Remark: this is not an answer but only a work-out based on Will's Pari/GP protocol
\\ Pari/GP-code
\ps 64 \\ define taylor-series-extension sufficiently high
f= taylor( (-1 + sqrt(1 + 4 * x))/2 , x )
\\ should be: x - x^2 + 2*x^3 - 5*x^4 + 14*x^5 - 42*x^6 + ...
fp = deriv(f)
\\ should be: 1 - 2*x + 6*x^2 - 20*x^3 + 70*x^4 - 252*x^5 + ...
listf = vectorv(24); \\ provide the required powers of f beforehand as constants
listf[1]=f;
for(k=2,#listf,listf[k] = listf[k-1]*f )
listx = vectorv(#listf,r,x^r) \\ that list for powers of x is not really needed
valpha = vectorv(#listf); \\ shall get the sought coefficents
valpha[1]=0; valpha[2]=-1 \\ known constants at the beginning
{for(j=2,#listf-1,
L = sum(k=2,j,va[k]*listf[k]) + 'a*listf[j+1];
R = sum(k=2,j,va[k]*listx[k]) + 'a*listx[j+1];
Compare = L-fp*R;
coefx = polcoeff(Compare,j+2);print(coefx);
ac=-polcoeff(coefx,0)/polcoeff(coefx,1);
valpha[j+1]=ac;
);}
Now check this:
valpha \\ display coefficients
/* should be:
[0, -1, 1, -3/2, 8/3, -31/6, 157/15, -649/30, 9427/210, -19423/210,
6576/35, -2627/7, 853627/1155, -2007055/1386, 3682190/1287, -29646689/5148,
212029715/18018, -1077705008/45045, 3291567542/69615, -4216011601/46410,
1728974695307/9699690, -3696738921829/9699690, 12315245049166/14549535,
-8505662174957/5290740]~
*/
alpha=Ser(valpha)
/* comes out to be:
-x + x^2 - 3/2*x^3 + 8/3*x^4 - 31/6*x^5 + 157/15*x^6 - 649/30*x^7 +
9427/210*x^8 - 19423/210*x^9 + 6576/35*x^10 - 2627/7*x^11 + 853627/1155*x^12
+ O(x^13)
*/
However, I didn't catch it how to proceed now...
Ok, I got it now working. Only I had to do one "magic step", indicated by (**) in the comment; ( I missed one link from that coefficients by Will's above procedure to arrive at
R and
S).
Now as it is working, it is really miraculous... ;-)
\\ I found heuristically examining your document, that it must be
result = intformal( 1/( x*alpha ) + 1/x ) \\ (**)
\\ the +1/x in the expressions allows "formal integration" for Pari/GP
coeffs_abel=Vec(result) \\ put the result into a coefficientsvector
#coeffs_abel \\ = 63 in my example
\\ getting : [1, 0, 1/2, -1/3, 13/36, -113/240] for x^-1,x^0,x^1,...
\\ your example-function f(x)
myf(x,h=0)=for(k=1,h,x=(-1+sqrt(1+4*x))/2);x
\\ then the Abel-function alpha(x) as given in the beginning of your example
{fAbel(x,n=0)=local(xn); xn = myf(x,n); \\ here n -> infty, but n~20 suffices
sum(k=-1,#coeffs_abel-2,coeffs_abel[2+k]*xn^k) - log(xn) - n }
Now test the functions:
\\ testing:
maxn=20 \\ try some sufficient n (=maxn) for the Abel-function
x0 = 0.125
x12 = myf(x0,12) \\ x12=0.0521939337419 is 12 iterations from x0
a0=fAbel(x0 , maxn) \\ =10.1373406515
a1=fAbel(x12 , maxn) \\ =22.1373406515
a1-a0 \\ comes out to be =12.0000000000
\\ how to find the 0.5-iterate from x0=0.25 (with a0=Abel(x0))
x_05=solve(x=0.01,x0-0.001, (fAbel(x,maxn)-a0) -1/2)
\\ comes out to be 0.118366472264
\\check
a0 - fAbel(x_05,maxn) \\ comes out to be -0.5
(a0 - fAbel(x_05,maxn)) - (-1/2)
\\ < 5e-201 using internal float precision of 200 digits
@Will: Could you make the missing step visible in your protocol; my move in the integral-expression using $x*alpha$ was simply a heuristic.
