So I've started running more advanced code. I haven't changed the code at all, but I've been running more advanced trials.
For this, I'd like to paste a bunch of commands you can run, which then produce the same output I will paste. I will focus solely on first degree variations. This is an explanation of the semi_operators program.
To begin initialization, once in pari-gp, read the program and declare A as this.
This is precisely the equation:
\[
A = 2 <1+s>_{f} 2+w\\
\]
But we only care about the first two terms of the Taylor series in \(f\) and \(w\), each centered about zero.
Now we introduce the numbers this produces:
From here we talk about:
\[
B = 2 <s>_{g} A\\
\]
Which becomes:
Now the goal from here is to relate the final result to \(f,g\). These play the role of \(\varphi_2,\varphi_1\) respectively. We will design our solution such that \(\varphi_3\) is a function of the above two. At least, we will design the solution in this instance. We will say \(h\) is a function of \(f,g\) so that:
\[
2<s>_g\left(2<s+1>_{f}2+w\right) = 2<s+1>_h 3+w\\
\]
The value \(h\) is then called with this code:
This is the exact functional equation I made above. Remember that I'm only asking pari to check the first term in every variable other than \(s\).
And now the goal is to check, that up to first order terms:
\[
2<s>_g\left(2<s+1>_{f}2+w\right) -2<s+1>_h (3+w) = 0\\
\]
We run this check with:
Which is 0 in all variables, up to 8 digits. I ran this at low precision, because the moment you ask this to be 50 digits you better be ready to melt your computer. I've set a default at 30 series precision, and 40 digit precision; so through all of these equations we lose like 25 digits of accuracy, or something like that.
There exists a differential equation here that is a much simpler form than what I started with. But I don't see it yet. But there's a differential equation which solves all of my troubles. Either way, the code I posted works for this shit.
For this, I'd like to paste a bunch of commands you can run, which then produce the same output I will paste. I will focus solely on first degree variations. This is an explanation of the semi_operators program.
To begin initialization, once in pari-gp, read the program and declare A as this.
Code:
\r semi_operators
A = Ibennet(1+s,2,2+w+O(w^2),f+O(f^2))This is precisely the equation:
\[
A = 2 <1+s>_{f} 2+w\\
\]
But we only care about the first two terms of the Taylor series in \(f\) and \(w\), each centered about zero.
Now we introduce the numbers this produces:
Code:
%11 = ((4.000000000000000000000000000000000000000 + 1.386294361119890618834464242916353136151*f + O(f^2)) + (2.000000000000000000000000000000000000000 + 1.000000000000000000000000000000000000000*f + O(f^2))*w + O(w^2)) + ((4.028429993553611161 E-43 + 0.4528112327564592919426380883134781216581*f + O(f^2)) + (0.9580900168644644000649219792917077097099 - 0.1356667678466469569688361793634485948803*f + O(f^2))*w + O(w^2))*s + ((6.579116249093475050 E-44 + 0.07395183096062961295613131306770006209139*f + O(f^2)) + (-0.09757488571442024085704436528909808885552 - 0.1314752626782190355417664773490717959648*f + O(f^2))*w + O(w^2))*s^2 + ((7.163215891112436507 E-45 + 0.008051733859954727214726890947358116898670*f + O(f^2)) + (-0.06932117930707267344880303791788394197782 - 0.05392397722344977943503659016787524597137*f + O(f^2))*w + O(w^2))*s^3 + ((5.849379304145433158 E-46 + 0.0006574930327220950519769745338508746188389*f + O(f^2)) + (-0.01583215451145976116656433947903545843674 - 0.01734912798323829586380034471450982423896*f + O(f^2))*w + O(w^2))*s^4 + ((3.821215360683631440 E-47 + 4.295195003681144004114867909746573876425 E-5*f + O(f^2)) + (-0.002444332250449864366921702815873886462024 - 0.004305681437306917432108522766354604094587*f + O(f^2))*w + O(w^2))*s^5 + ((2.0802330859242300099 E-48 + 2.338263069149667219796167858305565345683 E-6*f + O(f^2)) + (-0.0002959644872275539864420689126690125585809 - 0.0008401974800674236013287136324014847675665*f + O(f^2))*w + O(w^2))*s^6 + ((9.706791351474238090 E-50 + 1.091081181751779435490420298220746956322 E-7*f + O(f^2)) + (-2.994057135719400337679700228929075018729 E-5 - 0.0001340837717811466774764455199877469224530*f + O(f^2))*w + O(w^2))*s^7 + ((3.963213262315457120 E-51 + 4.454806179721509602629562334846970016409 E-9*f + O(f^2)) + (-2.620033049418160161413540043074308064995 E-6 - 1.812641173160424769156356958489636153262 E-5*f + O(f^2))*w + O(w^2))*s^8 + ((1.4383569234139090268 E-52 + 1.616769244288898090374658488227443062511 E-[+++]From here we talk about:
\[
B = 2 <s>_{g} A\\
\]
Which becomes:
Code:
B = Ibennet(s,2,A,g+O(g^2))
%12 = (((6.000000000000000000000000000000000000000 + 1.000000000000000000000000000000000000000*g + O(g^2)) + (1.386294361119890618834464242916353136151 - 5.613967822811889894 E-46*g + O(g^2))*f + O(f^2)) + ((2.000000000000000000000000000000000000000 - 8.089529890453804219 E-46*g + O(g^2)) + (1.000000000000000000000000000000000000000 - 1.2780703748038775050 E-44*g + O(g^2))*f + O(f^2))*w + O(w^2)) + (((1.189949083815681480393430888415606037397 + 0.8417315202049436792677019359063981245388*g + O(g^2)) + (1.000719794693837950862764734580566482002 + 0.1469473776340554402038720597133570834643*g + O(g^2))*f + O(f^2)) + ((1.748554982032124044032368915689313800842 + 0.2120002529842895601913183207125649505424*g + O(g^2)) + (-0.04373588768808528901181784667221274173166 - 0.05458972429145882245883848409293278725907*g + O(g^2))*f + O(f^2))*w + O(w^2))*s + (((0.5008088256433267578529147867862586443978 + 0.4898149264719639103174144458346616286488*g + O(g^2)) + (0.6675480426182790657218391397189526167333 + 0.2286945117977071640326024100678923402640*g + O(g^2))*f + O(f^2)) + ((0.8792787821023755987942472648656339966780 + 0.3622475552788372384714479063158105266271*g + O(g^2)) + (-0.1333431007554850822479490261212832581241 - 0.1745944775957549810056107935950344863934*g + O(g^2))*f + O(f^2))*w + O(w^2))*s^2 + (((0.1942849409981862815449175271024313895223 + 0.2456453146530388712583021862265536282185*g + O(g^2)) + (0.3998559258485188889027993342244624720898 + 0.2171633103305817165950812905984866299433*g + O(g^2))*f + O(f^2)) + ((0.5063705716324703251059901499869113497935 + 0.3313802751353067258060520803433950314312*g + O(g^2)) + (0.05650644674046454733268988387440112126009 - 0.1593880733599506333918730958105527370538*g + O(g^2))*f + O(f^2))*w + O(w^2))*s^3 + (((0.07307635427874960048126815750544007013580 + 0.1129435824959106148489227445288041493028*g + O(g^2)) + (0.2107034201691668111303555331928121547808 + 0.1598975709948236622798142534183039736352*g + O(g^2))*f + O(f^2)) + ((0.244[+++]Now the goal from here is to relate the final result to \(f,g\). These play the role of \(\varphi_2,\varphi_1\) respectively. We will design our solution such that \(\varphi_3\) is a function of the above two. At least, we will design the solution in this instance. We will say \(h\) is a function of \(f,g\) so that:
\[
2<s>_g\left(2<s+1>_{f}2+w\right) = 2<s+1>_h 3+w\\
\]
The value \(h\) is then called with this code:
Code:
h = Iexp(-s-1,B,3+w+O(w^2)) - Iexp(-s-1,2,3+w+O(w^2)) - 3 - w -O(w^2)
