Bernoulli numbers and large factorials - Printable Version +- HP Forums (https://www.hpmuseum.org/forum) +-- Forum: HP Calculators (and very old HP Computers) (/forum-3.html) +--- Forum: General Forum (/forum-4.html) +--- Thread: Bernoulli numbers and large factorials (/thread-623.html) 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 RTN The 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         7 This 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