Data of the experiment:
x_0
- the initial value
x_1
- the correct value by one integer iteration using the original formula
abel_x_05
- "half-iterate" using the Abel-mechanism
abel_x_10
- "unit-iterate" by applying "half-iteration" to the abel_x_05
should equal the original x_1
h
- the "height" of iteration = 0.5, thus: "half-iterate"
a0
- the Abel-function-value of x_0
a05
- the Abel function-value of the half-iterate x_05
a05-a0-1/2
- the difference between the abel-values should be 1/2. This is the error
x_1-abel_x_10
- if the difference is zero, then the Abel-function is exact. This is the error
The table:
x_0 x_1 abel_x_05 abel_x_1 h a_0 a_05 a05-a0-1/2 x_1-abel_x_1
0.0100000000000 0.00990195135928 0.00995073533545 0.00990195135928 1/2 104.610137209 105.110137209 1.11696228987E-201 -2.85779229102E-97
0.0200000000000 0.0196152422707 0.0198057704819 0.0196152422707 1/2 53.9218924877 54.4218924877 3.97098709435E-202 -6.15809353856E-82
0.0300000000000 0.0291502622129 0.0295691127718 0.0291502622129 1/2 36.8546006147 37.3546006147 4.97268862342E-202 -4.06098551075E-74
0.0400000000000 0.0385164807135 0.0392444803983 0.0385164807135 1/2 28.2383644612 28.7383644612 -3.54446782891E-200 -3.59148072904E-69
0.0500000000000 0.0477225575052 0.0488353314257 0.0477225575052 1/2 23.0199413289 23.5199413289 -1.92438083583E-202 -1.07323790193E-65
0.0600000000000 0.0567764362830 0.0583448891277 0.0567764362830 1/2 19.5089497541 20.0089497541 3.82913315022E-200 -4.30261434261E-63
0.0700000000000 0.0656854249492 0.0677761642099 0.0656854249492 1/2 16.9784545543 17.4784545543 2.30176349353E-200 -4.68144861850E-61
0.0800000000000 0.0744562646538 0.0771319743721 0.0744562646538 1/2 15.0637628558 15.5637628558 -1.959630265E-200 -2.06820942631E-59
0.0900000000000 0.0830951894845 0.0864149615923 0.0830951894845 1/2 13.5615925326 14.0615925326 0.E-202 -4.75931307811E-58
0.100000000000 0.0916079783100 0.0956276074506 0.0916079783100 1/2 12.3495715644 12.8495715644 2.612840354E-200 -6.71587352419E-57
0.110000000000 0.100000000000 0.104772246757 0.100000000000 1/2 11.3495715644 11.8495715644 0.E-202 -6.49893010190E-56
0.120000000000 0.108276253030 0.113851079713 0.108276253030 1/2 10.5093372632 11.0093372632 -1.469722699E-200 -4.66951632156E-55
0.130000000000 0.116441400297 0.122866182786 0.116441400297 1/2 9.79257475074 10.2925747507 -6.53210088E-201 -2.64025433320E-54
0.140000000000 0.124499799840 0.131819518477 0.124499799840 1/2 9.17327627451 9.67327627451 -1.143117654E-200 -1.22717201784E-53
0.150000000000 0.132455532034 0.140712944100 0.132455532034 1/2 8.63230833801 9.13230833801 3.266050442E-201 -4.84797799860E-53
0.160000000000 0.140312423743 0.149548219701 0.140312423743 1/2 8.15527503721 8.65527503721 9.79815132E-201 -1.67081681025E-52
0.170000000000 0.148074069841 0.158327015221 0.148074069841 1/2 7.73113278533 8.23113278533 8.16512610E-201 -5.12835185230E-52
0.180000000000 0.155743852430 0.167050916985 0.155743852430 1/2 7.35126498055 7.85126498055 9.79815132E-201 -1.42532084917E-51
0.190000000000 0.163324958071 0.175721433593 0.163324958071 1/2 7.00884764373 7.50884764373 -1.633025221E-201 -3.63574721484E-51
0.200000000000 0.170820393250 0.184340001282 0.170820393250 1/2 6.69840449769 7.19840449769 1.469722699E-200 -8.60676914865E-51
0.210000000000 0.178232998313 0.192907988820 0.178232998313 1/2 6.41548854806 6.91548854806 -1.633025221E-201 -1.90833380748E-50
