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Bernoulli numbers and large factorials - Dieter - 02-09-2014 04:59 PM
In December Namir posted a program that evaluates the Bernoulli numbers on the HP-41. It can be found in the HP-41C Software Library. In January I suggested another version based on the same approach:
\( B_n = 2 n! {(2\pi)^{-n}} \sum\limits_{i=1}^{\infty}i^{-n} \)
For large n the sum rapidly approaches 1 so that at some point it may be omitted. On the other hand, for n as low as 10 only a few terms are required for the usual 10 or 12 digit precision. \(B_{0...8}\), which would require a substantial number of terms, may be given directly. So this is the easy part.
Both Namir's and my solution suffer from a limitation due to overflow in the factorial function. For HP's 10-digit calculators with working range up to 9,999...E99 the limit is 69!, while most 12-digit devices with their upper limit at 9.999...E499 will work up to 253!. However, this is significantly less than the largest possible n that returns a Bernoulli number within the working range. Here, results up to \(B_{116}\) resp. \(B_{372}\) would be possible.
There is an elegant way to overcome this problem if the calculator offers a permutation function (nPr). The essential idea is to split n! into two factors a and b which both fall within the calculator's working range. If factorials up to 253! are possible (the largest value below 9,999...E499) this can be done as follows:
\( \begin{align*}
n! &= \frac{n!}{253!} · 253!\\
&= \frac{n!}{(n - (n - 253))!} · 253!\\
&= Perm(n, n - 253) · 253!
\end{align*} \)
If n is less than 253, this constant is simply replaced with 0:
\( Perm(n, n - 0) · 0! = n! · 1 = n!\)
This method may also be written as follows:
let r = max(0, n - 253)
let a = Perm(n, r)
let b = (n-r)!
Then n! = a · b
This way the factor \(2 n! {(2\pi)^{-n}}\) can be evaluated as \(2 a {(2\pi)^{-n} b}\) to avoid overflow. On calculators with a limit at 9,999...E99, for instance the 15C, simply use 69 instead of 253.
Here is a complete program for the HP-35s. It evaluates all Bernoulli numbers within the working rage, i.e. from \(B_0\) to \(B_{372}\).
Code:
B001 LBL B
B002 ABS
B003 IP
B004 STO N
B005 1
B006 x>y?
B007 RTN ' n=0: B = 1
B008 RCL N
B009 x≠y?
B010 GTO B015
B011 +
B012 1/x
B013 +/-
B014 RTN ' n=1: B = -1/2
B015 2
B016 RMDR
B017 -
B018 x=0?
B019 RTN ' n is odd: B = 0
B020 9
B021 RCL N
B022 x>y?
B023 GTO B035
B024 RCL N
B025 6
B026 -
B027 x²
B028 3
B029 x
B030 42
B031 -
B032 ABS ' n = 2, 4, 6, 8:
B033 1/x ' |B| = 1/6, 1/30, 1/42, 1/6
B034 GTO B076
B035 RCL N ' n ≥ 10: use Bernoulli formula
B036 253
B037 -
B038 x<0?
B039 CLx
B040 nPr
B041 RCL N
B042 LASTx
B043 -
B044 !
B045 π
B046 ENTER
B047 +
B048 RCL N
B049 +/-
B050 y^x
B051 5E-14 ' adjust (2pi)^-n
B052 RCLx N
B053 x<>y
B054 x
B055 LASTx
B056 +
B057 x
B058 x
B059 ENTER
B060 +
B061 1E-12
B062 RCL N
B063 +/-
B064 x√y
B065 IP
B066 STO I ' largest i where i^(-n) > 0,1 ULP
B067 RCL- I ' sum = 0
B068 RCL I
B069 RCL N
B070 +/-
B071 y^x
B072 +
B073 DSE I
B074 GTO B068
B075 x
B076 RCL N ' adjust sign
B077 RCL N
B078 4
B079 RMDR
B080 x=0?
B081 RCL- N
B082 SGN
B083 R↑
B084 x
B085 RTNThe adjustment in line 051...056 tries to reduce the error in \((2\pi)^{-n}\). On the 35s I could not find errors larger than some units in the last place. Other calculators may require a different correction, or it may even be omitted completely if a somewhat larger error is acceptable.
Usage:
Enter n [XEQ] B [ENTER]
=> display shows n and Bn.
Execution time:
within approx. 3 seconds (at n = 10).
