Proceedings of the 8th WSEAS International Conference on EDUCATION and EDUCATIONAL TECHNOLOGY
On approximations to the factorial
GIANFRANCO CHICCO
Dipartimento di Ingegneria Elettrica
Politecnico di Torino
Corso Duca degli Abruzzi 24, 10129 Torino
ITALY
[email protected]
http://www.polito.it
Abstract: - Analytic formulations for approximating the factorial can be helpful to be used in continuous problems. The DeMoivre-Stirling’s formula is the most classical one. However, the literature has provided some examples of continuous
functions, that generally can be structured as products of the De Moivre-Stirling’s formula times a correction function, thus
investigating the characteristics of the correction function to check whether it is possible to get a better representation of the
factorial for natural numbers. This paper provides a tutorial illustration of the characteristics of various approximations to the
factorial, as well as the proposal of a novel continuous function, included within the comparisons. This function has relative
approximation errors with respect to the factorial very low and always positive for any natural number. These characteristics
enable the novel function to be used as an upper bound to the factorial in continuous problems.
Key-Words: - Factorial, Stirling, approximation, asymptotic convergence, correction function, relative approximation error.
1 Introduction
The factorial of a natural number n ∈ ℵ can be written by
using the Gamma function [1] as n ! = Γ(n + 1) or, in the
integral form,
n! =
∫
∞
x n e − x dx
(1)
0
However, the integral formulation makes Equation (1)
difficult to handle. Simple continuous explicit functions
have then been proposed for an easier approximation to the
factorial. A typical approximation is the well-known De
Moivre-Stirling’s formula [2,3], often called in short
Stirling’s formula, expressed (using a natural number
argument n ∈ ℵ) as
s (n ) = n
n+
1
2
e − n 2π
2 Approximations to the factorial
Different approximations to the factorial by using explicit
continuous functions have been proposed. Some of them
can be written by multiplying the Stirling’s formula to a
correction function ζ (n ) , such that
ψ (n ) = ζ (n ) s(n )
(3)
A common formulation uses as corrective function the
first terms of the asymptotic series for the Gamma function
[16-18]. Using the first two terms of the series, the
correction function becomes
(2)
Proofs of the asymptotic convergence of the Stirling’s
formula to n! for n → ∞ and related discussions have been
presented in several references [3-15]. As such, the
Stirling’s formula is typically used for approximating the
factorial n! for very large values of n. However, its accuracy
for relatively low values of n is limited and can be further
improved also for high values of n.
This paper recalls different approximations to the
factorial leading to lower approximation errors than the
Stirling’s formula for small and large values of n and
compare their behavior for high and relatively low values of
n. An original analytical formula with intermediate
ISSN: 1790-5109
characteristics is proposed and included within the
comparisons.
⎛
ζ b (n ) = ⎜⎜1 +
⎝
1 ⎞
⎟
b n ⎟⎠
(4)
with b = 12. Another formulation uses an exponential
correction function [5,9,19,20]:
ζ e (n ) = eτ (n )
(5)
The exponential correction function has been used by
Robbins [5] to introduce suitable expressions to be used as
lower and upper bounds to the factorial, by respectively
using τ l (n ) =
273
1
1
and τ u (n ) =
, such that:
12 n + 1
12 n
ISBN: 978-960-474-128-1
Proceedings of the 8th WSEAS International Conference on EDUCATION and EDUCATIONAL TECHNOLOGY
s (n ) τ l (n ) < n !< s (n ) τ u (n )
A variant to the lower boundary, τ l (n ) =
for which all RAE values are nonnegative. For b > b* the
RAE becomes negative for low values of n.
Results of a more detailed analysis are shown in Fig. 3,
where the maximum RAE is reported for different values of
n in function of the parameter b. Fig. 3 shows that only the
cases n = 1, n = 4 and n = 5 are involved in defining the
maximum RAE as the parameter b changes. Let’s then
compute the minimum value of the maximum RAE:
(6)
1
3
12 n +
2 (2n + 1)
,
has been successively proposed by Maria [20].
