(Last update at13.Feb.2011 remark: I changed the use of the symbol U of the previous version to get a clearer distinction between symbols for the function and that for the matrices)
Here I consider the formal powerseries for iterates of the function d(x) = exp(x)-1 .
I'll use the notations
d(x,
1) = d(x) = exp(x) – 1
d(x, h+1) = d( d(x,1),h)
d(x, 0) = x
I call the number-of-iterations "height" (of the powertower) and use the letter h for that parameter.
For negative integer h we have the iterates of log(1+x):
d(x,–1) = log(1+x)
d(x,–2) = log(1+log(1+x))
...
It is possible to define formal powerseries for fractional iterations consistent with the addition of (iteration-) heights:
d( d(x, h1), h2) = d(x, h1+h2)
A procedure for the computation can be found for instance in L. Comtet. Comtet introduces the self-composition using the Newton-binomial-formula for the formal powerseries.
In this article I simply use the matrix-logarithm of a small modification of the Bell-matrix for the d(x)
Let
U1 the column-vector of coefficients of the formal powerseries for
d(x)
U1 =
column(0,1,1/2!,1/3!,…)
Then
let U0 the column-vector for d(x)0 = 1
U0 =
column(1,0,0,0,…)
Let
Uc the column-vector for the c'th power of d(x)c (not the iterate!)
Uc = column( )
Then let U the matrix of concatenation of all Uc-vectors. U is then the factorially scaled matrix of the Stirling-numbers 2nd kind, as, for instance defined in [A&S]
|
Matrixoperator U:
|
Matrix of Stirlingnumbers 2nd
kind
|
Further, let's define a general type of Vandermondevector
V(x) = row(1, x, x2, x3, …)
With this we can construct the (iterable) matrix-product:
V(x) * U = V(d(x)) = V(d(x,1))
V(x) * U2 =
V(d(x,1))*U = V(d(x,2))
V(x) * Uh =
V(d(x,h))
This expression is so far only meant as expression for the formal composition of the powerseries, which we find in the second column of V(d(x,h)). This means, to get that powerseries we need only consider the second column of the h'th power of U, let's write it this way:
d(x,h) = V(x) * Uh [,1]
We can get formal powerseries for fractional iterates analoguosly if we can define fractional powers of the matrix U.
Because the matrix has a unit-diagonal the best method to compute a fractional power is to use the matrix-logarithm defined by the mercator-series using U–I as parameter.
UL = Log( U ) = (U–I) – (U–I)2/2 – (U–I)3/3 + … – …
The argument U–I is nilpotent to any selected truncation size, so that series gives finite expressions for the entries of UL at any finite row/column-index and these evaluate then to exact entries in rational numbers. This allows a meaningful interpretation of that matrix-logarithm.
To compute fractional powers of U we can then similarly use the matrix-exponential
Uh = Exp( h * Log(U))
And then, as above we get the formal powerseries for the fractional iterate by the second column of the h'th power of U:
d(x,h) = V(x) * Uh [,1] // also valid for all h
For instance, for h=1/2 we get the matrix U1/2
|
Matrixoperator U1/2:
|
|
giving the formal powerseries
whose coefficients increase strongly such that the radius of convergence of this series is zero.
The first 20 coefficients of that formal powerseries are
0,
1, 2, 2, 0, 8, -56, 32, 10176, -215808, -78784, 150990912, -3405688576,
-139041794560, 10385778676736, 130003936220160,
-43016304236761088,
526545841919713280, 266085261164348628992,
-12347306589339686547456
and it seems, that the above scaling ensures that those coefficients are all integer.
