The coefficients of the formal powerseries in all the following tables are listed in the four groups for indexes k=0+4*j, k=1+4*j, k=2+4*j and k=3+4*j so the powerseries must be constructed by reading the coefficientstable rowwise: (see table 1:)
f(x) = 0 + 1*x + 0.5*x^2 + 0.25*x^3+0*x^4-0.125*x^5 + …
The values are rounded to at most 20 digits precision
The coefficients are computed by solving the system of linear equations given by the formal-powerseries expression and the functional equation and stored in a columnvector C.
Pari/Gp: msep(Mat(1.0 * C),4) // generate 4 columns
|
j |
k=0+4*j |
k=1+4*j |
k=2+4*j |
k=3 + 4*j |
|
0 |
0 |
1.0 |
0.5 |
0.25 |
|
1 |
0 |
-0.125 |
0 |
0.203125 |
|
2 |
0 |
-0.56640625 |
0 |
2.2734375 |
|
3 |
0 |
-12.154296875 |
0 |
82.94464111328125 |
|
4 |
0 |
-703.072265625 |
0 |
7256.3267326354980469 |
|
5 |
0 |
-89745.217952728271484 |
0 |
1312224.1918640136719 |
|
6 |
0 |
-22417090.807624161243 |
0 |
442786094.82607311010 |
|
7 |
0 |
-10019069234.956951141 |
0 |
257568676676.25336117 |
|
8 |
0 |
-7467678385190.8180471 |
0 |
242566269546205.76846 |
|
9 |
0 |
-8774848164193169.9917 |
0 |
351620564083979346.32 |
|
10 |
0 |
-15531448551067605373. |
0 |
7.5286878628995499649E20 |
|
11 |
0 |
-3.9886735381186916520E22 |
0 |
2.3010051165013368823E24 |
|
12 |
0 |
-1.4404342653983849935E26 |
0 |
9.7538778343572586847E27 |
|
13 |
0 |
-7.1234822648979373949E29 |
0 |
5.5956801953184238188E31 |
|
14 |
0 |
-4.7157972738557446783E33 |
0 |
4.2537213112625261693E35 |
|
15 |
0 |
-4.0976158782126604755E37 |
0 |
4.2066740672814747817E39 |
|
16 |
0 |
-4.5935028310265258395E41 |
0 |
5.3253586643677463923E43 |
|
17 |
0 |
-6.5433717457400760496E45 |
0 |
8.5073113921491861915E47 |
|
18 |
0 |
-1.1685568507605221349E50 |
0 |
1.6933178958860089169E52 |
|
19 |
0 |
-2.5849698234049366885E54 |
0 |
4.1517227317727836772E56 |
|
20 |
0 |
-7.0066881488349954826E58 |
0 |
1.2410538803185083949E61 |
|
21 |
0 |
-2.3044537664409700782E63 |
0 |
4.4809863305922725774E65 |
|
22 |
0 |
-9.1150142765464911615E67 |
0 |
1.9377100450137473797E70 |
|
23 |
0 |
-4.3008667358048960096E72 |
0 |
9.9578206430529793498E74 |
|
24 |
0 |
-2.4028979829281814754E77 |
0 |
6.0381785049956473522E79 |
|
25 |
0 |
-1.5788008990704301692E82 |
0 |
4.2920456296404958396E84 |
|
26 |
0 |
-1.2122560075037148162E87 |
0 |
3.5547327262491254371E89 |
|
27 |
0 |
-1.0814416165454142218E92 |
0 |
3.4110997229268980855E94 |
|
28 |
0 |
-1.1148151395051822886E97 |
0 |
3.7727668121842937777E99 |
|
29 |
0 |
-1.3213152999403604849E102 |
0 |
4.7862105959415043382E104 |
|
30 |
0 |
-1.7921473201822500315E107 |
0 |
6.9329453179377657584E109 |
|
31 |
0 |
-2.7694728223830348788E112 |
0 |
1.1418050316520052996E115 |
The integer part of the log10 of a number |ck| gives the number of digits in the decimal expansion before the decimal-dot. The increase of the logarithms should be at most linear if a formal powerseries shall have nonzero radius of convergence.
