6.1 Gamma, Beta, and Related Functions 213
Hart, J.F., et al. 1968, Computer Approximations (New York: Wiley).
Hastings, C. 1955, Approximations for Digital Computers (Princeton: Princeton University Press).
Luke, Y.L. 1975, Mathematical Functions and Their Approximations (New York: Academic Press).
6.1 Gamma Function, Beta Function, Factorials,
Binomial Coefficients
The gamma function is defined by the integral
"
(z) = tz-1e-tdt (6.1.1)
0
When the argument z is an integer, the gamma function is just the familiar factorial
function, but offset by one,
n! =(n +1) (6.1.2)
The gamma function satisfies the recurrence relation
(z +1) =z(z)(6.1.3)
If the function is known for arguments z >1 or, more generally, in the half complex
plane Re(z) > 1 it can be obtained for z <1 or Re (z) < 1 by the reflection formula
Ą Ąz
(1 - z) = = (6.1.4)
(z)sin(Ąz) (1 + z)sin(Ąz)
Notice that (z) has a pole at z =0, and at all negative integer values of z.
There are a variety of methods in use for calculating the function (z)
[1]
numerically, but none is quite as neat as the approximation derived by Lanczos .
This scheme is entirely specific to the gamma function, seemingly plucked from
thin air. We will not attempt to derive the approximation, but only state the
resulting formula: For certain integer choices of ł and N, and for certain coefficients
c1, c2, . . . , cN , the gamma function is given by
1 1
1
2 2
(z +1) =(z + ł + )z+ e-(z+ł+ )
2
(6.1.5)
"
c1 c2 cN
2Ą c0 + + + + + (z >0)
z +1 z +2 z + N
You can see that this is a sort of take-off on Stirling s approximation, but with a
series of corrections that take into account the first few poles in the left complex
plane. The constant c0 is very nearly equal to 1. The error term is parametrized by .
For ł =5, N =6, and a certain set of c s, the error is smaller than | | < 2 10-10.
Impressed? If not, then perhaps you will be impressed by the fact that (with these
same parameters) the formula (6.1.5) and bound on apply for the complex gamma
function, everywhere in the half complex plane Re z > 0.
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214 Chapter 6. Special Functions
It is better to implement ln (x) than (x), since the latter will overflow many
computers floating-point representation at quite modest values of x. Often the
gamma function is used in calculations where the large values of (x) are divided by
other large numbers, with the result being a perfectly ordinary value. Such operations
would normally be coded as subtraction of logarithms. With (6.1.5) in hand, we can
compute the logarithm of the gamma function with two calls to a logarithm and 25
or so arithmetic operations. This makes it not much more difficult than other built-in
functions that we take for granted, such as sin x or ex:
#include
float gammln(float xx)
Returns the value ln[(xx)] forxx> 0.
{
Internal arithmetic will be done in double precision, a nicety that you can omit if five-figure
accuracy is good enough.
double x,y,tmp,ser;
static double cof[6]={76.18009172947146,-86.50532032941677,
24.01409824083091,-1.231739572450155,
0.1208650973866179e-2,-0.5395239384953e-5};
int j;
y=x=xx;
tmp=x+5.5;
tmp -= (x+0.5)*log(tmp);
ser=1.000000000190015;
for (j=0;j<=5;j++) ser += cof[j]/++y;
return -tmp+log(2.5066282746310005*ser/x);
}
How shall we write a routine for the factorial function n!? Generally the
factorial function will be called for small integer values (for large values it will
overflow anyway!), and in most applications the same integer value will be called for
many times. It is a profligate waste of computer time to callexp(gammln(n+1.0))
for each required factorial. Better to go back to basics, holdinggammlnin reserve
for unlikely calls:
#include
float factrl(int n)
Returns the valuen! as a floating-point number.
{
float gammln(float xx);
void nrerror(char error_text[]);
static int ntop=4;
static float a[33]={1.0,1.0,2.0,6.0,24.0}; Fill in table only as required.
int j;
if (n < 0) nrerror("Negative factorial in routine factrl");
if (n > 32) return exp(gammln(n+1.0));
Larger value than size of table is required. Actually, this big a value is going to overflow
on many computers, but no harm in trying.
while (ntopj=ntop++;
a[ntop]=a[j]*ntop;
}
return a[n];
}
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readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
6.1 Gamma, Beta, and Related Functions 215
A useful point is thatfactrlwill be exact for the smaller values ofn, since
floating-point multiplies on small integers are exact on all computers. This exactness
will not hold if we turn to the logarithm of the factorials. For binomial coefficients,
however, we must do exactly this, since the individual factorials in a binomial
coefficient will overflow long before the coefficient itself will.
The binomial coefficient is defined by
n n!
= 0 d" k d" n (6.1.6)
k k!(n - k)!
#include
float bico(int n, int k)
n
Returns the binomial coefficient as a floating-point number.
k
{
float factln(int n);
return floor(0.5+exp(factln(n)-factln(k)-factln(n-k)));
Thefloorfunction cleans up roundoff error for smaller values ofnandk.
}
which uses
float factln(int n)
Returns ln(n!).
{
float gammln(float xx);
void nrerror(char error_text[]);
static float a[101]; A static array is automatically initialized to zero.
if (n < 0) nrerror("Negative factorial in routine factln");
if (n <= 1) return 0.0;
if (n <= 100) return a[n] ? a[n] : (a[n]=gammln(n+1.0)); In range of table.
else return gammln(n+1.0); Out of range of table.
}
If your problem requires a series of related binomial coefficients, a good idea
is to use recurrence relations, for example
n +1 n +1 n n n
= = +
k n - k +1 k k k - 1
(6.1.7)
n n - k n
=
k +1 k +1 k
Finally, turning away from the combinatorial functions with integer valued
arguments, we come to the beta function,
1
B(z, w) =B(w, z) = tz-1(1 - t)w-1dt (6.1.8)
0
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readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website
Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-
Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software.
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
216 Chapter 6. Special Functions
which is related to the gamma function by
(z)(w)
B(z, w) = (6.1.9)
(z + w)
hence
#include
float beta(float z, float w)
Returns the value of the beta function B(z, w).
{
float gammln(float xx);
return exp(gammln(z)+gammln(w)-gammln(z+w));
}
CITED REFERENCES AND FURTHER READING:
Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathe-
matics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by
Dover Publications, New York), Chapter 6.
Lanczos, C. 1964, SIAM Journal on Numerical Analysis, ser. B, vol. 1, pp. 86 96. [1]
6.2 Incomplete Gamma Function, Error
Function, Chi-Square Probability Function,
Cumulative Poisson Function
The incomplete gamma function is defined by
x
ł(a, x) 1
P (a, x) a" a" e-tta-1dt (a >0) (6.2.1)
(a) (a)
0
It has the limiting values
P (a, 0) = 0 and P (a, ") =1 (6.2.2)
The incomplete gamma function P (a, x) is monotonic and (for a greater than one or
so) rises from near-zero to near-unity in a range of x centered on about a - 1,
"
and of width about a (see Figure 6.2.1).
The complement of P (a, x) is also confusingly called an incomplete gamma
function,
"
(a, x) 1
Q(a, x) a" 1 - P (a, x) a" a" e-tta-1dt (a >0) (6.2.3)
(a) (a)
x
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readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website
Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-
Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software.
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
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