Because of the convenient properties of the logarithm of the matrix U we can express the coefficients of the powerseries depending on a variable h in terms of polynomials of that h:
d(x,h)
=
1x
+ (1/2*h)x2
+ ( 1/4*h2
- 1/12*h) x3
+ ( 1/8*h3
- 5/48*h2 + 1/48*h) x4
+ ( 1/16*h4
- 13/144*h3 + 1/24*h2 - 1/180*h) x5
+ …
or, attempting to get integer coefficients
From this we can write the two-parameteric function d(x,h) formally as matrix-product introducing the matrix POLY of coefficients of that polynomials
d(x,h) = V(x) * POLY * V(h) ~
where the top-left truncation of POLY looks like
|
Matrix POLY of coefficients for the
two-parametric function d(x,h):
|
|
For a given height h this means, that we have to multiply POLY with the transpose V(h)~ to get the column-vector U1 corresponding to Uh[,1] which contains the coefficients for the formal powerseries for the h'th iterate:
U1 = Uh[,1] = POLY * V(h)~
and then
d(x,h) = V(x) * U1
An eyeball-check gives for h=1 : POLY * V(1)~ for the first few rows:
1
= 1 row 1 for coefficient
at x
1/2*h = 1/2 row 2 for
coefficient at x2
-1/12*h+1/4*h2
= 1/6 row 3 for
coefficient at x3
1/48–5/48+1/8 = 1/24 row 4 for coefficient at x4
… giving coefficients for
exp(x)-1
For h=-1 : POLY * V(-1)~ for the first few rows:
1
= 1 row 1 for coefficient
at x
1/2*h = -1/2 row 2 for coefficient at x2
-1/12*h+1/4*h2
= 1/3 row 3 for coefficient at x3
-1/48–5/48-1/8 =– 1/4 row
4 for coefficient at x4
… giving coefficients for
log(1+x)
(truncation to 32 x 12)
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1.00000000000 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0.500000000000 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-0.0833333333333 |
0.250000000000 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0.0208333333333 |
-0.104166666667 |
0.125000000000 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-0.00555555555556 |
0.0416666666667 |
-0.0902777777778 |
0.0625000000000 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0.00127314814815 |
-0.0157986111111 |
0.0515046296296 |
-0.0668402777778 |
0.0312500000000 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-0.000148809523810 |
0.00526620370370 |
-0.0258680555556 |
0.0506365740741 |
-0.0453125000000 |
0.0156250000000 |
0 |
0 |
0 |
0 |
0 |
0 |
-0.0000454695767196 |
-0.00133308531746 |
0.0113136574074 |
-0.0321180555556 |
0.0434317129630 |
-0.0290364583333 |
0.00781250000000 |
0 |
0 |
0 |
0 |
0 |
0.0000199790564374 |
0.000158110119048 |
-0.00405953758818 |
0.0174816743827 |
-0.0336202739198 |
0.0340108989198 |
-0.0178943452381 |
0.00390625000000 |
0 |
0 |
0 |
0 |
0.0000113214653145 |
0.0000198297876249 |
0.00106265615814 |
-0.00798717321612 |
0.0220179880401 |
-0.0313159079218 |
0.0249550471230 |
-0.0107166108631 |
0.00195312500000 |
0 |
0 |
0 |
-0.0000113193776388 |
0.0000281142067166 |
-0.000188055739271 |
0.00292184945207 |
-0.0121852839322 |
0.0240907720872 |
-0.0267906803443 |
0.0174372597346 |
-0.00627919353505 |
0.000976562500000 |
0 |
0 |
-0.00000172664698128 |
-0.0000398044512447 |
0.0000929097888191 |
-0.000857463809011 |
0.00558563546987 |
-0.0156722021953 |
0.0237718056857 |
-0.0214808783063 |
0.0117297621631 |
-0.00361631686095 |
0.000488281250000 |
0 |
0.00000705612171179 |
0.00000110606798251 |
-0.0000973900278552 |
0.000319743843236 |
-0.00214332382319 |
0.00857213099601 |
-0.0178075901564 |
0.0216702768669 |
-0.0163663738175 |
0.00765414014275 |
-0.00205391667794 |
0 |
-0.00000101301777217 |
0.0000255407498408 |
0.0000193146034564 |
-0.000221106056740 |
0.000853415265246 |
-0.00399194850843 |
0.0112958472089 |
-0.0184076755112 |
0.0185526492239 |
-0.0119658680773 |
0.00487200212379 |
0 |
-0.00000559541003662 |
-0.00000844104671673 |
0.0000565866905508 |
0.0000736013774889 |
-0.000480293863401 |
0.00179037366013 |
-0.00615026857187 |