|
j |
k=1+4*j |
k=3 + 4*j |
|
0 |
0. |
-0.60205999132796239043 |
|
1 |
-0.90308998699194358564 |
-0.69223662167705040208 |
|
2 |
-0.24687196307687466959 |
0.35668301933803892258 |
|
3 |
1.0847298400086397687 |
1.9187883323722939419 |
|
4 |
2.8469999664837994578 |
3.8607168295903467520 |
|
5 |
4.9530113166144502286 |
6.1180080400484722581 |
|
6 |
7.3505792510700896345 |
8.6461939739159709878 |
|
7 |
10.000827377729057688 |
11.410893046677686591 |
|
8 |
12.873185605669029671 |
14.384830409191622998 |
|
9 |
15.943239610377341108 |
17.546074266307983625 |
|
10 |
19.191211962386503845 |
20.876719291797270959 |
|
11 |
22.600828491963983429 |
24.361917584364352553 |
|
12 |
26.158493443924211440 |
27.989177311825900496 |
|
13 |
29.852692347390201900 |
31.747852885702326108 |
|
14 |
33.673555127055327685 |
35.628769033064220105 |
|
15 |
37.612531244013553107 |
39.623938864224113932 |
|
16 |
41.662143988371838519 |
43.726348862948373431 |
|
17 |
45.815801594383245261 |
47.929792329527255024 |
|
18 |
50.067649845843340696 |
52.228738498255940763 |
|
19 |
54.412455477563340641 |
56.618228341930496667 |
|
20 |
58.845512788500562283 |
61.093790636790470763 |
|
21 |
63.362567999429086196 |
65.651373619084343911 |
|
22 |
67.959757353212663361 |
70.287288790634403958 |
|
23 |
72.633555985962319119 |
74.998164299642405573 |
|
24 |
77.380735332854732323 |
79.780905947804118377 |
|
25 |
82.198327365038663615 |
84.632664330394505376 |
|
26 |
87.083594344999769575 |
89.550806952497907686 |
|
27 |
92.034003078285899425 |
94.532894416119152577 |
|
28 |
97.047202857923320777 |
99.576659963082981232 |
|
29 |
102.12100646385606091 |
104.67999180389779403 |
|
30 |
107.25337370717457670 |
109.84091777481127122 |
|
31 |
112.44239710764952223 |
115.05759195250310926 |
The difference of the logarithms should be at most constant (if not decreasing) if a formal powerseries shall have nonzero radius of convergence. (Here the differences are shown as differences between each fourth coefficent: simply the differences along the columns in the above table).
That first 64 differences increase so far and this suggests that the formal powerseries will have convergence-radius zero.
Caveat: I did not yet perform deeper analysis of that rate of increase of differences of second and higher order and it is not impossible that the rate of increase approximates a constant upper bound and thus the powerseries of f(x) had in fact nonzero radius of convergence.
|
j |
k=1+4*j |
k=3 + 4*j |
|
0 |
0.0 |
-0.60 |
|
1 |
-0.90 |
-0.09 |
|
2 |
0.65 |
1.04 |
|
3 |
1.33 |
1.56 |
|
4 |
1.76 |
1.94 |
|
5 |
2.10 |
2.25 |
|
6 |
2.39 |
2.52 |
|
7 |
2.65 |
2.76 |
|
8 |
2.87 |
2.97 |
|
9 |
3.07 |
3.16 |
|
10 |
3.24 |
3.33 |
|
11 |
3.40 |
3.48 |
|
12 |
3.55 |
3.62 |
|
13 |
3.69 |
3.75 |
|
14 |
3.82 |
3.88 |
|
15 |
3.93 |
3.99 |
|
16 |
4.04 |
4.10 |
|
17 |
4.15 |
4.20 |
|
18 |
4.25 |
4.29 |
|
19 |
4.34 |
4.38 |
|
20 |
4.43 |
4.47 |
|
21 |
4.51 |
4.55 |
|
22 |
4.59 |
4.63 |
|
23 |
4.67 |
4.71 |
|
24 |
4.74 |
4.78 |
|
25 |
4.81 |
4.85 |
|
26 |
4.88 |
4.91 |
|
27 |
4.95 |
4.98 |
|
28 |
5.01 |
5.04 |
|
29 |
5.07 |
5.10 |
|
30 |
5.13 |
5.16 |
|
31 |
5.18 |
5.21 |
The original coefficients are transformed by a matrix-transformation according to the scheme of Noerlund-summation. The process is defined in two steps:
a) transform the sequence of coefficients in a column-vector C to the sequence of partial sums in a vector S . This can be done by a matrix-multiplication with a lower triangular matrix DR consisting of 1 only: S = DR * C
b) transform the partial sums in S using a lower triangular matrix M (with all rowsums=1 and certain other required properties) into the transformed vector T by T = M*S . If the sequence of coefficients in T converge they converge to the Noerlund-sum of f(1).