0.220000000000 0.185565460040 0.201426701971 0.185565460040 1/2 6.15645005622 6.65645005622 9.79815132E-201 -3.99383229632E-50
0.230000000000 0.192820323028 0.209897387587 0.192820323028 1/2 5.91826470908 6.41826470908 1.633025221E-201 -7.94107754605E-50
0.240000000000 0.200000000000 0.218321237354 0.200000000000 1/2 5.69840449769 6.19840449769 4.899075662E-201 -1.50846028308E-49
0.250000000000 0.207106781187 0.226699391244 0.207106781187 1/2 5.49473939600 5.99473939600 6.53210088E-201 -2.75054364650E-49
0.260000000000 0.214142842854 0.235032940678 0.214142842854 1/2 5.30546158398 5.80546158398 -1.143117654E-200 -4.83408189236E-49
0.270000000000 0.221110255093 0.243322931449 0.221110255093 1/2 5.12902639712 5.62902639712 -4.899075662E-201 -8.21796258865E-49
0.280000000000 0.228010988928 0.251570366421 0.228010988928 1/2 4.96410584104 5.46410584104 2.612840354E-200 -1.35554771981E-48
0.290000000000 0.234846922835 0.259776208015 0.234846922835 1/2 4.80955165341 5.30955165341 1.633025221E-201 -2.17542886532E-48
0.300000000000 0.241619848710 0.267941380520 0.241619848710 1/2 4.66436569742 5.16436569742 -1.143117654E-200 -3.40480227973E-48
0.310000000000 0.248331477355 0.276066772226 0.248331477355 1/2 4.52767604024 5.02767604024 -1.143117654E-200 -5.20802211274E-48
0.320000000000 0.254983443527 0.284153237414 0.254983443527 1/2 4.39871747998 4.89871747998 6.53210088E-201 -7.80012141557E-48
0.330000000000 0.261577310586 0.292201598193 0.261577310586 1/2 4.27681558319 4.77681558319 -9.79815132E-201 -1.14578285473E-47
0.340000000000 0.268114574787 0.300212646221 0.268114574787 1/2 4.16137351452 4.66137351452 -1.143117654E-200 -1.65319303514E-47
0.350000000000 0.274596669241 0.308187144298 0.274596669241 1/2 4.05186110361 4.55186110361 2.449537831E-200 -2.34609807510E-47
0.360000000000 0.281024967591 0.316125827860 0.281024967591 1/2 3.94780571723 4.44780571723 3.266050442E-201 -3.27863351540E-47
0.370000000000 0.287400787401 0.324029406368 0.287400787401 1/2 3.84878459717 4.34878459717 0.E-202 -4.51684740363E-47
0.380000000000 0.293725393319 0.331898564609 0.293725393319 1/2 3.75441839607 4.25441839607 1.633025221E-201 -6.14045635954E-47
0.390000000000 0.300000000000 0.339733963915 0.300000000000 1/2 3.66436569742 4.16436569742 -1.633025221E-201 -8.24471876728E-47
0.400000000000 0.306225774830 0.347536243297 0.306225774830 1/2 3.57831834906 4.07831834906 6.53210088E-201 -1.09424173357E-46
0.410000000000 0.312403840464 0.355306020520 0.312403840464 1/2 3.49599747214 3.99599747214 -9.79815132E-201 -1.43659422864E-46
0.420000000000 0.318535277187 0.363043893101 0.318535277187 1/2 3.41715003390 3.91715003390 -6.53210088E-201 -1.86694656416E-46
0.430000000000 0.324621125124 0.370750439252 0.324621125124 1/2 3.34154589332 3.84154589332 4.899075662E-201 -2.40311964896E-46
0.440000000000 0.330662386292 0.378426218767 0.330662386292 1/2 3.26897524487 3.76897524487 -3.266050442E-201 -3.06557066840E-46
0.450000000000 0.336660026534 0.386071773851 0.336660026534 1/2 3.19924639910 3.69924639910 1.633025221E-201 -3.87763161862E-46
0.460000000000 0.342614977318 0.393687629910 0.342614977318 1/2 3.13218384914 3.63218384914 1.633025221E-201 -4.86575271568E-46
0.470000000000 0.348528137424 0.401274296286 0.348528137424 1/2 3.06762658100 3.56762658100 -9.79815132E-201 -6.05974959408E-46
0.480000000000 0.354400374532 0.408832266957 0.354400374532 1/2 3.00542659239 3.50542659239 1.143117654E-200 -7.49305322423E-46
