The Museum of HP Calculators

Bernoulli Numbers for the HP-41

This program is supplied without representation or warranty of any kind. Jean-Marc Baillard and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.

Overview

-The Bernoulli numbers could be computed by the relations:

B(0) = 1 ; B(0) + C_n+1¹ B(1) + C_n+1² B(2) + ...... + C_n+1ⁿ B(n) = 0 where C_n^k = n!/(k!(n-k)!) are the binomial coefficients

-However, this recurrence relation is unstable and the results are quite inaccurate for large n ( for n = 20 , only 4 digits are correct! )
-The following program uses a series expansion instead:

B(n) = (-1)^-1+n/2 2n!/(2pi)ⁿ ( 1/1ⁿ + 1/2ⁿ + ...... + 1/kⁿ + ...... ) if n is even and B(0) = 1 ; B(1) = -1/2 ; B(2n+1) = 0 if n > 0

-Actually, B(2) = 1/6 ; B(4) = -1/30 ; B(6) = 1/42 are given directly ( lines 32 to 39 may be deleted without a great loss of speed )

Program listing

Data Registers: R00 to R02: temp
Flags: /
Subroutine: "ZETA" ( cf "Miscellaneous Functions for the HP-41" )

01 LBL "BERN"
02 STO 02
03 1
04 X>Y?
05 RTN
06 ST+ X
07 X<=Y?
08 GTO 00
09 1/X
10 CHS
11 RTN
12 LBL 00
13 X#Y?
14 GTO 00
15 6
16 1/X
17 RTN
18 LBL 00
19 MOD
20 0
21 X#Y?
22 RTN
23 4
24 RCL 02
25 X#Y?
26 GTO 00
27 30
28 1/X
29 CHS
30 RTN
31 LBL 00
32 6
33 X#Y?
34 GTO 00
35 42
36 1/X
37 RTN
38 LBL 00
39 X<>Y
40 FIX 9
41 XEQ "ZETA"
42 ST+ X
43 1
44 CHS
45 RCL 02
46 2
47 /
48 Y^X
49 *
50 CHS
51 PI
52 ST+ X
53 E-9
54 -
55 RCL 02
56 Y^X
57 /
58 LBL 01
59 RCL 02
60 *
61 DSE 02
62 GTO 01
63 END

( 91 bytes / SIZE 003 )

STACK INPUTS OUTPUTS

X n B(n)

Example: 116 XEQ "BERN" gives B(116) = -1.748892190 10⁹⁸ ( in 24 seconds )

References: John H. Conway & Richard K. Guy , "The Book of Numbers" - Springer Verlag - ISBN 0-387-97993-X
Abramowitz and Stegun , "Handbook of Mathematical Functions" - Dover Publications - ISBN 0-486-61272-4

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