Bi7740: Scientific computing
Introductory considerations - additional slides
Vlad Popovici
popovici@iba.muni.cz
Institute of Biostatistics and Analyses
Masaryk University, Brno
Vlad Bi7740: Scientific computing
Stirling’s approximation
1 % Script for Stirling's approximation.
2 %
3
4 s = @(n) sqrt(2*pi.*n).*(n./exp(1)).^n;
5 % ^−−− check "anonymous functions"!
6
7 x = 1:10;
8 f = factorial(x);
9 plot(x, (s(x) − f)./f);
10
11 return
Vlad Bi7740: Scientific computing
Taylor series for Euler’s number
∞
n=0
1
n!
= e
For correct 3 decimals, the tail (the rest of the sum, not computed)
must be upper bounded by 0.0005 (why?)
∞
n=1
1
n!
=
k
n=1
1
n!
+
∞
n=k+1
1
n!
=
k
n=1
1
n!
+
1
(k + 1)!
1 +
1
k + 2
+
1
(k + 2)(k + 3)
+ . . .
<
k
n=1
1
n!
+
1
(k + 1)!
1 +
1
k + 1
+
1
(k + 1)2
+ . . .
=
k
n=1
1
n!
+
1
(k + 1)!


1
1 − 1
k+1

 =
k
n=1
1
n!
+
1
k · k!
Vlad Bi7740: Scientific computing
1 % real code would check for proper n...
2 e = @(n) (1 + sum(1 ./ factorial(1:n)));
3 fprintf('%2s\t%10s\t%10s\n', 'k', 'approx', 'tail');
4 for k = 1:10
5 fprintf('%d\t%10.8f\t%10.8f\n', k, e(k), ...
1/(k*factorial(k)));
6 end
1 k approx tail
2 1 2.00000000 1.00000000
3 2 2.50000000 0.25000000
4 3 2.66666667 0.05555556
5 4 2.70833333 0.01041667
6 5 2.71666667 0.00166667
7 6 2.71805556 0.00023148 <−−−−−
8 7 2.71825397 0.00002834
9 8 2.71827877 0.00000310
10 9 2.71828153 0.00000031
11 10 2.71828180 0.00000003
Vlad Bi7740: Scientific computing