0.490000000000 0.360232526704 0.416362021194 0.360232526704 1/2 2.94544759052 3.44544759052 -9.79815132E-201 -9.20296150448E-46
0.500000000000 0.366025403784 0.423864024184 0.366025403784 1/2 2.88756384413 3.38756384413 0.E-202 -1.12308915176E-45
0.510000000000 0.371779788708 0.431338727620 0.371779788708 1/2 2.83165916874 3.33165916874 -8.16512610E-201 -1.36226314832E-45
0.520000000000 0.377496438739 0.438786570254 0.377496438739 1/2 2.77762602736 3.27762602736 -1.143117654E-200 -1.64285914844E-45
0.530000000000 0.383176086633 0.446207978426 0.383176086633 1/2 2.72536473159 3.22536473159 -8.16512610E-201 -1.97040520998E-45
0.540000000000 0.388819441732 0.453603366565 0.388819441732 1/2 2.67478273021 3.17478273021 0.E-202 -2.35094101264E-45
0.550000000000 0.394427191000 0.460973137658 0.394427191000 1/2 2.62579397425 3.12579397425 3.266050442E-201 -2.79104206351E-45
0.560000000000 0.400000000000 0.468317683702 0.400000000000 1/2 2.57831834906 3.07831834906 -1.633025221E-201 -3.29784346620E-45
0.570000000000 0.405538513814 0.475637386133 0.405538513814 1/2 2.53228116531 3.03228116531 -6.53210088E-201 -3.87906318943E-45
0.580000000000 0.411043357914 0.482932616224 0.411043357914 1/2 2.48761270178 2.98761270178 -1.633025221E-201 -4.54302477715E-45
0.590000000000 0.416515138991 0.490203735478 0.416515138991 1/2 2.44424779394 2.94424779394 3.266050442E-201 -5.29867944782E-45
0.600000000000 0.421954445729 0.497451095989 0.421954445729 1/2 2.40212546307 2.90212546307 6.53210088E-201 -6.15562753640E-45
0.610000000000 0.427361849550 0.504675040790 0.427361849550 1/2 2.36118858117 2.86118858117 6.53210088E-201 -7.12413923790E-45
0.620000000000 0.432737905309 0.511875904189 0.432737905309 1/2 2.32138356786 2.82138356786 -1.143117654E-200 -8.21517461673E-45
0.630000000000 0.438083151965 0.519054012082 0.438083151965 1/2 2.28266011564 2.78266011564 6.53210088E-201 -9.44040285135E-45
0.640000000000 0.443398113206 0.526209682255 0.443398113206 1/2 2.24497094044 2.74497094044 -4.899075662E-201 -1.08122206882E-44
0.650000000000 0.448683298051 0.533343224672 0.448683298051 1/2 2.20827155486 2.70827155486 -1.633025221E-201 -1.23437700836E-44
0.660000000000 0.453939201417 0.540454941749 0.453939201417 1/2 2.17252006161 2.67252006161 -1.633025221E-201 -1.40489550174E-44
0.670000000000 0.459166304663 0.547545128614 0.459166304663 1/2 2.13767696515 2.63767696515 8.16512610E-201 -1.59424574642E-44
0.680000000000 0.464365076099 0.554614073360 0.464365076099 1/2 2.10370499971 2.60370499971 9.79815132E-201 -1.80397525144E-44
0.690000000000 0.469535971483 0.561662057284 0.469535971483 1/2 2.07056897183 2.57056897183 -4.899075662E-201 -2.03571226387E-44
0.700000000000 0.474679434481 0.568689355110 0.474679434481 1/2 2.03823561638 2.53823561638 -1.633025221E-201 -2.29116710935E-44
0.710000000000 0.479795897113 0.575696235217 0.479795897113 1/2 2.00667346430 2.50667346430 9.79815132E-201 -2.57213344685E-44
0.720000000000 0.484885780180 0.582682959838 0.484885780180 1/2 1.97585272133 2.47585272133 -6.53210088E-201 -2.88048943794E-44
0.730000000000 0.489949493661 0.589649785270 0.489949493661 1/2 1.94574515637 2.44574515637 -3.266050442E-201 -3.21819883116E-44
0.740000000000 0.494987437107 0.596596962058 0.494987437107 1/2 1.91632399887 2.41632399887 1.633025221E-201 -3.58731196224E-44