Examples:
4 [XEQ] B [ENTER]
=> -0,333333333333
32 [XEQ] B [ENTER]
=> -15.116.315.767,1
100 [XEQ] B [ENTER]
=> 2,83322495708 E+78
372 [XEQ] B [ENTER]
=> -5,58475372908 E+499
Up to n = 26 the result may also be viewed in fraction mode. If no limitations are set (Flag 8 and 9 clear, /c ≥ 2730) the display shows the exact representation of the respective Bernoulli number:
22 [XEQ] B [ENTER]
=> 6.192,12318839
[FDISP] => 6192 17 / 138 ' exact result is \(6192 \frac{17}{138}\) or \(\frac{854513}{138}\)
[RND] ' round to exact result
[FDISP] => 6.192,12318841
Dieter
RE: Bernoulli numbers and large factorials - Tugdual - 02-09-2014 07:47 PM
Nice job
I checked the values on Wolfram Alpha and you are prettyu close (for the 10 first digits over 500 ;-) ).
The Prime has Bernouilli numbers calculated internally but unfortunately the Bn function is not exposed in the library. So we can use the formula Bernoulli(x) := –x*Zeta(1–x) (Thanks Joe Horn)
I tried it and couldn't calculate B372; the bigger number is B370.
RE: Bernoulli numbers and large factorials - Marcus von Cube - 02-09-2014 08:01 PM
The WP 34S has Bn built-in. In double precision mode I can get B2122. Larger arguments return 0 (which can be considered a bug).
RE: Bernoulli numbers and large factorials - Bunuel66 - 02-09-2014 08:42 PM
Another approach would be to compute iteratively the factorial using 1/2π factor at every step. As the factorial as the same number of factor as the term (2π)^−n this would limit the overflow.
My two cents...
RE: Bernoulli numbers and large factorials - Dieter - 02-09-2014 08:43 PM
(02-09-2014 07:47 PM)Tugdual Wrote: Nice job
I checked the values on Wolfram Alpha and you are prettyu close (for the 10 first digits over 500 ;-) ).
Thank you. :-) I just compared all possible 188 non-zero results with the correctly rounded 12-digit values. If I got it right, more than 80% are within ±1 ULP and more than 95% within ±2 ULP. The rest (<5%) is off by ±3 or 4 ULP.
Dieter
RE: Bernoulli numbers and large factorials - Dieter - 02-09-2014 09:26 PM
(02-09-2014 08:01 PM)Marcus von Cube Wrote: The WP 34S has Bn built-in. In double precision mode I can get B2122. Larger arguments return 0 (which can be considered a bug).
On my 34s (v. 3.2 3405) both B2124 and Zeta (-2123) still return a result, while beyond that a "+∞ Error" is displayed. Do you really get a zero here?
If 11 valid digits are sufficient, the 35s program does a good job and, compared to the 34s, it is really fast. I wonder how the 34s will perform with the same algorithm in user code. OK, 34 digits for n as low as 10 or 12 will take somewhat longer. ;-)
EDIT: The reason for the 34s limit at B2122 probably is the same as the one mentioned in my original post: it's the factorial function. In DP mode the 34s still can handle 2122! but 2124! will cause an overflow. This could be overcome by using the permutation function. A quick-and-dirty test confirmed that B2776 = 1,01268...E+6140 can be done. The result is returned in about a second.
Dieter
RE: Bernoulli numbers and large factorials - Paul Dale - 02-09-2014 09:44 PM
The 34S is computing the Bernoulli numbers from the zeta function. A short series expansion like you've got here will be faster I expect.
- Pauli
RE: Bernoulli numbers and large factorials - Dieter - 02-12-2014 08:56 PM
(02-09-2014 09:44 PM)Paul Dale Wrote: The 34S is computing the Bernoulli numbers from the zeta function. A short series expansion like you've got here will be faster I expect.
Well, this series expansion, i.e. the sum in the formula, actually is the Zeta function. ;-)
\( \begin{align*}
B_n &= 2 n! {(2\pi)^{-n}} \sum\limits_{i=1}^{\infty}i^{-n}\\
&= 2 n! {(2\pi)^{-n}} \zeta (n)
\end{align*} \)
The 35s program works so fast because the larger n gets, the less terms are required. The following table shows the number of terms needed for an error of at most 0,1 ULP in Zeta:
Code:
n 10 12 16 34 digits
---------------------------------------
8 17 31 100 17782
10 10 15 39 2511
12 6 10 21 681
14 5 7 13 268
16 4 5 10 133
20 3 3 6 50
24 2 3 4 26
30 2 2 3 13
40 1 1 2 7This also explains why up to n = 8 the result is given directly. Otherwise the number of required terms would increase rapidly.
I tried a program with the same algorithm on the 34s. In SP mode it is much faster than the internal Bernoulli function. For large n the result appears within a fraction of a second. As you will expect after a look at the above table, DP mode with 34 digit precision is a different story, at least for small n. ;-)
Dieter