02/18/2016, 06:51 PM
(This post was last modified: 02/19/2016, 04:28 AM by sheldonison.)

(02/16/2016, 03:17 AM)tommy1729 Wrote: I need the ratio's of the real a_n vs fake a_n for the function exp(x) , where the fake is the limit of the Tommy-Sheldon iterations.

From post#141, we have the Gaussian approximation error for a_n for exp(x) is the same as the error in Stirling's approximation for n!, or approximately

Numerical results using the iterated Gaussian, post#150, do not show improved behavior, over Gaussian, for exp(x), for a_n where n>39, although it is an improvement for smaller values of n. I have no idea what the expected behavior is for the iterated Gaussian approach in post#150 is, which Tommy refers to as "Tommy-Sheldon iterations". One interesting note on post#150, if F_n were exactly equal to exp(x), than F_n+1 is the Gaussian approximation...

a_n for n=1-50 error term for Gaussian, vs 20 iterations of post#150, using 5000 term Taylor series approximation; here the error term ratio is printed as , so Gaussian looks like instead of

Code:

`n a_n Gaussian error a_n post#150 error`

1 0.0844375514192275 0.0683398515477305

2 0.0844142416333461 0.0607258884762405

3 0.0841935537563679 0.0539730385655355

4 0.0840332149853975 0.0478920432435974

5 0.0839199291390415 0.0422747285213201

6 0.0838370948924301 0.0369976779272284

7 0.0837743008426081 0.0319840348760891

8 0.0837252084955692 0.0271826096143000

9 0.0836858377963707 0.0225573573477991

10 0.0836535913240025 0.0180817574806996

11 0.0836267114154591 0.0137356044170443

12 0.0836039699542637 0.00950306495393020

13 0.0835844845780208 0.00537144303115174

14 0.0835676057991404 0.00133036115853485

15 0.0835528453405350 -0.00262880126422681

16 0.0835398292592345 -0.00651330495603565

17 0.0835282664554722 -0.0103293386111823

18 0.0835179270079519 -0.0140822322763638

19 0.0835086269484465 -0.0177766186034792

20 0.0835002173553064 -0.0214165568027041

21 0.0834925764050155 -0.0250056293731362

22 0.0834856034885243 -0.0285470186139815

23 0.0834792147938839 -0.0320435678797153

24 0.0834733399466525 -0.0354978311600175

25 0.0834679194244035 -0.0389121136095030

26 0.0834629025452544 -0.0422885049797763

27 0.0834582458872422 -0.0456289074255043

28 0.0834539120347256 -0.0489350588073065

29 0.0834498685755973 -0.0522085523576732

30 0.0834460872927128 -0.0554508533850614

31 0.0834425435070680 -0.0586633135474178

32 0.0834392155405340 -0.0618471831168360

33 0.0834360842735316 -0.0650036215728523

34 0.0834331327786455 -0.0681337067965566

35 0.0834303460154027 -0.0712384430865797

36 0.0834277105746333 -0.0743187681777040

37 0.0834252144632697 -0.0773755594108080

38 0.0834228469223184 -0.0804096391772155

39 0.0834205982721904 -0.0834217797398696

40 0.0834184597807136 -0.0864127075170098

41 0.0834164235500404 -0.0893831069003792

42 0.0834144824193701 -0.0923336236687975

43 0.0834126298809658 -0.0952648680487083

44 0.0834108600073941 -0.0981774174656687

45 0.0834091673882766 -0.101071819024386

46 0.0834075470751343 -0.103948591749587

47 0.0834059945331426 -0.106808228615549

48 0.0834045055988069 -0.109651198388331

49 0.0834030764427299 -0.112477947301598

50 0.0834017035367683 -0.115288900584174

- Sheldon