0.750000000000 0.500000000000 0.603524735182 0.500000000000 1/2 1.88756384413 2.38756384413 9.79815132E-201 -3.98996667126E-44
0.760000000000 0.504987562112 0.610433344234 0.504987562112 1/2 1.85944056601 2.35944056601 -1.469722699E-200 -4.42838913794E-44
0.770000000000 0.509950493836 0.617323023586 0.509950493836 1/2 1.83193123628 2.33193123628 3.266050442E-201 -4.90489463626E-44
0.780000000000 0.514889156509 0.624194002553 0.514889156509 1/2 1.80501405007 2.30501405007 0.E-202 -5.42188821009E-44
0.790000000000 0.519803902719 0.631046505547 0.519803902719 1/2 1.77866825684 2.27866825684 3.266050442E-201 -5.98186527137E-44
0.800000000000 0.524695076596 0.637880752227 0.524695076596 1/2 1.75287409642 2.25287409642 8.16512610E-201 -6.58741212246E-44
0.810000000000 0.529563014099 0.644696957644 0.529563014099 1/2 1.72761273971 2.22761273971 8.16512610E-201 -7.24120640468E-44
0.820000000000 0.534408043279 0.651495332378 0.534408043279 1/2 1.70286623365 2.20286623365 1.796327743E-200 -7.94601747474E-44
0.830000000000 0.539230484541 0.658276082669 0.539230484541 1/2 1.67861744997 2.17861744997 3.266050442E-201 -8.70470671114E-44
0.840000000000 0.544030650891 0.665039410547 0.544030650891 1/2 1.65485003771 2.15485003771 -4.899075662E-201 -9.52022775248E-44
0.850000000000 0.548808848170 0.671785513954 0.548808848170 1/2 1.63154837883 2.13154837883 3.266050442E-201 -1.03956266698E-43
0.860000000000 0.553565375285 0.678514586862 0.553565375285 1/2 1.60869754695 2.10869754695 6.53210088E-201 -1.13340420751E-43
0.870000000000 0.558300524426 0.685226819385 0.558300524426 1/2 1.58628326890 2.08628326890 0.E-202 -1.23387051676E-43
0.880000000000 0.563014581273 0.691922397891 0.563014581273 1/2 1.56429188873 2.06429188873 -6.53210088E-201 -1.34129397209E-43
0.890000000000 0.567707825203 0.698601505104 0.567707825203 1/2 1.54271033417 2.04271033417 -1.633025221E-201 -1.45601620124E-43
0.900000000000 0.572380529476 0.705264320212 0.572380529476 1/2 1.52152608528 2.02152608528 9.79815132E-201 -1.57838806969E-43
0.910000000000 0.577032961427 0.711911018956 0.577032961427 1/2 1.50072714504 2.00072714504 -1.633025221E-201 -1.70876966271E-43
0.920000000000 0.581665382639 0.718541773732 0.581665382639 1/2 1.48030201191 1.98030201191 -8.16512610E-201 -1.84753026232E-43
0.930000000000 0.586278049120 0.725156753679 0.586278049120 1/2 1.46023965409 1.96023965409 -1.633025221E-201 -1.99504831930E-43
0.940000000000 0.590871211464 0.731756124764 0.590871211464 1/2 1.44052948530 1.94052948530 -6.53210088E-201 -2.15171142049E-43
0.950000000000 0.595445115010 0.738340049873 0.595445115010 1/2 1.42116134220 1.92116134220 9.79815132E-201 -2.31791625155E-43
0.960000000000 0.600000000000 0.744908688889 0.600000000000 1/2 1.40212546307 1.90212546307 -3.266050442E-201 -2.49406855556E-43
0.970000000000 0.604536101719 0.751462198770 0.604536101719 1/2 1.38341246783 1.88341246783 -1.633025221E-201 -2.68058308730E-43
0.980000000000 0.609053650641 0.758000733628 0.609053650641 1/2 1.36501333924 1.86501333924 -1.143117654E-200 -2.87788356377E-43
0.990000000000 0.613552872566 0.764524444801 0.613552872566 1/2 1.34691940522 1.84691940522 8.16512610E-201 -3.08640261096E-43
1.00000000000 0.618033988750 0.771033480925 0.618033988750 1/2 1.32912232216 1.82912232216 -8.16512610E-201 -3.30658170700E-43
[update]: Another protocol, as requested by Will Jagy is at my website (to save space here) at
go.helms-net.de