Taylor polynomial. System of equations for the coefficients.

[marraco]

This is a numerical example for base a=e

I had shown how to get 2 sides of a system of equations for solving the coefficients. One side is blue (the linear one), and the other is red (nonlinear, and I still don't know the pattern to generate the complete system of equations. I only got up to x^7, and I can get more, but of course, what I need is a more general expression.

For base e, you get a system of equations.

you get this blue side for 7 coefficients (it is independent of the base):

Code:

(21:23) gp > Size=9
%3 = 9
(21:24) gp > m=matrix(Size, Size,n,i,binomial(i-1,n-1))
%4 =
[1 1 1 1 1  1  1  1  1]
[0 1 2 3 4  5  6  7  8]
[0 0 1 3 6 10 15 21 28]
[0 0 0 1 4 10 20 35 56]
[0 0 0 0 1  5 15 35 70]
[0 0 0 0 0  1  6 21 56]
[0 0 0 0 0  0  1  7 28]
[0 0 0 0 0  0  0  1  8]
[0 0 0 0 0  0  0  0  1]

and this is the red side:


\( {\color{Red} {

e^{^xe}=
e \,+\,
e.a_1 \,.\, x \,+\,
(e.a_2 + \frac{e}{2}.a_1^2) \,.\, x^2 \,+\,
(e.a_3 + e.a_1.a_2 +\frac{e}{6}.a_1^3) \,.\, x^3 \,+\,
(e.a_4 + \frac{e}{2}.a_2^2 +e.a_3.a_1+\frac{e}{2}.a_2.a_1^2+\frac{e}{24}.a_1^4) \,.\,x^4 \\

+(e.a_5 + {e}.a_3.a_2 + e.a_1.a_4+\frac{e}{2}.a_1.a_2^2+\frac{e}{2}.a_3.a_1^2+\frac{e}{6}.a_2.a_1^3+ \frac{e}{120} .a_1^5) \,.\,x^5 \\


+(e.a_6 + \frac{e}{2}.a_3^2 + e.a_2.a_4+\frac{e}{6}.a_2^3+ {e}.a_1.a_5+ {e}.a_1.a_3.a_2+\frac{e}{2}.a_4.a_1^2+\frac{e}{4}.a_2^2 . a_1^2+\frac{e}{6}.a_3.a_1^3+\frac{e}{24}.a_2.a_1^4+\frac{e}{ 720}.a_1^6) \,.\,x^6 \\
} \)

\( {\color{Red} {

+e.(a_7 + 1^2. a_4.a_3 + a_5.a_2 + \frac{1}{2}a_3.a_2^2 + a_6.a_1 + \frac{1}{2}a_3^2.a_1 + a_4.a_2.a_1 + \frac{1}{6}a_2^3.a_1 + \frac{1}{2}a_5.a_1^2 + \frac{1}{2} a_3.a_2.a_1^2 + \frac{1}{6}a_4.a_1^3 + \frac{1}{12} a_2^2.a_1^3 + \frac{1}{24}a_3.a_1^4 + \frac{1}{120}a_2.a_1^5 + \frac{1}{5040}.a_1^7 )\,.\,x^7

} \)

So you get this systems of equations (blue to the left, and red to the right):

Code:

[1 1 1 1 1  1  1  1  1]    [ 1]   [e]
[0 1 2 3 4  5  6  7  8]    [a₁]   [e.a₁]
[0 0 1 3 6 10 15 21 28]    [a₂]   [e.a₂+e/2.a₁²]
[0 0 0 1 4 10 20 35 56]    [a₃]   [e.a₃+e.a₁.a₂+e/6.a₁³]
[0 0 0 0 1  5 15 35 70]  * [a₄] = [e.a₄+e/2.a₂²+e.a₃.a₁+e/2.a₂.a1²+e/24.a₁⁴]
[0 0 0 0 0  1  6 21 56]    [a₅]   [...]
[0 0 0 0 0  0  1  7 28]    [a₆]   [...]
[0 0 0 0 0  0  0  1  8]    [a₇]   [...]
[0 0 0 0 0  0  0  0  1]    [a₈]   [...]

