Solve,
a_0*x^0 + a_1*x^1 + a_2*x^2 + ... +a_n*x^n = 0
x = ?
Let
Q=(a_0,a_1,a_2,...,a_n)
U=(1,1,1,...,1)
E=(1,1,1/2!,1/3!,1/4!,...,1/n!)
P=(1,x,x^2,x^3,....,x^n)
u=U/|U|
by vector analysis,
P*u=|Q|^2(E*P)/[(E*Q)(Q*u)]
express P as ratios of Q,U,E
qQ+rU+sE=P
dot both sides by Q,u,E and note that P*E=e^x.
Solve the matrix,
Q*Q Q*U Q*E | 0
Q*u U*u E*u | {|Q|^2/[(E*Q)(Q*u)]}e^x
Q*E E*U E*E | e^x
P/|P| = (s_0,s_1,s_2...) e^x cancels
P=P_u/s_0 because x^0=1
P=(p_0,p_1,p_2,p_3,...,p_n)=(1,x,x^2,x^3,..,x^n)
x=a+bi
(a+bi)^(1/n)
=
{a^2+b^2}^(.5/n)[cos((arctan(b/a)/n)] real
+{a^2+b^2}^(.5/n)[sin((arctan(b/a)/n)]i complex
by DeMoirve
x_n+1=x_n-
(a_0+a_1*x+a_2*x^2+a_3*x^3+...+a_n*x^n)
/
(a_1+2*a_2*x+3*a_3*x^2+...+n*a_n*x^(n-1) )
by Newton
The math behind this is at,
http://mypeoplepc.com/members/jon8338/math/id11.html
I also included a link to download an Excel polynomial calculator for up
to
137 dimensions. The calculator is slow and takes about 30 seconds for
each
set of coefficients (a_0,a_1,a_2,..,a_n). If the calculator generates an
overflow error, it means the values you entered are too big to calculate.
Also, Newton's Method omits one dimension in the denominator. I haven't
figured out yet how to deal with that, but it is a problem in the
spreadsheet that I need to fix.... probably as well as other errors.
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