*** log: Warning: increasing stack size to 16000000.
%13 = (((-4.993567076550786624856049516800516312200 E-10 + 0.4551196131538278335227906183724725879997*g + O(g^2)) + (0.6309297533502175232422057906869026880288 - 0.1051549589200787456803620238772458073549*g + O(g^2))*f + O(f^2)) + ((3.543397536875013131988607015654339882352 E-8 - 0.1380892300724521722372012563450441842459*g + O(g^2)) + (0.2636872921730001624603967617469509578375 - 0.008896226977222334731860334301022909213812*g + O(g^2))*f + O(f^2))*w + O(w^2)) + (((-0.07768558393188282342115547358779740775592 - 0.01015344185200757851025127121996980849904*g + O(g^2)) + (-0.08970191377896801856281812185334754508365 - 0.06141814843793742831710323647194593304319*g + O(g^2))*f + O(f^2)) + ((0.04918565647026930110099196500453115445801 - 0.1065173717674995882109371953568912347709*g + O(g^2)) + (-0.6716227325727445260977542869162375520244 + 0.05965190728832847433737714435779781136407*g + O(g^2))*f + O(f^2))*w + O(w^2))*s + (((0.08252415909077602415472930552111683061178 + 0.01248579915994197062667945382975394706707*g + O(g^2)) + (0.07742856714620473248450754980829984371186 + 0.002615207622404777499200964124979216939756*g + O(g^2))*f + O(f^2)) + ((-0.08901111934548888374832533444093805256379 + 0.006867486033378285216751176346463146563634*g + O(g^2)) + (-0.01037995218916216468293471665016699477237 + 0.04844999500533523427556484357205950302367*g + O(g^2))*f + O(f^2))*w + O(w^2))*s^2 + (((-0.007020215702594515380164121231704846731659 - 0.002631184343897851646355404663080024544446*g + O(g^2)) + (0.007829887393306312960052145960192432909575 - 0.001950730365551465780848676898844163507034*g + O(g^2))*f + O(f^2)) + ((0.05364123779427932696347221092482634071556 + 0.01255478455261672717193317739436683665404*g + O(g^2)) + (0.001785790365460631865975588101929528360753 - 0.001978829162948244458338459868584428891072*g + O(g^2))*f + O(f^2))*w + O(w^2))*s^3 + (((0.002359456171603009420132877252429216608766 + 0.0003843534947151930638314944520071316729913*g + O(g^2)) + (0.0043290220474440[+++]This is the exact functional equation I made above. Remember that I'm only asking pari to check the first term in every variable other than \(s\).
And now the goal is to check, that up to first order terms:
\[
2<s>_g\left(2<s+1>_{f}2+w\right) -2<s+1>_h (3+w) = 0\\
\]
We run this check with:
Code:
B - Ibennet(1+s,2,3+w+O(w^2),h)
*** log: Warning: increasing stack size to 16000000.
%16 = (((1.097198830817115840580640383740426105346 E-9 + 5.335234388653624176126351941311194771086 E-10*g + O(g^2)) + (7.396205348243446130059830035065780100486 E-10 + 1.433670230476728327818058387004388550141 E-10*g + O(g^2))*f + O(f^2)) + ((-7.752349706696407465510397411648075463539 E-8 - 3.823679586907149411923075600094968883301 E-8*g + O(g^2)) + (-5.247393106172077730471169094032216921336 E-8 - 1.037201762553142918609860579552292670284 E-8*g + O(g^2))*f + O(f^2))*w + O(w^2)) + (((7.916717181790533867820027857920293452362 E-10 + 6.561601955365761690827143577917803392156 E-10*g + O(g^2)) + (8.209767281168871684138449979351427186223 E-10 + 3.079293025448427775001022847589823762440 E-10*g + O(g^2))*f + O(f^2)) + ((-5.518542542422709936196748114259743075607 E-8 - 4.663778484674427092988315411890056052904 E-8*g + O(g^2)) + (-5.827302856245977431549306248837868963595 E-8 - 2.223683439631890100008727112315780612967 E-8*g + O(g^2))*f + O(f^2))*w + O(w^2))*s + (((4.452378780099874635612548098715402982312 E-10 + 5.035278599165156491403799417503666241698 E-10*g + O(g^2)) + (6.575036890453595230403040984504917560050 E-10 + 3.787711689056494287191792808246850539164 E-10*g + O(g^2))*f + O(f^2)) + ((-3.082750185203135359157015393672450861482 E-8 - 3.551309064897895084302374043923006774901 E-8*g + O(g^2)) + (-4.671337498729872955564733070778658889302 E-8 - 2.733869995004656929501998483482988880244 E-8*g + O(g^2))*f + O(f^2))*w + O(w^2))*s^2 + (((2.203917275845259628744966439945247657971 E-10 + 3.117927660141714454224385568600062300231 E-10*g + O(g^2)) + (4.502252583065156472977668171370664419663 E-10 + 3.516625137387558164346354075930654313331 E-10*g + O(g^2))*f + O(f^2)) + ((-1.512024102446443131058097835961820175504 E-8 - 2.180739465138867015518149311894135354548 E-8*g + O(g^2)) + (-3.188865758065010989314802401209628163679 E-8 - 2.532089299817959287259590818245032503508 E-8*g + O(g^2))*f + O(f^2))*w + O(w^2))*s^3 + (((1.004143152172660312944934132436136179849 E-10 + 1.69[+++]Which is 0 in all variables, up to 8 digits. I ran this at low precision, because the moment you ask this to be 50 digits you better be ready to melt your computer. I've set a default at 30 series precision, and 40 digit precision; so through all of these equations we lose like 25 digits of accuracy, or something like that.
There exists a differential equation here that is a much simpler form than what I started with. But I don't see it yet. But there's a differential equation which solves all of my troubles. Either way, the code I posted works for this shit.