open problems survey

[Gottfried]

My examples with truncations of the Carlemanmatrices from sizes \(4 \times 4,8 \times 8,16 \times 16,32 \times 32 \) up to \(64 \times 64 \) with some random chosen small bases (like \(b=\sqrt 2 , b=4 , b=e \) ) show nice improvement of approximations to 5 or 10 ore more correct digits.

sorry to ask here , but has it been proven that all rising integer sequences (of size) give the same result ?

In all my heuristics increasing the sizes of the truncations of square(*) Carlemanmatrices the coefficients as well as the results seem to approximate limit values. However, due to extreme need of internal digits-precision I can with reasonable effort only go up to \( 64 \times 64 \) matrices, sometimes a bit more (see fractional iteration of the sine-function, in MO), so this does not really mean much...

(*) for triangular Carlemanmatrix this problem is of much lesser relevance

Example: Carlemanmatrix half-iterate of \(f(x)= 4^x \) for sizes \( d \times d \) for \(d =4,8,12,16,20,24 \) , coefficients c_k for the powerseries seem to converge to limit-values when d is increased:

Code:

.         4x4              8x8                   12x12                16x16              20x20             24x24       |    Kneser(*)    
------------------------------------------------------------------------------------------------------------------------------------------
   0.458786484402      0.457506956745      0.457297994170      0.457244494586      0.457226376487      0.457219121381  |  0.457213238216
    1.01416292699       1.00585184963       1.00570266416       1.00576279604       1.00580782971       1.00583514094  |  1.00588535950
   0.354910001792      0.372463884665      0.374338685963      0.374762250136      0.374890508012      0.374935814278  |  0.374953659250
  0.0423749372335     0.0485034871162     0.0483181418142     0.0480895085885     0.0479650214772     0.0478979652601  |  0.0477946354397
                .  -0.000531678816504   -0.00167964829897   -0.00193989480094   -0.00201097333281   -0.00203167751111  |  -0.00202366092283
                .   0.000528586184651   0.000787045655542   0.000917125824152   0.000980591763273    0.00101283363819  |  0.00105579202994
                .   -3.94783452855E-5    8.17659708660E-5   0.000124080100384   0.000130492347057   0.000128471429488  |  0.000111287079756
                .   -4.44632351059E-6  -0.000137772165702  -0.000205048904088  -0.000233715355744  -0.000246849844561  |  -0.000259506482216
                .                   .    4.27004369289E-5    5.64191495551E-5    6.41783344678E-5    6.96058835299E-5  |  8.20289987826E-5
                .                   .   -2.97008358857E-6    1.57094892640E-5    2.55689613195E-5    2.99812302337E-5  |  3.25772461877E-5
                .                   .   -1.23612708539E-6   -1.52680772749E-5   -2.18517047449E-5   -2.54670434954E-5  |  -3.19939947128E-5
                .                   .    2.28200841656E-7    3.19014711939E-6    1.42388118905E-6    3.52461380591E-7  |  3.94504668958E-7
                .                   .                   .    7.54744807314E-7    4.18053003922E-6    6.04642021637E-6  |  9.15787915121E-6
                .                   .                   .   -5.49383830555E-7   -1.68109328381E-6   -1.70179488684E-6  |  -2.15328520187E-6
                .                   .                   .    1.16400876115E-7   -2.15881955255E-7   -1.03326553912E-6  |  -2.50691103240E-6
                .                   .                   .   -9.14068746483E-9    3.92005894211E-7    7.18657474565E-7  |  1.11459855070E-6
                .                   .                   .                   .   -1.46087335360E-7    2.28851750529E-8  |  7.31756775619E-7
                .                   .                   .                   .    2.60282057394E-8   -1.80477053045E-7  
                .                   .                   .                   .   -1.96351794647E-9    7.64568418432E-8  
                .                   .                   .                   .   9.26756456931E-12   -7.14784297921E-9  
                .                   .                   .                   .                   .   -4.93066217462E-9  
                .                   .                   .                   .                   .    2.09545681167E-9  
                .                   .                   .                   .                   .  -3.50584306552E-10  
                .                   .                   .                   .                   .   2.31284640845E-11

"Kneser(*)" here means: fatou.gp 1708 computed with internal precision 60 digits using "derivnum" by Pari/GP :

Code:

.    for(d=0,16,print(d," ",derivnum(x=0,SLtet(x,0.5),d)/d!))

Note: the "derivnum()"-procedure becomes inaccurate for higher derivatives, so only 16 are documented here!