0.0132851855803 |
-0.0176563735180 |
0.0150939818079 |
-0.00845607452096 |
0 |
0.00000278500897643 |
-0.0000218764183589 |
-0.0000308783186148 |
0.0000918274988497 |
0.000200451899171 |
-0.000956523820017 |
0.00311688150890 |
-0.00826461703283 |
0.0142971284796 |
-0.0159372946748 |
0.0117720730416 |
0 |
0.00000556362245182 |
0.0000161788895796 |
-0.0000534680155645 |
-0.0000764323476234 |
0.000107980977407 |
0.000450686111708 |
-0.00170513688559 |
0.00469959885089 |
-0.0100009496866 |
0.0143229295042 |
-0.0136786842136 |
0 |
-0.00000548903126386 |
0.0000232632702353 |
0.0000552525079060 |
-0.0000999706949845 |
-0.000145443330382 |
0.0000656096485154 |
0.000876233786845 |
-0.00272141966761 |
0.00632621838743 |
-0.0111319804655 |
0.0135249843031 |
0 |
-0.00000669291680506 |
-0.0000312146749719 |
0.0000608496312550 |
0.000140289074251 |
-0.000154302742138 |
-0.000223099906968 |
-0.0000851073857983 |
0.00150933332628 |
-0.00392984366170 |
0.00776958374083 |
-0.0115704367446 |
0 |
0.0000116121812620 |
-0.0000293586208479 |
-0.000109807488995 |
0.000120164536704 |
0.000290176034739 |
-0.000206232231635 |
-0.000276467343816 |
-0.000391754045290 |
0.00234414377342 |
-0.00520000569233 |
0.00884602555284 |
0 |
0.00000922094512360 |
0.0000684549777770 |
-0.0000798140615651 |
-0.000293289356200 |
0.000190169428300 |
0.000514515807562 |
-0.000249794344110 |
-0.000258744046274 |
-0.000884113280245 |
0.00332974863404 |
-0.00637985999133 |
0 |
-0.0000281326395956 |
0.0000407197995304 |
0.000254478523660 |
-0.000159415648134 |
-0.000643681175683 |
0.000244712936664 |
0.000808996669112 |
-0.000290514027042 |
-0.000120237194298 |
-0.00156152904832 |
0.00437759973853 |
0 |
-0.0000130744820957 |
-0.000174765277447 |
0.000107022219976 |
0.000723693305397 |
-0.000241293720810 |
-0.00121349305469 |
0.000247074481467 |
0.00115660459637 |
-0.000348228712042 |
0.000177531440051 |
-0.00238806973087 |
0 |
0.0000790678539611 |
-0.0000509153645345 |
-0.000691411780357 |
0.000185376174105 |
0.00169611543620 |
-0.000257736012247 |
-0.00202316585911 |
0.000159489699569 |
0.00153386513052 |
-0.000453577912925 |
0.000649502319867 |
0 |
0.0000125598083643 |
0.000520031268526 |
-0.0000878204186092 |
-0.00209831597086 |
0.000159818142150 |
0.00341809068358 |
-0.000102695878258 |
-0.00304489925501 |
-0.0000465076302668 |
0.00191880195570 |
-0.000639295364230 |
0 |
-0.000257444626675 |
-0.00000560163583526 |
0.00219132519453 |
0.0000774885874687 |
-0.00524993565328 |
-0.000238197148364 |
0.00609224515367 |
0.000347091607245 |
-0.00419971113343 |
-0.000382906372544 |
0.00229692073560 |
0 |
0.0000399463258851 |
-0.00179321941211 |
-0.000393926763669 |
0.00708866051564 |
0.00107649022665 |
-0.0112876740738 |
-0.00145536270256 |
0.00979926697250 |
0.00119787229708 |
-0.00536903759965 |
-0.000843150907368 |
0 |
0.000965154912485 |
0.000597931194494 |
-0.00803525350006 |
-0.00251127673418 |
0.0188906613302 |
0.00429420339159 |
-0.0214416358316 |
-0.00407840860992 |
0.0144407003456 |
0.00250329169161 |
-0.00641682419913 |
0 |
-0.000479398220790 |
0.00710879598324 |
0.00415391248559 |
-0.0276350570478 |
-0.0101591117622 |
0.0432073573046 |
0.0121303625147 |
-0.0367072852041 |
-0.00872608327064 |
0.0197254703316 |
0.00423995118032 |
0 |
-0.00413400035554 |
-0.00492686165030 |
0.0337822921646 |
0.0196419756453 |
-0.0781999520194 |
-0.0316735109546 |
0.0871838557014 |
0.0280285757842 |
-0.0574862641505 |
-0.0158949998739 |
0.0252069454158 |
0 |
0.00364643030317 |
-0.0321199817356 |
-0.0303987588850 |
0.123132859442 |
0.0717993863914 |
-0.189632330949 |
-0.0821936577995 |
0.158294897310 |
0.0561459477663 |
-0.0833102527369 |
-0.0258044757139 |
0 |
0.0200644656154 |
0.0355155275043 |
-0.161371990105 |
-0.138129365513 |
0.368719723749 |
0.216573813194 |
-0.405011847681 |
-0.185172406164 |
