NAG Library Routine Document

g01tcf (inv_cdf_chisq_vector)


    1  Purpose
    7  Accuracy


g01tcf returns a number of deviates associated with the given probabilities of the χ2-distribution with real degrees of freedom.


Fortran Interface
Subroutine g01tcf ( ltail, tail, lp, p, ldf, df, x, ivalid, ifail)
Integer, Intent (In):: ltail, lp, ldf
Integer, Intent (Inout):: ifail
Integer, Intent (Out):: ivalid(*)
Real (Kind=nag_wp), Intent (In):: p(lp), df(ldf)
Real (Kind=nag_wp), Intent (Out):: x(*)
Character (1), Intent (In):: tail(ltail)
C Header Interface
#include nagmk26.h
void  g01tcf_ (const Integer *ltail, const char tail[], const Integer *lp, const double p[], const Integer *ldf, const double df[], double x[], Integer ivalid[], Integer *ifail, const Charlen length_tail)


The deviate, xpi, associated with the lower tail probability pi of the χ2-distribution with νi degrees of freedom is defined as the solution to
P Xi xpi :νi = pi = 1 2 νi/2 Γ νi/2 0 xpi e -Xi/2 Xi vi / 2 - 1 dXi ,   0 xpi < ; ​ νi > 0 .  
The required xpi is found by using the relationship between a χ2-distribution and a gamma distribution, i.e., a χ2-distribution with νi degrees of freedom is equal to a gamma distribution with scale parameter 2 and shape parameter νi/2.
For very large values of νi, greater than 105, Wilson and Hilferty's Normal approximation to the χ2 is used; see Kendall and Stuart (1969).
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.


Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2 distribution Appl. Statist. 24 385–388
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin


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. For j= i-1 mod ltail +1 , for i=1,2,,maxltail,lp,ldf:
The lower tail probability, i.e., pi = P Xi xpi :νi .
The upper tail probability, i.e., pi = P Xi xpi :νi .
Constraint: tailj='L' or 'U', 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 probability of the required χ2-distribution as defined by tail with pi=pj, j=i-1 mod lp+1.
  • if tailk='L', 0.0pj<1.0;
  • otherwise 0.0<pj1.0.
Where k=i-1 mod ltail+1 and j=i-1 mod lp+1.
5:     ldf – IntegerInput
On entry: the length of the array df.
Constraint: ldf>0.
6:     dfldf – Real (Kind=nag_wp) arrayInput
On entry: νi, the degrees of freedom of the χ2-distribution with νi=dfj, j=i-1 mod ldf+1.
Constraint: dfj>0.0, for j=1,2,,ldf.
7:     x* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array x must be at least maxltail,lp,ldf.
On exit: xpi, the deviates for the χ2-distribution.
8:     ivalid* – Integer arrayOutput
Note: the dimension of the array ivalid must be at least maxltail,lp,ldf.
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,invalid value for pi.
On entry,νi0.0.
pi is too close to 0.0 or 1.0 for the result to be calculated.
The solution has failed to converge. The result should be a reasonable approximation.
9:     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, p or df was invalid, or the solution failed to converge.
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: ldf>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 results should be accurate to five significant digits for most argument values. Some accuracy is lost for pi close to 0.0 or 1.0.

Parallelism and Performance

g01tcf is not threaded in any implementation.

Further Comments

For higher accuracy the relationship described in Section 3 may be used and a direct call to g01tff made.


This example reads lower tail probabilities for several χ2-distributions, and calculates and prints the corresponding deviates.

Program Text

Program Text (g01tcfe.f90)

Program Data

Program Data (g01tcfe.d)

Program Results

Program Results (g01tcfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017