Borel summation

[Gottfried]

practically not better than my handy matrix-summation method, ready made for use in Pari/GP.        

What is that method? Does it also compute regular iteration?

No. It is just a (parametrizable) summation matrix, say \( M( order, dim=128 ) \) which by  M( 1.4 ,128 )  * A  performs summation (in the Noerlund style) of the fixed sequence in the cells of a column-vector \( A \), taking the first 128 terms, showing the first 128 partial sums (of the transformed sequence in \(A \) ).

If in \( A \) I have geometrical growth, I take the matrix M for the Euler-summation, and if the growth is factorially I use the matrix M for Noerlund-summation (with parameter "order" casewise adapted) and so on...

I have no such matrix M for the Borel-summation because then another parameter (the "x" from the description above) must be taken in respect too ... and too much fiddling ...

Example: my Pari/GP-command PkPowSum( 1.15, 1.0, 64) * Mat( G[1..64] ) uses the first 64 coefficients of the powerseries of \( g(z) \) which are stored in columnvector \( G \), and the PkPowSum() gives the transformation/summation-matrix for the Noerlund-summation with order 1.15 (which should capture sequences in G with at most factorially growth), and in a second window occur the leading 64 partial sums (which I can copy&paste to clipboard for documentation) :

Code:

                      0
  0.50000000000000000000
  0.80142587824806072141
  0.98432556940817345050
  1.0957027560716282926
  1.1636752922666828048
  1.2052223288847365712
  1.2306484763014693961
  1.2462257520559636128
  1.2557788388344492333
  1.2616432434973325406
  1.2652467543462965175
  1.2674631429332411599
(...)
  1.2710274138627236401
  1.2710274138726573085
  1.2710274138789613347
  1.2710274138829668630
  1.2710274138855097373
  1.2710274138871189746
  1.2710274138881450066
  1.2710274138888130869
  1.2710274138892329362
  1.2710274138894639965
  1.2710274138896221822
  1.2710274138898144174

Using the functional equation with \( z_{-40} = f°^{-40}(1) \approx 0.048759544221234150945 \)
then I can even use the simpler Euler-sum matrix with little order 0.25 partsums = ESum(0.25,64) * dV(z_40,64) * Mat( G[1..64] )   getting

Code:

                        0
  0.039007635376987320756
  0.047189561356810783097
  0.048903262872140813102
  0.049261713746775914190
  0.049336595521074853097
  0.049352220101960622416
  0.049355476647235224581
  0.049356154679576723518
(...)
  0.049356332694373519259
  0.049356332694373519273
  0.049356332694373519276
  0.049356332694373519276
  0.049356332694373519277
  0.049356332694373519277
  0.049356332694373519277
  0.049356332694373519277
  0.049356332694373519277
  0.049356332694373519277
  0.049356332694373519277
  0.049356332694373519277
  0.049356332694373519277

and the retransformation, iterating each of this partial sums 40 times forward  by     Mat( vectorv ( 64,r,  it ( partsums [r,1] , 40 )))  gives then the approximation-protocol

Code:

                0.E-808
  0.17007875242702049417
  0.62004003166073784848
  1.0553493687949762372
  1.2198838762606499392
  1.2600508274304994244
  1.2687263993243629131
  1.2705478466365730441
  1.2709276617668611349
  1.2710066884474990569
(...)
  1.2710274138899515195
  1.2710274138899515210
  1.2710274138899515213
  1.2710274138899515214
  1.2710274138899515214
  1.2710274138899515214
  1.2710274138899515214
  1.2710274138899515214
  1.2710274138899515214
  1.2710274138899515214
  1.2710274138899515214
  1.2710274138899515214

- - - -
Just to see "the workflow":

image