If we want to evaluate f(x) for x<>1 we have to adapt step a)
a1) The k'th entry in C must be cofactored by xk such that we get the vectors Cx and Sx
It is not trivial to find appropriate matrices M. I've an experimental method derived from Euler-summation which allows two parameters for the control of the "order" (which means power of the method) and seems to allow to sum hypergeometric series if they have alternating signs. I write Ms,t = NoerlundMat(s,t) where s and t are the parameters for the "order". Then the initial transform into partial sums by DR is also included into M, so we do not need the column vector Sx explicitely:
b2) Ms,t =
NoerlundMat(s,t)*DR
Tx = Ms,t * Cx
The parameters s and t can roughly be estimated by the growthrate of the coefficients in Cx and then be finetuned to get convergence for a certain truncation of the powerseries to, say, n=128 terms. Finally, if we have appropriate s and t and they are sufficient even for a certain interval of x we can use that configuration for the estimation of the function f(x) for that interval. Then we need no more the complete vector Tx "to see" the converging partial sums but only the last entry. So we can reformulate the evaluation of the function f(x) by the polynomial expression g(x) using the coefficients of the last row of Ms,t only:
c) g(x) = Ms,t [n , ]*Cx
Finally we can even define fixed coefficients dk for the polynomial g(x) such that
d) dk = Ms,t[n,k] * C[k]
g(x) = sum (
k=0..n, dk * xk )
and we shall have
g(x) ~ f(x) in a certain interval for 0 ≤ |x| ≤ <upper bound>
Here are such coefficients dk allowing a radius for x of about 0 ≤ |x| ≤ 2 . The parameters for the NoerlundMat were s=2.0, t=2.0 for my implementation of that summation-procedure and likely allow a wider range of x.
Coefficients dk for g(x) after Noerlund-transformation (2.0,2.0, n=128)
|
j |
k=0+4*j |
k=1+4*j |
k=2+4*j |
k=3 + 4*j |
|
0 |
0 |
0.99999852217335088445 |
0.49995234009056602358 |
0.24960666720092133995 |
|
1 |
0 |
-0.11854872816204695391 |
0 |
0.13909227071570742428 |
|
2 |
0 |
-0.16321790298810986304 |
0 |
0.14556668797777364888 |
|
3 |
0 |
-0.092659364396231715751 |
0 |
0.042360485241538665072 |
|
4 |
0 |
-0.014254133187443249926 |
0 |
0.0036257690515390767526 |
|
5 |
0 |
-0.00071408910950032942897 |
0 |
0.00011113114508112695202 |
|
6 |
0 |
-0.000013900223237317901918 |
0 |
0.0000014172804184334123117 |
|
7 |
0 |
-0.00000011920494381165783752 |
0 |
0.0000000083541175209951350161 |
|
8 |
0 |
-0.00000000049203633661723641130 |
0 |
2.4534755383571286964E-11 |
|
9 |
0 |
-1.0423558757682540781E-12 |
0 |
3.7939764987149730098E-14 |
|
10 |
0 |
-1.1887728266728606497E-15 |
0 |
3.2198778018906497803E-17 |
|
11 |
0 |
-7.5664203184159705866E-19 |
0 |
1.5474533020412982039E-20 |
|
12 |
0 |
-2.7618476850193236247E-22 |
0 |
4.3116934249302280231E-24 |
|
13 |
0 |
-5.8995590607700057576E-26 |
0 |
7.0864867870306307876E-28 |
|
14 |
0 |
-7.4828439403818582868E-30 |
0 |
6.9532384417708716722E-32 |
|
15 |
0 |
-5.6902912599374182172E-34 |
0 |
4.1033355332885080981E-36 |
|
16 |
0 |
-2.6080250354221044438E-38 |
0 |
1.4610528974835199775E-40 |
|
17 |
0 |
-7.2126888089847358030E-43 |
0 |
3.1361170366164466680E-45 |
|
18 |
0 |
-1.2001193378053455445E-47 |
0 |
4.0378440854142736461E-50 |
|
19 |
0 |
-1.1928899960931690810E-52 |
0 |
3.0894376347056581581E-55 |
|
20 |
0 |
-7.0008284811803382440E-58 |
0 |
1.3849111419296878497E-60 |
|
21 |
0 |
-2.3852982526906005953E-63 |
0 |
3.5659420835481077370E-66 |
|
22 |
0 |
-4.6107770596120193867E-69 |
0 |
5.1353230708805371805E-72 |
|
23 |
0 |
-4.9036082919980959221E-75 |
0 |
3.9927518493304392631E-78 |
|
24 |
0 |
-2.7550742267283330087E-81 |
0 |
1.5994484764619244911E-84 |
|
25 |
0 |
-7.7472094367919585292E-88 |
0 |
3.1002730536866830412E-91 |
|
26 |
0 |
-1.0132276980188306357E-94 |
0 |
2.6672853545167160316E-98 |
|
27 |
0 |
-5.5621072731599007995E-102 |
0 |
9.0014966023065726664E-106 |
|
28 |
0 |
-1.1018904236646584200E-109 |
0 |
9.8710019538348320894E-114 |
|
29 |
0 |
-6.1929313293258333032E-118 |
0 |
2.5595926901140559924E-122 |
|
30 |
0 |
-6.3637697733415820885E-127 |
0 |
8.1989616768106427630E-132 |
|
31 |
0 |
-4.0894689153726913675E-137 |
0 |
3.2919700322651948880E-143 |
so we have
g(x) =
0.99999852217335088445 x +0.49995234009056602358 x2
+0.24960666720092133995 x3
–
… + 3.2919700322651948880E-143 x127
≈f(x)
usable for the interval about 0 ≤ |x| ≤ 2
Gottfried Helms