0.262477331560 |
0.100651849527 |
-0.112748061977 |
0 |
-0.0266575258227 |
0.163984344506 |
0.217099842002 |
-0.621211818806 |
-0.502402984261 |
0.944593968266 |
0.561221939319 |
-0.777011178359 |
-0.372049587016 |
0.402058565471 |
0.164786181361 |
0 |
-0.109433975865 |
-0.260796020227 |
0.868040322611 |
0.997694463252 |
-1.96144527296 |
-1.53571144363 |
2.12752918160 |
1.28414365605 |
-1.35896459404 |
-0.679330436962 |
0.574102779542 |
0 |
0.203327436701 |
-0.937854250819 |
-1.62757702735 |
3.51646920918 |
3.71182584274 |
-5.28880579131 |
-4.07561297163 |
4.29651204725 |
2.64641867680 |
-2.19161222503 |
-1.14304812752 |
0 |
0.665130619209 |
2.03725117010 |
-5.21187900185 |
-7.69651265113 |
11.6634597966 |
11.6847026371 |
-12.5142190515 |
-9.61214522598 |
7.89504962424 |
4.98627814056 |
-3.28846954268 |
0 |
-1.66290268242 |
5.95678065193 |
13.1278524887 |
-22.1339082828 |
-29.5986840636 |
32.9725282990 |
32.0714426086 |
-26.4976106584 |
-20.5007171703 |
13.3508288173 |
8.69043910681 |
0 |
-4.46640908999 |
-17.2004065386 |
34.6184714675 |
64.3249478880 |
-76.8111306867 |
-96.5868487442 |
81.6300989964 |
78.4396151915 |
-50.9456676853 |
-40.0779077720 |
20.9616625377 |
0 |
14.7351954384 |
-41.6407654187 |
-114.983036865 |
153.482786768 |
256.820864049 |
-226.702159524 |
-275.300078287 |
180.447707027 |
173.785467539 |
-89.9407771586 |
-72.5919140556 |
0 |
32.8328822406 |
157.885595430 |
-251.974932794 |
-585.440530393 |
554.789212202 |
871.054820739 |
-584.569327444 |
-699.977362563 |
361.335696650 |
353.288864141 |
-147.066594023 |
0 |
-141.988015420 |
317.215867693 |
1096.93776669 |
-1160.66255957 |
-2430.64419617 |
1701.15729804 |
2582.17675228 |
-1342.39810662 |
-1613.21386356 |
662.614306472 |
665.804867226 |
0 |
-261.445537846 |
-1578.11102327 |
1988.14780518 |
5808.95742632 |
-4346.51106854 |
-8575.75205606 |
4543.98663605 |
6830.48490615 |
-2784.11676454 |
-3412.43948801 |
1121.99572399 |
0 |
1488.67144258 |
-2602.46237198 |
-11400.3914328 |
9457.12874113 |
25088.3012505 |
-13761.5135901 |
-26447.6015871 |
10772.1967423 |
16378.6965829 |
-5269.35407760 |
-6692.11726412 |
0 |
2225.38586814 |
17167.5422612 |
-16775.7623465 |
-62790.1318537 |
36429.2371165 |
92071.4180506 |
-37801.2355725 |
-72775.7762113 |
22968.1109026 |
36043.7099027 |
-9169.48708622 |
0 |
-16965.9992298 |
22632.4324966 |
128919.990125 |
-81699.3259569 |
-282001.209389 |
118053.557193 |
295291.413488 |
-91687.1256079 |
-181491.326093 |
44455.6923008 |
73520.4777118 |
0 |
-19864.8060373 |
-202949.957825 |
148447.802757 |
738101.788937 |
-320192.510162 |
-1075878.94572 |
329766.603527 |
844756.046306 |
-198680.621951 |
-415253.578974 |
78563.7447446 |
0 |
209816.100420 |
-203538.981603 |
-1583257.02501 |
729670.792682 |
3444745.92555 |
-1046623.20368 |
-3585816.98087 |
806147.907721 |
2189367.41953 |
-387205.992395 |
-880311.779326 |
0 |
179952.210614 |
2601889.48175 |
-1331939.30794 |
-9415064.91099 |
2851431.65424 |
13651227.5125 |
-2911995.29981 |
-10655777.4807 |
1737625.06275 |
5203689.86150 |
-679539.726003 |
0 |
-2809660.13947 |
1804361.41661 |
21068440.9439 |
-6414541.20408 |
-45620175.4042 |
9117979.35471 |
47239086.9788 |
-6950397.41814 |
-28674195.3232 |
3298413.50452 |
11454373.4953 |
0 |
-1537617.51817 |
-36091847.4033 |
11232709.1805 |
130008060.821 |
-23794693.2562 |
-187610831.806 |
24005965.8180 |
145678442.909 |
-14121437.7770 |
-70728434.2325 |
5429606.42913 |
0 |
40646569.5053 |
-13834857.2340 |
-303052770.285 |
48467482.2723 |
653379950.065 |
-67774658.5637 |
-673377701.275 |
50662222.3521 |
406613987.318 |
-23479413.7152 |
-161490843.705 |
0 |
9497336.70101 |
540392137.825 |
-67078112.2567 |
-1938575327.49 |
138038281.362 |
2785541778.69 |
-134418266.711 |
-2152796870.90 |
75631332.4795 |
1039789385.27 |
-27460376.7453 |
0 |
-633756457.512 |