Other approximations are built by modifying the
structure of the Stirling’s formula. An example is
g (n ) = n n e − n 2π n + q
n
f (n ) − n !
n!
(8)
Fig. 1 shows the RAE values reached by using the
Stirling’s formula and other approximations introduced
through Equations (3)-(4), Equation (5) with upper bound
from [5] and lower bounds from [5] and [20], and Equation
(7) with q = π/3. The results easily confirm that the
approximated formulas exhibit much better characteristics
and lower RAE than the Stirling’s formula, especially for
low values of n. However, in all these cases the maximum
RAE occurs for n = 1 and is significantly higher than the
errors reached for higher values of n.
The following section introduces and illustrates the
characteristics of a novel continuous formula that improves
the approximation accuracy in terms of limiting to a very
low value the maximum RAE over the entire range of
numbers n ∈ ℵ.
]
[
(10)
n−
1
2
e −n
(11)
The formulation of Equation (11) merges the simplicity
of representation with a very low value of the maximum
RAE and with RAE values always nonnegative for n ∈ ℵ.
The latter property allows for using Equation (11) as an
upper bound of the factorial for any n ∈ ℵ.
The asymptotic convergence of ν(n) to n! for n → ∞ is
guaranteed by the fact that ζ b (n ) tends to unity for n → ∞
and by the existing proof of convergence of the Stirling’s
formula to n! for n → ∞, such that
lim [ν (n )] = lim [s(n )] = n !
3 A novel continuous approximation to the
factorial
n→∞
(12)
4 Conclusions
Among the various continuous approximations to the
factorial, this paper has presented a novel simple formula
leading to an improved approximation to the factorial by
using an explicit continuous function. This formula shares
the asymptotic properties of the classical Stirling’s formula
and exhibits excellent performance for approximating the
factorial in the entire range of the natural numbers. Its
superiority with respect to the Stirling’s formula has been
shown with numerical evidence.
The relative approximation error reached with the proposed
formula in the entire range of the natural numbers is always
nonnegative and its maximum value (about 0.01%) is lower
than the one reached by using other common continuous
explicit functions as approximations to the factorial.
Let us consider Equation (4), for which the maximum RAE
is relatively low, as shown in Fig. 1, and let’s write the
corresponding approximation function as
(9)
A parametric study carried out by changing the (real)
value of the parameter b showed that the RAE is very
sensitive to the value of b. Fig. 2 indicates some results. In
2π
= 11.843, the RAE is
the particular case b = b * =
e − 2π
null for n = 1 and the corresponding maximum RAE is
0.0102% (for n = 5). This case corresponds to the limit case
ISSN: 1790-5109
}}
{
ν (n ) = e + (n − 1) 2π a
n→∞
ψ b (n ) = ζ b (n ) s(n )
b
%
For n ∈ ℵ, ε min
= 0.0085% occurs for n = 5 and b = b′ =
11.855. However, in this case the RAE values are negative
for n < 3 and positive for n ≥ 3 (Fig. 2).
Let us focus on the case with parameter b = b*, for
which the RAE is always nonnegative and the maximum
RAE is relatively close to its minimum value. By using
Equation (2) and Equation (4) with b = b* and substituting
into Equation (9), it is possible to represent the
approximation to the factorial in a simple form, for n ∈ ℵ:
corresponding for q = π/3 to the Gosper’s approximation
reported in [21].
In order to evaluate the accuracy of the approximation to
the factorial by using a function f(n), let’s define the relative
approximation error (RAE), expressed in percent, as
ε % (n, f (n )) = 100
{
%
ε min
= min max ε % (n,ψ b (n ))
(7)
274
ISBN: 978-960-474-128-1
Proceedings of the 8th WSEAS International Conference on EDUCATION and EDUCATIONAL TECHNOLOGY
Exponential correction (upper bound from [5])
Exponential correction (lower bound from [20])
approximation Eq. (4) with b = 12
Exponential correction (lower bound from [5])
Gosper's approximation [21]
Stirling's formula
relative approximation error (%)
0.3
0.2
0.1
0
0
5
10
15
20
25
30
35
40
-0.1
-0.2
-0.3
-0.4
-0.5
natural number
Fig. 1. Relative approximation errors for the Stirling’s formula and other approximations to the factorial.