It is a non linear system of equations, and the solution for this particular case is:

a₀= 1,00000000000000000
a₁= 1,09975111049169000
a₂= 0,24752638354178700
a₃= 0,15046151104294100
a₄= 0,12170896032120000
a₅= 0,16084324512292400
a₆= -0,02254254634348470
a₇= -0,10318144159688800
a₈= 0,06371479195361670

Compare with Sheldonison's kneser.gp solution:

Code:

a0=   1.000000000000000000000000000000000000000000000000000000000000000
a1=   1.091767351258320991801384550027151644384731177193748032852736750
a2=   0.2714832129016945953317066836235490061739872161990598817820694029
a3=   0.2124532481762562843089676377409482685666305767555868534429427250
a4=   0.06954037613998737372867423270746944766611954537384154515961000453
a5=   0.04429195209047330440644034438551461331201912452396122603445409319
a6=   0.01473674209638939115209628691553407087036283284802524310071901416
a7=   0.008668781817225260366380392529639960019193562974233192587504340111
a8=   0.002796479398385459694825991301149597269227881323260081302040577606
a9=   0.001610631290584272072162645164026110049792896861436056957822790182
a10=  0.0004899272314843773346986672258324787189179236966986597429129832332
a11=  0.0002881810711540458113452640412964722721968311050442105340877181595
a12=  8.009461253854333344427358300999263618841918537261023130790163984 E-5
a13=  5.029114179380540369459011462420351199253676250298573251465345775 E-5
a14=  1.218379034490009161619171109859335192564698424745403501454238589 E-5
a15=  8.665533667381574685245804554105291695042724874461047071879364312 E-6
a16=  1.687782319317538991789009317583785512852144820496380916395175482 E-6
a17=  1.493253248573492581066504431732751485456118359869377142782801855 E-6
a18=  1.987607642049274553198189794968196417348167228423221823766208771 E-7
a19=  2.608673560043263731645821608532933364457829738518570518951024854 E-7
a20=  1.470995414254190186141218818247639464249470033298457423356618453 E-8
a21=  4.683449732741350625509370993006576850859342869408936466444526422 E-8
a22= -1.549241665546769521805465176448252112209562873661024180269440066 E-9
a23=  8.741510781350935912992558117122349583609707251224417742494147299 E-9
a24= -1.125787310103062317575134515738378022835159203164294437887008675 E-9
a25=  1.707959267270728412565608778729726960430408931538582375779659881 E-9
a26= -3.778583154922985176492143492500296036058409993500345462555638121 E-10
a27=  3.495778765110216317873145649935531461006488191616787742019992521 E-10
a28= -1.053770123445001506629425792917061970255456556066414419570018689 E-10
a29=  7.459097147607505280732283202189701957070179874063753558099592924 E-11
a30= -2.717598206577734869329877172492724299204938962536131010671806592 E-11
a31=  1.646076610661447130388508182175742339593708736967841194185926886 E-11
a32= -6.741873152405052999147453463677312005415532501572568628993007553 E-12
a33=  3.725328723319468544317086960688595471124134410671562760851241992 E-12
a34= -1.639087326793590223458207893420015494534981157084934821663605376 E-12
a35=  8.583638311358568060488665543255939724011799192373411667290837403 E-13
a36= -3.943738739105384313579489883439749145625771519735734916959207765 E-13
a37=  2.002523128021887055893526704591127887891070857821009841312866823 E-13
a38= -9.441962242924065023715111579990954292712006432497954997730218600 E-14
a39=  4.712054745849371340817414393412613311684787369946584046802094568 E-14
a40= -2.256291882035597080043272706155245325625189406689252333379604487 E-14
a41=  1.115468850616536996293093710617388942901924987321949988173687870 E-14
a42= -5.390745557016350491840931638954438791176540897408393713471988968 E-15
a43=  2.652158491516681872817207767927177972525297352287457523410538087 E-15
a44= -1.288910765544553681933994492897094530731798239089453308045916388 E-15
a45=  6.326678501956660453007840302627916831673424077807564985330246473 E-16
a46= -3.085457150492335988961833456292213681855982339566944345179362092 E-16
a47=  1.513176771782740527337006888790219839276363051364244735670911612 E-16
a48= -7.396534137094751433579658688811828771999777094874553108587585999 E-17
a49=  3.626987671054187604858900781140139207944144420079830647639126832 E-17
a50= -1.775725598676298403622157433145771025580236336491145640801940287 E-17
a51=  8.709879544396054644216187847273723532343829279009721890829235905 E-18
a52= -4.269289282339156322311235369711703332985224326874010775539646161 E-18
a53=  2.095044162575528110270707271710047732715299297962567107013275560 E-18
a54= -1.027883709282258792101512551665977860702273246084671587434733006 E-18
a55=  5.046824247438176461629463228342973276669090061957443080345413943 E-19
a56= -2.478050595821552304759308268057851431159467131543548274881868429 E-19
a57=  1.217394203039331646009323993543125208245611419932029287538356629 E-19
a58= -5.981648632303782947485566789252901875389559922820136772179247709 E-20
a59=  2.940264344513896261748193069655017265616156575476060556667439589 E-20

This is a comparison of the coefficients obtained from both methods. Of course I only calculated (7+1dummy) coefficients, with excel precision, and Sheldonison code did 60 coefficients with 134 bits of precision.
(I estimate that I need at least between 20 and 30 coefficients, and have no clue on how to solve it).




The advantage of this method is that it does not depend of the base, so it will be able to solve negative bases.