36321161.5318 |
4700532956.13 |
-112411455.910 |
-10094645940.5 |
133313249.347 |
10359297792.8 |
-77628094.3870 |
-6226116877.91 |
23149218.9697 |
2460005438.55 |
0 |
55726510.4799 |
-8712493396.23 |
-464543661.478 |
31137786218.3 |
1087458698.49 |
-44568140091.6 |
-1222476193.24 |
34298090791.4 |
809405223.168 |
-16488568951.5 |
-354311889.107 |
0 |
10624909421.6 |
2258110308.89 |
-78427159370.5 |
-8325790733.68 |
167824702223. |
12318317185.2 |
-171557719099. |
-9839260929.73 |
102672679190. |
4931865970.96 |
-40378641054.5 |
0 |
-4479387389.03 |
150896297343. |
33830524377.7 |
-537443820225. |
-73730866273.1 |
766524921906. |
76950043000.1 |
-587612884016. |
-47153631547.1 |
281298641107. |
19049913758.0 |
0 |
-191085306212. |
-90492032637.6 |
1.40426147025E12 |
326278291737. |
-2.99510410092E12 |
-471578616715. |
3.05089282338E12 |
367059804326. |
-1.81884436225E12 |
-178813905496. |
712297299427. |
0 |
147339758784. |
-2.80099958068E12 |
-1.09524040267E12 |
9.94481572290E12 |
2.35767861839E12 |
-1.41375347119E13 |
-2.42709175388E12 |
1.07994799022E13 |
1.46464955320E12 |
-5.15000328031E12 |
-581666558076. |
0 |
3.67833350467E12 |
2.68324418152E12 |
-2.69212194339E13 |
-9.59393858609E12 |
5.72461167767E13 |
1.37437470904E13 |
-5.81230177576E13 |
-1.05919222433E13 |
3.45290228970E13 |
5.10280681255E12 |
-1.34704718730E13 |
0 |
-4.19319940120E12 |
5.55998508341E13 |
3.09092946611E13 |
-1.96830270039E14 |
-6.61078904851E13 |
2.78974129793E14 |
6.75678356972E13 |
-2.12413893151E14 |
-4.04490518050E13 |
1.00938902710E14 |
1.59204641663E13 |
0 |
-7.56224969314E13 |
-7.43691124956E13 |
5.51372049332E14 |
2.64508276903E14 |
-1.16918343295E15 |
-3.76818353935E14 |
1.18353893902E15 |
2.88614340474E14 |
-7.00827624795E14 |
-1.38089771104E14 |
2.72448367408E14 |
0 |
1.15852751376E14 |
-1.17766671407E15 |
-8.48897778129E14 |
4.15784511805E15 |
1.80736068720E15 |
-5.87669966081E15 |
-1.83806072834E15 |
4.46120741857E15 |
1.09425993761E15 |
-2.11311591829E15 |
-4.28046930851E14 |
0 |
1.65696934682E15 |
2.05186154784E15 |
-1.20384785431E16 |
-7.26873720573E15 |
2.54614960359E16 |
1.03117358326E16 |
-2.57027203038E16 |
-7.86161269750E15 |
1.51743397354E16 |
3.74228431873E15 |
-5.88004687148E15 |
... |
... |
... |
... |
... |
... |
... |
... |
... |
... |
|
|
1.00000000000 |
0.433045596018 |
0.175813571676 |
0.0685214533424 |
0.0259260061301 |
0.00958557929620 |
0.00347798716295 |
0.00124213189878 |
0.000437626092140 |
0.000152362209100 |
0.0000524902552539 |
0.0000179137522579 |
The interesting thing is here, whether we can change order of summation. This is then interesting, if we always assume x=1 in d(x,h).
Since each column is associated with one power of the h-parameter, we may sum up each column to have only one term for a powerseries in h.
We see, that the sequence of entries along a column diverge after a local minimum and looking at coefficients of higher index it seems that they will grow at least hypergeometrically (like the factorials) .
I constructed a Noerlund-means method of summation which I can apply to the powerseries if I want to estimate the column-sums meaning we apply x=1 at d(x,h) keeping h variable. That summation is parametrized to sum alternating series of order gamma(k)1.5 Remark: this is absolutely experimental and no proof for the correct value , for conversion to convergent series and/or regularity of the result is given so far. But because the results and also the intermediate values seem to be reasonable I give here the assumed column-sums in the brown row at the bottom of the table.
This would then give a powerseries for d(1,h) in terms of h:
d(1,h) = 1 + 0.4330 h + 0.1758 h2 + 0.06852 h3 + ...
and for h=1 this should be e - 1 ~ 1.718 ... and this value is good approximated by the row-sum of the last (brown-marked) row.
