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Guest
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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
This strategy has some flaws. You need n vectors to construct n dimensions,
but I only used 3. Also, it is only useful for large n in order to reflect
the contribution from E*U=e^x. It also generates numbers too large for
computing.
Its advantage is that it does produce some numbers to play around with
Newton's Method. Their initial values may or may not be accurate, but after
7 iterations and 50 repeats, does it really matter?
=(a_1+MMULT(s_2*TRANSPOSE(qd_2)*((r_iter_7d*TRANSPOSE(s_2))^TRANSPOSE(n_1_2))*COS(ang_iter_7d*TRANSPOSE(n_1_2)),n_2)) |
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