NAG Library Routine Document

g01taf (inv_cdf_normal_vector)


g01taf returns a number of deviates associated with given probabilities of the Normal distribution.


Fortran Interface
Subroutine g01taf ( ltail, tail, lp, p, lxmu, xmu, lxstd, xstd, x, ivalid, ifail)
Integer, Intent (In):: ltail, lp, lxmu, lxstd
Integer, Intent (Inout):: ifail
Integer, Intent (Out):: ivalid(*)
Real (Kind=nag_wp), Intent (In):: p(lp), xmu(lxmu), xstd(lxstd)
Real (Kind=nag_wp), Intent (Out):: x(*)
Character (1), Intent (In):: tail(ltail)
C Header Interface
#include <nagmk26.h>
void  g01taf_ (const Integer *ltail, const char tail[], const Integer *lp, const double p[], const Integer *lxmu, const double xmu[], const Integer *lxstd, const double xstd[], double x[], Integer ivalid[], Integer *ifail, const Charlen length_tail)


The deviate, xpi associated with the lower tail probability, pi, for the Normal distribution is defined as the solution to
ZiXi=12πσi2e-Xi-μi2/2σi2, ​-<Xi< .  
The method used is an extension of that of Wichura (1988). pi is first replaced by qi=pi-0.5.
(a) If qi0.3, zi is computed by a rational Chebyshev approximation
zi=siAisi2 Bisi2 ,  
where si=2πqi and Ai, Bi are polynomials of degree 7.
(b) If 0.3<qi0.42, zi is computed by a rational Chebyshev approximation
zi=signqi Citi Diti ,  
where ti=qi-0.3 and Ci, Di are polynomials of degree 5.
(c) If qi>0.42, zi is computed as
zi=signqi Eiui Fiui +ui ,  
where ui = -2 × log minpi,1-pi  and Ei, Fi are polynomials of degree 6.
xpi is then calculated from zi, using the relationsship zpi = xi - μi σi .
For the upper tail probability -xpi is returned, while for the two tail probabilities the value xipi* is returned, where pi* is the required tail probability computed from the input value of pi.
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.


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


1:     ltail – IntegerInput
On entry: the length of the array tail.
Constraint: ltail>0.
2:     tailltail – Character(1) arrayInput
On entry: indicates which tail the supplied probabilities represent. Letting Z denote a variate from a standard Normal distribution, and zi = xpi - μi σi , then for j= i-1 mod ltail +1 , for i=1,2,,maxltail,lp,lxmu,lxstd:
The lower tail probability, i.e., pi=PZzi.
The upper tail probability, i.e., pi=PZzi.
The two tail (confidence interval) probability, i.e., pi=PZzi-PZ-zi.
The two tail (significance level) probability, i.e., pi=PZzi+PZ-zi.
Constraint: tailj='L', 'U', 'C' or 'S', for j=1,2,,ltail.
3:     lp – IntegerInput
On entry: the length of the array p.
Constraint: lp>0.
4:     plp – Real (Kind=nag_wp) arrayInput
On entry: pi, the probabilities for the Normal distribution as defined by tail with pi=pj, j=i-1 mod lp+1.
Constraint: 0.0<pj<1.0, for j=1,2,,lp.
5:     lxmu – IntegerInput
On entry: the length of the array xmu.
Constraint: lxmu>0.
6:     xmulxmu – Real (Kind=nag_wp) arrayInput
On entry: μi, the means with μi=xmuj, j=i-1 mod lxmu+1.
7:     lxstd – IntegerInput
On entry: the length of the array xstd.
Constraint: lxstd>0.
8:     xstdlxstd – Real (Kind=nag_wp) arrayInput
On entry: σi, the standard deviations with σi=xstdj, j=i-1 mod lxstd+1.
Constraint: xstdj>0.0, for j=1,2,,lxstd.
9:     x* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array x must be at least maxltail,lxmu,lxstd,lp.
On exit: xpi, the deviates for the Normal distribution.
10:   ivalid* – Integer arrayOutput
Note: the dimension of the array ivalid must be at least maxltail,lxmu,lxstd,lp.
On exit: ivalidi indicates any errors with the input arguments, with
No error.
On entry,invalid value supplied in tail when calculating xpi.
On entry,pi0.0,
On entry,σi0.0.
11:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. 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).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, at least one value of tail, xstd or p was invalid.
Check ivalid for more information.
On entry, array size=value.
Constraint: ltail>0.
On entry, array size=value.
Constraint: lp>0.
On entry, array size=value.
Constraint: lxmu>0.
On entry, array size=value.
Constraint: lxstd>0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


The accuracy is mainly limited by the machine precision.

Parallelism and Performance

g01taf is not threaded in any implementation.

Further Comments



This example reads vectors of values for μi, σi and pi and prints the corresponding deviates.

Program Text

Program Text (g01tafe.f90)

Program Data

Program Data (g01tafe.d)

Program Results

Program Results (g01tafe.r)