A numerical check of
U1 = POLY * V(1)
Approx = Noerlund(0,0) * dV(1)* U1 // we need no acceleration for this because
it's simply the exponential-series
showing the convergence of approx_k to the final value:
left the coefficients in U1, right the Noerlund-summed partial sums
U1 |
Partial sums: Approx: |
0 |
0 |
1.00000000000 |
1.10803324100 |
0.500000000000 |
1.57457355300 |
0.166666666667 |
1.69632420459 |
0.0416666666667 |
1.71614032126 |
0.00833333333333 |
1.71816671191 |
0.00138888888889 |
1.71828087196 |
0.000198412698413 |
1.71828207238 |
0.0000248015873016 |
1.71828183500 |
0.00000275573192240 |
1.71828182789 |
0.000000275573192240 |
1.71828182845 |
0.0000000250521083854 |
1.71828182846 |
0.00000000208767569879 |
1.71828182846 |
1.60590438368E-10 |
1.71828182846 |
1.14707455977E-11 |
1.71828182846 |
7.64716373182E-13 |
1.71828182846 |
The same with h-parameter h= 2, so the second-iteration; d(x,2) = exp(exp(x)-1)-1 with x=1
U21 = POLY * V(2)~
Approx = Noerlund(0,0) * dV(1)* U21// again we need no
acceleration-procedure for this
U21 |
Partial sums: Approx: |
0 |
0 |
1.00000000000 |
1.60230732254 |
1.00000000000 |
2.77868485049 |
0.833333333333 |
3.64038609659 |
0.625000000000 |
4.13617161436 |
0.433333333333 |
4.38997896613 |
0.281944444444 |
4.50325607997 |
0.174007936508 |
4.54954546855 |
0.102678571429 |
4.56656903743 |
0.0582754629630 |
4.57238024057 |
0.0319596009700 |
4.57420919266 |
0.0169996091871 |
4.57474473371 |
0.00879662406138 |
4.57489192860 |
0.00443943225627 |
4.57492973020 |
0.00218975755744 |
4.57493886159 |
0.00105757104279 |
4.57494096334 |
0.000500896018318 |
4.57494141015 |
0.000232971039341 |
4.57494150301 |
0.000106534988039 |
4.57494152085 |
0.0000479488461554 |
4.57494152405 |
0.0000212602719146 |
4.57494152466 |
0.00000929459892442 |
4.57494152474 |
0.00000400953097900 |
4.57494152476 |
0.00000170787472022 |
4.57494152476 |
0.000000718768664806 |
4.57494152476 |
0.000000299047612615 |
4.57494152476 |
0.000000123065453419 |
4.57494152476 |
0.0000000501169612163 |
4.57494152476 |
The same with h-parameter h= -1, so the inverse-iteration; giving log(1+x) with x=1:
U -11 = POLY * V(-1)~
Approx = Noerlund(0,0.95) * dV(1)* U-11 // we use only small order of the
Noerlundtransformation to accelerate convergence
U-11 |
Partial sums: Approx: |
0 |
0 |
1.00000000000 |
0.444444444444 |
-0.500000000000 |
0.592592592593 |
0.333333333333 |
0.658436213992 |
-0.250000000000 |
0.680384087791 |
0.200000000000 |
0.688797439415 |
-0.166666666667 |
0.691601889956 |
0.142857142857 |
0.692623801540 |
-0.125000000000 |
0.692964438735 |
0.111111111111 |
0.693085511172 |
-0.100000000000 |
0.693125868651 |
0.0909090909091 |
0.693140005390 |
-0.0833333333333 |
0.693144717637 |
0.0769230769231 |
0.693146352716 |
-0.0714285714286 |
0.693146897743 |
0.0666666666667 |
0.693147085613 |
-0.0625000000000 |
0.693147148237 |
0.0588235294118 |
0.693147169718 |
-0.0555555555556 |
0.693147176879 |
0.0526315789474 |
0.693147179326 |
-0.0500000000000 |
0.693147180142 |
0.0476190476190 |
0.693147180420 |
-0.0454545454545 |
0.693147180513 |
0.0434782608696 |
0.693147180544 |
-0.0416666666667 |
0.693147180555 |
0.0400000000000 |
0.693147180558 |
-0.0384615384615 |
0.693147180559 |
0.0370370370370 |
0.693147180560 |
-0.0357142857143 |
0.693147180560 |
0.0344827586207 |
0.693147180560 |
-0.0333333333333 |
0.693147180560 |
The same with h-parameter h= 1/2, so giving the formal powerseries for the half-iteration, using x=1
The approximation is very difficult now since we have a divergent series in the terms, whose characteristic is not yet really known, but seems to be more than hypergeometric, so a high order for Noerlund-summation is needed. However, for size = 64 I got an approximation using Eulersummation of order ord=2.5 which is shown below.
