(DM42) Harmonic function based on Bernoulli  32 digits

12122022, 12:34 PM
Post: #1




(DM42) Harmonic function based on Bernoulli  32 digits
Harmonic function
H(n) = SUM[1/i] , i=1 ..n Direct calculation used for values of n < 40 H(n) = 1 + 1/2 + 1/3 ... 1/n Asymptotic expansion used for larger values of n >= 40 H(n) = ln(n) + gamma + 1/2n  SUM [B[k]/(2k n^(2k))] H(n) = ln(n) + gamma + 1/2n  ( B[1]/(2*n^2) + B[2]/(4*n^4) + B[3]/(6*n^6) ... ) A version with 32 digit accuracy using array of 10 precomputed Bernoulli numbers Largest error is less than 1E32 (for values below 1000), typical relative error is about 1E33 Should run on Free42, run in nstack mode, alternatively insert LNSTK after line 02 Note: / is division x is multiply 00 { 121Byte Prgm }@ DM42 01 LBL "H(n)" @ Harmonic function, H(n) = SUM[1/i] , i=1 ..n 02 FUNC 11 @ H(n) = 1 + 1/2 + 1/3 ... 1/n 03 40 04 X>Y @ uses asymptotic expansion with Bernoulli numbers 05 GTO 01 @ small numbers are calculated directly 06 CLX 07 RCL ST Y @ H(n) = 08 LN @ ln(n) 09 5.772156649015328606065120900824024E1 @ gamma with 34 DIGITS 10 2 11 RCLx ST T 12 1/X 13 R^N 4 14 X^2 15 1 16 INDEX "Bn" 17 LBL 00 18 RCLx ST Y 19 RCLEL 20 RCLIJ 21 CLX 22 2 23 x 24 / 25 RCL/ ST Y 26 STO ST T 27 DROP 28 I+ 29 FC? 77 30 GTO 00 31 DROPN 2 32 + 33 GTO 03 34 LBL 01 @ direct sum for small values 35 CLX 36 LBL 02 37 RCL ST Y 38 1/X 39 + 40 DSE ST Y 41 GTO 02 42 LBL 03 43 + 41 END Program to expand 10 Bernoulli numbers into an array, 00 { 124Byte Prgm } 01 Lbl "B10" 02 10 03 1 04 NEWMAT 05 EDIT 06 LBL 00 07 CLX 08 1 09 6 10 / 11 > 12 1 13 30 14 / 15 +/ 16 > 17 1 18 42 19 / 20 > 21 1 22 30 23 / 24 +/ 25 > 26 5 27 66 28 / 29 > 30 691 31 2730 32 / 33 +/ 34 > 35 7 36 6 37 / 38 > 39 3617 40 510 41 / 42 +/ 43 > 44 43867 45 798 46 / 47 > 48 174611 49 330 50 / 51 +/ 52 EXITALL 53 STO "Bn" 54 END Best regards Gjermund Skailand 20221212 

12122022, 03:19 PM
(This post was last modified: 12122022 03:20 PM by Werner.)
Post: #2




RE: (DM42) Harmonic function based on Bernoulli  32 digits
Hi Gjermund!
Doing the same with a Hornerlike scheme, independent of stack mode (I think). I also stored gamma in a variable.. 00 { 75Byte Prgm } 01▸LBL "Hn" 02 FUNC 11 03 40 04 X>Y? 05 GTO 00 06 R↓ 07 X^2 08 0 09 INDEX "B10" 10 I 11▸LBL 01 12 RCLIJ 13 STO+ ST X 14 × 15 RCLEL 16 X<>Y 17 ÷ 18 + 19 RCL÷ ST Y @ in 4STK mode, R^ / would work as well 20 I 21 FC? 77 22 GTO 01 23 X<>Y 24 SQRT @ n 25 LN 26 LASTX 27 STO+ ST X 28 1/X 29 + 30 RCL+ "GAMMA" 31 X<>Y 32  33 RTN 34▸LBL 00 35 CLX 36▸LBL 02 37 RCL ST Y 38 1/X 39 + 40 DSE ST Y 41 GTO 02 42 END Cheers, Werner 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

12122022, 05:41 PM
(This post was last modified: 12122022 06:00 PM by Gjermund Skailand.)
Post: #3




RE: (DM42) Harmonic function based on Bernoulli  32 digits
Hi Werner,
as always, a shorter and faster version from you. I did test the accuracy by expanding all part sums on the stack ( I love the large stack ), and summing in reverse order, from smallest to largest. But apart from the first three items I found that it did not matter. DM42 is my new favourite calculator. Wold anyone happen to know if it is possible to permanently store variables in any other way than storing the entire state? br Gjermund 

12132022, 04:03 PM
Post: #4




RE: (DM42) Harmonic function based on Bernoulli  32 digits
Hi Gjermund.
Sadly, copy/paste works only for programs, and whatever is in the Xregister. You can't export variables. Would be nice to have though. Cheers, Werner 41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE 

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