i.e., $P (X \leq x_{p})$ .
$tail = Nag_UpperTail$: The upper probability, i.e., $P (X \geq x_{p})$ .
$tail = Nag_TwoTailSignif$: The two tail (significance level) probability, i.e., $P (X \geq | x_{p} |) + P (X \leq - | x_{p} |)$ .
$tail = Nag_TwoTailConfid$: The two tail (confidence interval) probability, i.e., $P (X \leq | x_{p} |) - P (X \leq - | x_{p} |)$ .

Constraint:

tail = Nag_LowerTail

Nag_UpperTail

Nag_TwoTailSignif

Nag_TwoTailConfid

2: $p$ – double Input

On entry:

p

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

Constraint:

0.0 < p < 1.0

3: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If on exit

fail . code =

NE_NOERROR, then g01fac returns

0.0

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARG_GE: On entry, $p = ⟨ value ⟩$ .
Constraint: $p < 1.0$ .
NE_REAL_ARG_LE: On entry, $p = ⟨ value ⟩$ .
Constraint: $p > 0.0$ .

7 Accuracy

The accuracy is mainly limited by the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g01fac 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 PY MB

NAG CL Interfaceg01fac (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 CL Interface
g01fac (inv_cdf_normal)