U0.51 = POLY * V(1/2)
Approx = Noerlund(0, 2.5) * dV(1)* U0.51 // medium order of
Noerlundsummation provides convergence – but it is only local
U0.51 |
Partial sums: Approx: |
0 |
0 |
1.00000000000 |
0.162591050989 |
0.250000000000 |
0.373041564566 |
0.0208333333333 |
0.576651361395 |
0.E-810 |
0.751214327591 |
0.000260416666667 |
0.891123237110 |
-0.0000759548611111 |
0.998492144926 |
0.00000155009920635 |
1.07840852624 |
0.0000154041108631 |
1.13654884139 |
-0.00000907453910384 |
1.17810073526 |
-0.0000000828199706170 |
1.20737494332 |
0.00000360740727676 |
1.22775702511 |
-0.00000169514972633 |
1.24180758666 |
-0.00000133089916348 |
1.25141142987 |
0.00000177521444910 |
1.25792762288 |
0.000000370353976658 |
1.26232035257 |
-0.00000191475684776 |
1.26526469485 |
0.000000344673434042 |
1.26722815289 |
0.00000241913411616 |
1.26853149270 |
-0.00000147705874041 |
1.26939305279 |
-0.00000360462602023 |
1.26996042399 |
0.00000426030599723 |
1.27033276648 |
0.00000619401781838 |
1.27057634173 |
-0.0000126252925336 |
1.27073521306 |
-0.0000117360887110 |
1.27083855431 |
0.0000413952285774 |
1.27090560466 |
0.0000222030302120 |
1.27094900588 |
-0.000153108566769 |
1.27097703715 |
-0.0000278327871472 |
1.27099510405 |
0.000641018661899 |
1.27100672598 |
-0.000111307516319 |
1.27101418835 |
-0.00303026666274 |
1.27101897160 |
0.00167666962999 |
1.27102203256 |
0.0160951154545 |
1.27102398833 |
-0.0157084159784 |
1.27102523611 |
-0.0954804645039 |
1.27102603109 |
0.139489606827 |
1.27102653689 |
0.628520649401 |
1.27102685831 |
-1.27694165810 |
1.27102706231 |
-4.55956399021 |
1.27102719163 |
12.4027724564 |
1.27102727353 |
36.1854546816 |
1.27102732533 |
-129.305594720 |
1.27102735806 |
-311.608441223 |
1.27102737873 |
1453.71643376 |
1.27102739177 |
2883.75499733 |
1.27102739998 |
-17648.6056027 |
1.27102740515 |
-28323.2666121 |
1.27102740840 |
231312.842070 |
1.27102741045 |
289837.706925 |
1.27102741173 |
-3269335.96562 |
1.27102741254 |
-2992168.60724 |
1.27102741304 |
49750634.1587 |
1.27102741336 |
28980063.0330 |
1.27102741356 |
-813616473.772 |
1.27102741368 |
-201961594.949 |
1.27102741376 |
14271686431.9 |
1.27102741381 |
-1325490857.72 |
1.27102741384 |
-267978508283. |
1.27102741386 |
119319788076. |
1.27102741387 |
5.37563669599E12 |
1.27102741388 |
-4.37013046468E12 |
1.27102741388 |
-1.14977800862E14 |
1.27102741388 |
1.37951986894E14 |
1.27102741388 |
But using more terms it is obvious that that convergence is only local; So I used an experimental stronger parametrization of the Noerlund-summation (in my implementation with two parameters), nonetheless apparently arriving at the same value
Approx = Noerlund(1.4, 1.3) * dV(1)* U0.51 //
we need Noerlund-summation for this because series is strongly diverging
Using 256 terms we find the following approximation :
U0.51 |
Partial sums: Approx: |
0 |
0 |
0 |
1 |
1.00000000000 |
0.4166666666666666666666667 |
2 |
0.250000000000 |
0.6802187646414115767312961 |
3 |
0.0208333333333 |
0.8543367076839841850527931 |
4 |
1.063167461E-204 |
0.9728122533370859110058801 |
5 |
0.000260416666667 |
1.055158890718194379537863 |
6 |
-0.0000759548611111 |
1.113322049145836683676599 |
7 |
0.00000155009920635 |
1.154927026644590189403069 |
8 |
0.0000154041108631 |
1.184995099232425260962484 |
9 |
-0.00000907453910384 |
1.206912382008609448210350 |
10 |
-0.0000000828199706170 |
1.223005413341123686437128 |
11 |
0.00000360740727676 |
1.234896974845728865717737 |
12 |
-0.00000169514972633 |
1.243733093659537186974143 |
13 |
-0.00000133089916348 |
1.250331573819650138051143 |
14 |
0.00000177521444910 |
1.255281207630202865239360 |
15 |
0.000000370353976658 |
1.259009183115809527489827 |
16 |
-0.00000191475684776 |
