Cheta with base-change: preliminary results

[Gottfried]

Actually, since I'm trying to find the Taylor series, I don't worry about the factorials. So I work directly with the matrix above, where the first row is [1 -2 4 -8 16].

Inverting this and multiplying by 4!, or (n-1)! in general, I get back a matrix of integer coefficients. For this 5x5 matrix, I get:

Code:

[  0   0  24   0   0]
[  2 -16   0  16  -2]
[ -1  16 -30  16  -1]
[ -2   4   0  -4   2]
[  1  -4   6  -4   1]

Yes, exactly that's wat XI is for.
Using

Code:

´
  dim=5
  ScI = XI *FacI*PI *(dim-1)!

I get ScI (= Sc^-1 without general matrix-inversion!)

Code:

´
   0    0   24   0   0
   2  -16    0  16  -2
  -1   16  -30  16  -1
  -2    4    0  -4   2
   1   -4    6  -4   1

If I use dim=128 I need at most one second to get XI, dim=256 it's at most two seconds .
PI needs the signed binomials, and FacI the reciprocals of factorials, which is "wired" in the CAS we use.

Just for fun. The top 12x2 of the matrix ScI for the dim=63-problem

Code:

0                                                                       .
  -2181131468794922353615366650200339706856997013317222400000000000000   139737822767461358788291156722835097219304941986523381760000000000000
     70359079638545882374689246780656119576032161719910400000000000000    -4657927425582045292943038557427836573976831399550779392000000000000
   3516311353922279385648333502150600539038512466240565411840000000000  -225268491925020959287679489529214235962391937424715012898816000000000
   -113429398513621915666075274262922598033500402136792432640000000000     7508949730834031976255982984307141198746397914157167096627200000000
  -1654079054684419920279666586121037397819476185831565793689600000000   105955453869348853785751995418154264616278795206219780639948800000000
     53357388860787739363860212455517335413531489865534380441600000000    -3531848462311628459525066513938475487209293173540659354664960000000
    359946168888362231354382569977986774675428522317679104622592000000   -23053094875016314001471918871866472674486775969756723710577868800000
    -11611166738334265527560728063806024989529952332828358213632000000      768436495833877133382397295728882422482892532325224123685928960000
    -44332175994731622306985432635107929937606776681530699263508480000     2838596603460567513445078978963484415564939643140404602334085120000
      1430070193378439429257594601132513868955057312307441911726080000      -94619886782018917114835965965449480518831321438013486744469504000
      3462849129077008473136231592189678512700299197020396105131622400     -221654683043115153762232400649777616025545242953274974194850856960

...

and for a quick crosscheck also the 5 bottom rows.

Code:

...
  42061513  -2640168816  81493009963  -1648811309600  24593123988525  -288370678815056  2768098801017967  -22366853583637056
    293105    -17699760    525083425    -10199902720    145903801905    -1638716546480    15047425388225    -116142579256320
     -9455       589992    -18106325       364282240     -5403844515       63027559480     -601897015529       4839274135680
       -31         1860       -54839         1058960       -15061815         168246052       -1536862975         11803107648
         1          -62         1891          -37820          557845          -6471002          61474519          -491796152

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