On 11 May, 17:59, "Jon G." <jon8...@[EMAIL PROTECTED]
> wrote:
> Solve,
>
> a_0*x^0 + a_1*x^1 + a_2*x^2 + ... +a_n*x^n =3D 0
>
x =3D 0
> Let
>
> Q=3D(a_0,a_1,a_2,...,a_n)
> U=3D(1,1,1,...,1)
> E=3D(1,1,1/2!,1/3!,1/4!,...,1/n!)
> P=3D(1,x,x^2,x^3,....,x^n)
>
> u=3DU/|U|
>
> by vector analysis,
>
> P*u=3D|Q|^2(E*P)/[(E*Q)(Q*u)]
>
> express P as ratios of Q,U,E
>
> qQ+rU+sE=3DP
>
> dot both sides by Q,u,E and note that P*E=3De^x.
> Solve the matrix,
>
> Q*Q =A0Q*U =A0Q*E =A0| 0
> Q*u =A0U*u =A0E*u =A0| {|Q|^2/[(E*Q)(Q*u)]}e^x
> Q*E =A0E*U =A0E*E =A0| e^x
>
> P/|P| =3D (s_0,s_1,s_2...) =A0e^x cancels
>
> P=3DP_u/s_0 because x^0=3D1
>
> P=3D(p_0,p_1,p_2,p_3,...,p_n)=3D(1,x,x^2,x^3,..,x^n)
>
> x=3Da+bi
>
> (a+bi)^(1/n)
> =3D
> {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=3Dx_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) =A0)
>
> 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
t=
o
> 137 dimensions. =A0The calculator is slow and takes about 30 seconds for
e=
ach
> set of coefficients (a_0,a_1,a_2,..,a_n). =A0If the calculator generates
a=
n
> overflow error, it means the values you entered are too big to
calculate.
>
> Also, Newton's Method omits one dimension in the denominator. =A0I
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.
|
