g01ff returns the deviate associated with the given lower tail probability of the gamma distribution.


public static double g01ff(
	double p,
	double a,
	double b,
	double tol,
	out int ifail
Visual Basic
Public Shared Function g01ff ( _
	p As Double, _
	a As Double, _
	b As Double, _
	tol As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double g01ff(
	double p, 
	double a, 
	double b, 
	double tol, 
	[OutAttribute] int% ifail
static member g01ff : 
        p : float * 
        a : float * 
        b : float * 
        tol : float * 
        ifail : int byref -> float 


Type: System..::..Double
On entry: p, the lower tail probability from the required gamma distribution.
Constraint: 0.0p<1.0.
Type: System..::..Double
On entry: α, the shape parameter of the gamma distribution.
Constraint: 0.0<a106.
Type: System..::..Double
On entry: β, the scale parameter of the gamma distribution.
Constraint: b>0.0.
Type: System..::..Double
On entry: the relative accuracy required by you in the results. The smallest recommended value is 50×δ, where δ=max10-18,machine precision. If g01ff is entered with tol less than 50×δ or greater or equal to 1.0, then 50×δ is used instead.
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01ff returns the deviate associated with the given lower tail probability of the gamma distribution.


The deviate, gp, associated with the lower tail probability, p, of the gamma distribution with shape parameter α and scale parameter β, is defined as the solution to
PGgp:α,β=p=1βαΓα0gpe-G/βGα-1dG,  0gp<;α,β>0.
The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the χ2-distribution.
Let y=2gpβ. The required y is found from the Taylor series expansion
where y0 is a starting approximation
  • C1u=1,
  • Cr+1u=rΨ+dduCru,
  • Ψ=12-α-1u,
  • E=p-0y0ϕudu,
  • ϕu=12αΓαe-u/2uα-1.
For most values of p and α the starting value
is used, where z is the deviate associated with a lower tail probability of p for the standard Normal distribution.
For p close to zero,
is used.
For large p values, when y01>4.4α+6.0,
is found to be a better starting value than y01.
For small α α0.16, p is expressed in terms of an approximation to the exponential integral and y04 is found by Newton–Raphson iterations.
Seven terms of the Taylor series are used to refine the starting approximation, repeating the process if necessary until the required accuracy is obtained.


Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2 distribution Appl. Statist. 24 385–388

Error Indicators and Warnings

Note: g01ff may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
If on exit ifail=123 or 5, then g01ff returns 0.0.
On entry,p<0.0,
On entry,a0.0,
p is too close to 0.0 or 1.0 to enable the result to be calculated.
The solution has failed to converge in 100 iterations. A larger value of tol should be tried. The result may be a reasonable approximation.
The series to calculate the gamma function has failed to converge. This is an unlikely error exit.
An error occured, see message report.


In most cases the relative accuracy of the results should be as specified by tol. However, for very small values of α or very small values of p there may be some loss of accuracy.

Parallelism and Performance


Further Comments



This example reads lower tail probabilities for several gamma distributions, and calculates and prints the corresponding deviates until the end of data is reached.

Example program (C#): g01ffe.cs

Example program data: g01ffe.d

Example program results: g01ffe.r

See Also