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137 Degree Calculator

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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