3bfe6e81-d39f-43ba-961f-ff8e563eead2 42 4 5 decimal

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Source: Hart et al. "Computer Approximation" Copyright © 1968 by John Wiley & Sons,Inc. All rights reserved. No reproduction in any form of this book in whole or in part (except for brief quotation in critical articles or reviews), may be made whitout written authorization from the publisher.

JZERO 5852

|ε| = 23.22 decimals, [x] 0..8

p 1641556014884554375346147.435 394355966476729663601261.6471 - 21720183859245393139822.87997 481485995206981764828.5245941 - 5345457598841972345.381674607 33015389256896376.8646542622 - 118739068121104.2949874031474 247985116789.614443968987514 - 280314894.0831953934479400118 133662.5500481224741885945416 - 101 mat :

q 1641556014884554385346137.617 16033037244408932735390.45602 79130437776464052403.23616203 2613165313325.153278086066185 642960791882601.77592892131 1237672982083.40790348317773 1893012093.677918995179541438 2263381.356781110003609399116 1974.019272727281783930443513 1101 mat :

JONE 6054

|ε| = 25.02 decimals, [x] 0..8

a 1921176307760798128049021.316 222609203138739625477137.5773 - 7894463902082476734673.226741 126942437345360606543.6561036 - 1092152214043184787.101134641 5454629264396819.144157448868 - 16346594875712.84628830445048 29096627853.81647825756152444 - 28534334.51054763915026471449 11977.05712815379389149134705 - 101 mat :

b 3842352615521596256098041.912 35075670662720281057988.68716 161133431163341434400.7062889 492961231395985031.9632645381 1117536965288162.684489793105 1969278625584.719037168592923 2735606122.949877990248154504 2940957.355049651347475558106 2274.736606126590905134610965 1101 mat :

x J0 u x 2^11 line : p m el u m^* (m 110 sum q n el u n^* (n 110 sum / 21 line :

x J1 u x 2^11 line : x a m el u m^* (m 110 sum b n el u n^* (n 110 sum / * 21 line :

JONE can be safely extended to ~ twice its prescribed accuracy compared wrt Matlab "mwbesselj". This last one to be confirmed ?

|ε|= 23.22 decimals , [x] 0..8

x J0 x J1 2 1 sys

r x J1 x 1 - 18 solve :

rmw 1 x mwbesselj x 018 solve :

r 0 3.832 7.016 10.173 13.324 16.469 61 mat

rmw 0 3.832 7.016 10.173 13.324 16.471 61 mat

YZERO 6249

|ε| = 22.77 decimals, [x] 0..8

c 92710403643317780659970.77567 - 222179517146473495946103.3921 17975015353510263237433.32233 - 486723504289816378030.9467558 6166463755967882455.848436005 - 41985352648960441.27221217976 163038870471903.3222414474226 - 362611237102.3781263507858047 432207011.1754509283576952951 - 215700.05807808836412437323737 101 mat :

d 1256165423798014008016381.865 12506322316013967865289.69274 63102434611199419819.38406355 212799024783102857.564506927 536532884662417.5107302324905 1060973970586.885270806820407 1672209200.2084829725185024 2068779.667854516822965618638 1877.893230795989427041920381 1101 mat :

YONE 6449

|ε| = 24.58 decimals, [x] 0..8

f 586265542436344399293893.17 - 157066834199232845820836.4904 7351681299005467428400.402479 - 139065878575908011148.5190942 1339544201526785345.938109179 - 7290257386242270.629526344379 23405756030570.15935501295099 - 44115161991.85230690878878903 45421287.38770213026987060358 - 19886.12563465350530472715888 101 mat :

g 2990279721605116022908679.994 27802850103578030581271.75655 130268747460735555319.2845146 407133037223916434.9602952937 944661186508657.0116528399283 1707657951197.456205887347694 2440358986.882941823431612517 2708852.767034077697963790196 2174.361138333330803617969305 1101 mat :

Y0, Y1 rational generatrices

x Y0 u x 2^11 line : c m el u m^* (m 110 sum d n el u n^* (n 110 sum / 21 line :

x Y1 u x 2^11 line : x f m el u m^* (m 110 sum g n el u n^* (n 110 sum / * 21 line :

Y0, Y1 completed functions

x Y0 x Y0 2 π / x J0 * x ln * + :

x Y1 x Y1 2 π / x J1 x ln * 1 x / - (* + :

Smath complete Y0, Y1

x Y0 x Y1 2 1 sys

r x Y0 x 020 solve :

r 0.8936 3.9572 7.0706 10.1237 13.082 16.0428 19.1445 71 mat

http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/introductions/Bessels/05/

limited to 10

n x J 1 - (k^k! (k n + 1 + Gamma * / x 2 / (n 2 k * +^* (k 011 sum :

0 x J 1 x J 2 1 sys

n - t J 1 - (n^n t J * ≡

[p: 143]

n t - J 1 - (n^n t J * ≡

n - t - Y 1 - (n^n t Y * ≡

ε 109 -^:

Sanity check as far as casio is trustable ?

http://keisan.casio.com/exec/system/1180573474

xx 0.5 :

xx J0 xx Y0 21 mat 0.938469809569368 0.444518734398445 - 21 mat

xx J1 xx Y1 21 mat 0.242268457674873 1.47147239267024 - 21 mat

xx 2 :

xx J0 xx Y0 21 mat 0.223892972738013 0.510376187233793 21 mat

xx J1 xx Y1 21 mat 0.576724807757837 0.10703243154051 - 21 mat

Practical limit J(0,x), J(1,x) n x J 1 - (k^k! (k n + (! (* / x 2 / (2 k * n +^* (k 025 sum eval :

113 mwbesselj 0.070318 -

113 J eval 0.070318 -

Σ (21) too high for J(2,x) 213 J eval 0.217744 -

0 x J 1 x J 2 1 sys

213 mwbesselj 0.217744 -

Does NOT solve Σ(25) r 1 x J x 015 solve :

r 0 3.832 7.016 10.173 13.324 51 mat

0 3.832 7.016 10.173 13.324 51 mat

Seemingly, Matlab supports higher Σ limit

1 x mwbesselj 1 x - mwbesselj - 2 1 sys

Does NOT solve Σ(25) 1 - (k^k! (k n + (! (* / x 2 / (2 k * n +^* (k 025 sum eval

solves Σ(21) 1 - (k^k! (k n + (! (* / x 2 / (2 k * n +^* (k 021 sum

2 x mwbesselj 1 x mwbesselj /

Practical limit J(0,x), J(1,x) Σ(25) n x J 1 - (k^k! (k n + (! (* / x 2 / (2 k * n +^* (k 025 sum eval :

0 x J 1 x J 2 1 sys

115 mwbesselj 115 J eval 21 mat 0.205104038613523 0.205104033315055 21 mat

515 mwbesselj 515 J eval 21 mat 0.13045613456503 0.130456134416378 21 mat

Smath built-in Γ(x) is ± ~ 4/5 decimals n x J 1 - (k^k! (k n + 1 + Gamma * / x 2 / (n 2 k * +^* k 011 sum :

0.5 x J 1.5 x J 2 1 sys

x Γ