1.261827541981969525892891 |
17 |
0.000000344673434042 |
1.263965589964040800596140 |
18 |
0.00000241913411616 |
1.265592744460861863146730 |
19 |
-0.00000147705874041 |
1.266834787928400056087080 |
20 |
-0.00000360462602023 |
1.267785523004340843822270 |
21 |
0.00000426030599723 |
1.268515189788157734675139 |
22 |
0.00000619401781838 |
1.269076583543275748684661 |
23 |
-0.0000126252925336 |
1.269509526672548941751353 |
24 |
-0.0000117360887110 |
1.269844154811896911030546 |
25 |
0.0000413952285774 |
1.270103343096363738965483 |
26 |
0.0000222030302120 |
1.270304505488932369876442 |
27 |
-0.000153108566769 |
1.270460934623827685972399 |
28 |
-0.0000278327871472 |
1.270582803292161357418455 |
29 |
0.000641018661899 |
1.270677915673107836555729 |
30 |
-0.000111307516319 |
1.270752272718673998185568 |
31 |
-0.00303026666274 |
1.270810498998789451747707 |
32 |
0.00167666962999 |
1.270856165904167737114013 |
33 |
0.0160951154545 |
1.270892037054775901782266 |
34 |
-0.0157084159784 |
1.270920255131584852660284 |
35 |
-0.0954804645039 |
1.270942484470710617007680 |
36 |
0.139489606827 |
1.270960020154843030156470 |
37 |
0.628520649401 |
1.270973871664026359126703 |
38 |
-1.27694165810 |
1.270984827158658745978229 |
39 |
-4.55956399021 |
1.270993502982183114841320 |
40 |
12.4027724564 |
1.271000381858227299430625 |
41 |
36.1854546816 |
1.271005842420900556793057 |
42 |
-129.305594720 |
1.271010182086980195864213 |
|
|
|
… |
… |
… |
|
|
|
242 |
1.99391580834E202 |
1.271027413889951521419677 |
243 |
-1.89349634501E202 |
1.271027413889951521420391 |
244 |
-7.29686595931E204 |
1.271027413889951521421002 |
245 |
9.45085000863E204 |
1.271027413889951521421524 |
246 |
2.71412390180E207 |
1.271027413889951521421972 |
247 |
-4.46257691886E207 |
1.271027413889951521422355 |
248 |
-1.02596030513E210 |
1.271027413889951521422683 |
249 |
2.04869812900E210 |
1.271027413889951521422964 |
250 |
3.94079346151E212 |
1.271027413889951521423204 |
251 |
-9.27402089917E212 |
1.271027413889951521423411 |
252 |
-1.53792247833E215 |
1.271027413889951521423587 |
253 |
4.17365586728E215 |
1.271027413889951521423739 |
254 |
6.09718075121E217 |
1.271027413889951521423868 |
255 |
-1.87702245373E218 |
1.271027413889951521423980 |
We seem to arrive at a value d( 1, 0.5) ~ 1.27102741388995152142.., which reinserted into that powerseries (and again be Noerlund-summed) gives a reasonable approximation to d(d(1,0.5),0.5) = d(1,1) .
However, it is possible that this is still only a local approximation. It seems, that the growthrate of the terms is even more than hypergeometric; a plot of the quotients of successive coefficients indicates strongly, that in the average the coefficients grow dependent on the index k like O((k!)1+δ) (with oscillations) where if δ>0 significantly the used Noerlund-summation is not strong enough. The following plot shows the growthrate by the formula
Note: I used dxp(x,h) in the program for the drawing of the graphic instead of d(x,h)
Trend: (estimated using Excel)
ln(ck/ck-1)
= 1.0722ln(k)-2.9185
ck ~(-0.054)k (k!)1.07
Gottfried Helms
(Last update at 12.2.2011 // 3.Nov.2010)
[Comtet] Comtet,
L;
Advanced Combinatorics (1974 edition)
D. Reidel Publishing Company, Dordrecht - Holland
[A&S] Abramowitz,
M. and Stegun, I.A.;
Handbook of mathematical functions 9th printing
page 824; online at http://www.math.sfu.ca/~cbm/aands/
[Erdös] Erdös,
Paul , Jabotinsky, Eri;
On analytic iteration
J. Anal. Math. 8, 361-376 (1961) (also
online at digicenter göttingen)
[Baker] Baker, I.N.;
Zusammensetzung ganzer Funktionen,
Math Zeitschr. Bd. 69 pp 121-163 (1958)
(also online at digicenter göttingen)
Projectindex
http://go.helms-net.de/math
http://go.helms-net.de/math/tetdocs