The Power Series,
a_0 + a_1*x + a_2*x^2 + a_3*x^3 + ... + a_n*x^n = 0
May be expressed as the dot product Q*P where
Q=(a_0,a_1,a_2,a_3,...,a_n) and
P=(x^0,x^1,x^2,x^3,...,x^n)
The exponential function e^x may be expressed as the dot
product E*P where
E=(1/0!,1/1!,1/2!,1/3!,...,1/n!)=(w_0,w_1,w_2,w_3,...,w_n)
P may be expressed as ratios of Q and E:
qQ + sE = P
Taking the dot product of both sides by Q and then by E leads to,
|Q|^2 E*Q | 0
E*Q |E|^2 | e^x
This matrix has the solution,
q=-[(Q*E)e^x]/[(Q*E)^2-|Q|^2|E|^2]
s=[|Q|^2 e^x]/[(Q*E)^2-|Q|^2|E|^2]
Substituting these,
P=[|Q|^2 E-(Q*E) Q]e^x /[(Q*E)^2-|Q|^2|E|^2] eqn 1
for which
x^0 = [|Q|^2 w_0 - (Q*E) a_0]e^x / [(Q*E)^2-|Q|^2|E|^2]
Dividing the other components by this instance cancels e^x and
x^n = [|Q|^2 E-(Q*E) Q]/[|Q|^2 w_0 - (Q*E) a_0] eqn 2
APPLICATION
The power series, 1+x+x^2+x^3+...+x^n=0 diverges when
x >=1 and has the nth root,
x = ln{[n!(n+1)|E|^2 - n!e^2]/[n!e - (n+1)]
Solve eqn 1 for e^x and use the nth component of E and Q:
e^x = [|Q|^2 |E|^2 - (Q*E)^2] / [(Q*E)a_n -|Q|^2 w_n]
Q=(1,1,1,1,...,1)
E=(1/0!,1/1!,1/2!,1/3!,...,1/n!)
|Q|^2 = n+1
|E|^2 = (1/0!)^2 + (1/1!)^2 + (1/2!)^2 + ... + (1/n!)^2
Q*E = e
a_n = 1 w_n = 1/n!
substituting leads to,
x = lim[n->INF] of ln{[n!(n+1)|E|^2 - n!e^2]/[n!e - (n+1)]=INF
(the series diverges)
Proof
dividing 1+x+x^2+x^3+...+x^n=0 by x^(2n),
x^0/x^(2n) + x^1/x^(2n) + x^2/x^(2n) + ... + x^n/x^(2n) = 0
When x=INF, 0=0
Discussion
Since the exponential function is used in the solution, it fails for small
values of n and is only useful for large values of n, in order for the
exponential to reflect in the solution.
For a problem that has, say, 170 dimensions, this solves 2-dimensional
slices of the 170, so it's appropriate to express P in terms of only 2
vectors, so long as they aren't parallel.
Two things must be true:
I. If P=(1,x,x^2,x^3,...,x^n)=(p_0,p_1,p_2,p_3,...,p_n), when solving eqn
1,
x^0 =[|Q|^2 w_0-(Q*E) q_0]e^x /[(Q*E)^2-|Q|^2|E|^2]
for x, this must be one solution.
II. All solutions must be true for,
{
x^n = [|Q|^2 w[n]-(Q*E) q[n]]
over
x^(n-p) = [|Q|^2 w[n-p]-(Q*E) q[n-p]]
}^(1/p)
for instance,
x^170/x^(170-8)=x^170/x^162=x^8
{x^8}^(1/8) = x
{
x^170 = [|Q|^2 w[170]-(Q*E) q[170]]
over
x^162 = [|Q|^2 w[162]-(Q*E) q[162]]
}^(1/8)
= x
Fot the 170th degree equation, there are many possible combinations, more
than there are unknowns. There should not be more than 170 solutions. It
is desirable for the solutions selected to reflect the nonlinear nature of
the problem, such as p=1,2,3,4,...,, It is also desirable that this
reflection begin at the maximum power of the equation, such as
n=170,169,168,167,..., p=170/169,170/168,170/167,... to incor****ate the
essence of the exponential function.
Excel only solves factorials up to 171! or 1/171!, which is about
1E+/-300.
III. These solutions also apply:
If [|Q|^2 w_n-(Q*E) q_n]/[|Q|^2 w_0 - (Q*E) a_0]>0 (eqn 2)
x=[|Q|^2 w_n-(Q*E) q_n]/[|Q|^2 w_0 - (Q*E) a_0]^(1/n)
*[cos(2pi*k/n)+isin(2pi*k/n)], k=0,1,2,3,...,,n-1
If [|Q|^2 w_n-(Q*E) q_n]/[|Q|^2 w_0 - (Q*E) a_0]<0, (eqn 2)
x = abs[|Q|^2 w_n-(Q*E) q_n]/[|Q|^2 w_0 - (Q*E) a_0]^(1/n)
*[cos((2k+1)pi/n)+isin((2k+1)pi/n], k=0,1,2,3,.., n-1
******
Jon G.
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