0.1
relative approximation error (%)
bb = 11.700
11.700
0.08
*
11.843(proposed)
(proposed)
bb*
==11.843
0.06
b' = 11.855
0.04
bb == 12.000
12.000
0.02
0
-10
-0.02
0
10
20
30
40
-0.04
-0.06
-0.08
-0.1
natural number
Fig. 2. Relative approximation errors for different values of the parameter b.
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275
ISBN: 978-960-474-128-1
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Proceedings of the 8th WSEAS International Conference on EDUCATION and EDUCATIONAL TECHNOLOGY
absolute value of the relative
approximation error (%)
0.025
0.02
maximum error
0.015
0.01
n =6
n =7
n =5
n =4
0.005
n =1
n =3
n =2
0
11.81
11.82
11.83
11.84
11.85
11.86
11.87
parameter b
Fig. 3. Absolute values of the relative approximation error.
[12] N.G. de Bruijn, Asymptotic Methods in Analysis,
Dover, New York, 1981.
[13] P. Diaconis and D. Freedman, An Elementary Proof of
Stirling's Formula, American Math. Monthly, Vol. 93,
1986, pp. 123-125.
[14] C.R. Blyth and P.K. Pathak, A Note on Easy Proofs of
Stirling's Theorem, American Math. Monthly, Vol. 93,
1986, pp. 376-379.
[15] J.M. Patin, A Very Short Proof of Stirling's Formula,
American Math. Monthly, Vol. 96, 1989, 41-42.
[16] M. Abramowitz and I.A. Stegun (editors), Handbook
of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, Dover, New York, 1974.
[17] V. Namias, A Simple Derivation of Stirling's
Asymptotic Series, American Math. Monthly, Vol. 93,
1986, pp. 25-29.
[18] G. Marsaglia and J.C.W. Marsaglia, A New Derivation
of Stirling's Approximation to n!, American Math.
Monthly, Vol. 97, 1990, pp. 826-829.
[19] T.S. Nanjundiah, Note on Stirling's Formula, American
Math. Monthly, Vol. 66, 1959, 701-703.
[20] A.J. Maria, A Remark on Stirling's Formula, American
Math. Monthly, Vol. 72, 1965, pp. 1096-1098.
[21] E.W. Weisstein, Stirling’s Approximation, Web site
http://mathworld.wolfram.com/StirlingsApproximation
.html.
References:
[1] D. Wells, The Penguin Dictionary of Curious and
Interesting Numbers, Penguin Books, Middlesex,
England, 1986, p. 45.
[2] A. de Moivre, Miscellanea Analytica de Seriebus et
Quadraturis, London, 1730.
[3] J. Stirling, Methodus Differentialis, London, 1730.
[4] M.I. Aissen, Some remarks on Stirling formula,
American Math. Monthly, Vol. 61, 1954, pp. 687-691.
[5] H. Robbins, A Remark on Stirling’s Formula,
American Math. Monthly, Vol. 62, 1955, pp. 26-29.
[6] E.T. Whittaker and G. Robinson, Stirling’s
Approximation to the Factorial, in The Calculus of
Observations: A Treatise on Numerical Mathematics,
Dover, New York, 1967, pp. 138-140.
[7] W. Feller, A Direct Proof of Stirling's Formula,
American Math. Monthly, Vol. 74, 1967, pp. 12231225.
[8] W. Feller, Correction to "A Direct Proof of Stirling's
Formula", American Math. Monthly, Vol. 75, 1968, p.
518.
[9] W. Feller, Stirling’s Formula, in An Introduction to
Probability Theory and its Applications, Vol.1, Wiley,
New York, 1968, pp. 50-53.
[10] R.A. Khan, A Probabilistic Proof of Stirling's Formula,
American Math. Monthly, Vol. 81, 1974, pp. 366-369.
[11] M. Woodrofe, Probability Theory with Applications,
McGraw-Hill, New York, 1975, pp. 127-128.
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ISBN: 978-960-474-128-1