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the inconsistency depending on fixpoint-selection
#21
An interesting result occurs, if e<>1, thus if rescaling is allowed.
I tried to find one e, for which the resulting powerseries are linear scalings of each other, such that

g0(x)=c * g1(x)

This is possible.

Assume again, that, denoting the fixpoints p and q, such that
f(x) = x + (x - p) (x - q)
Denote also the substitution
z = x' = x/e - d
and the backsubstitution
x" = (x + d)*e
Recall, that for a fixpoint p, d and e must be selected that p = e*d

If I select
Code:
´
    e0 = 1 + (p-q)    d0=p/e0          // first fixpoint
    e1 = 1 + (q-p)    d1=q/e1          // second fixpoint
then the resulting G0 and G1-matrices provide the same powerseries with a linear scaling.

Example
Code:
´
    using fixpoint p                     using fixpoint q                              
    e0 = 1 + (p-q)                       e1 = 1 + (q-p)                              
    z0 =  x/e0 - d0 = (x-p)/e0           z1 =  x/e1 - d1 = (x-q)/e1                  
                                                                                  
    g0(z) = e0 z^2 + (1+(p-q))z          g1(z) = e1 z^2 + (1+(q-p))z



The matrices and eigenmatrices for g0 and g1 are as follows:
Code:
´
W0                                       W1              
  1        0       0    0     0     0  |  1            0         0      0      0       0
  0        1       0    0     0     0  |  0            1         0      0      0       0
  0        2       1    0     0     0  |  0           -2         1      0      0       0
  0      8/3       4    1     0     0  |  0         24/5        -4      1      0       0
  0     24/7    28/3    6     1     0  |  0     -1176/95      68/5     -6      1       0
  0  416/105  368/21   20     8     1  |  0  204768/6175  -4176/95  132/5     -8       1
  ...                                      ...

D0                                       D1  
  1      1/2     1/4  1/8  1/16  1/32  |  1          3/2       9/4   27/8  81/16  243/32

W0^-1                                    W1^-1
  1        0       0    0     0     0  |  1            0         0      0      0       0
  0        1       0    0     0     0  |  0            1         0      0      0       0
  0       -2       1    0     0     0  |  0            2         1      0      0       0
  0     16/3      -4    1     0     0  |  0         16/5         4      1      0       0
  0  -352/21    44/3   -6     1     0  |  0       416/95      52/5      6      1       0
  0  2048/35  -384/7   28    -8     1  |  0   32768/6175   2048/95  108/5      8       1
  ...                                      ...
--------------------------------------- -------------------------------------------------
G0                                      G1
  1        0       0    0     0     0  |  1            0         0      0      0       0
  0      1/2       0    0     0     0  |  0          3/2         0      0      0       0
  0      1/2     1/4    0     0     0  |  0          3/2       9/4      0      0       0
  0        0     1/2  1/8     0     0  |  0            0       9/2   27/8      0       0
  0        0     1/4  3/8  1/16     0  |  0            0       9/4   81/8  81/16       0
  0        0       0  3/8   1/4  1/32  |  0            0         0   81/8   81/4  243/32
  ...                                      ...

The additionally interesting thing is, that we get by this column-scaled binomial-matrices for G0 and G1
X = G0 * dV(2) = G1* dV(2/3)
Code:
X=
  1  .  .  .  .   .  .  .
  0  1  .  .  .   .  .  .
  0  1  1  .  .   .  .  .
  0  0  2  1  .   .  .  .
  0  0  1  3  1   .  .  .
  0  0  0  3  4   1  .  .
  0  0  0  1  6   5  1  .
  0  0  0  0  4  10  6  1
so maybe I have just introduced some triviality.

Anyway- this does not (yet?) provide a linear scaling for the half-powers. Perhaps one can find an x->x' substitution, which provides such a linear scaling - but this is only an idea, don't know, whether this is even possible.

Gottfried
Gottfried Helms, Kassel
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Messages In This Thread
RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 01:22 PM

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