🚀 go-pugleaf

If T*N+a[0] is the nth degree polynomial where T=(t,t^2,t^3,...,t^n), 
N=(a[1],a[2],a[3],...,a[n]), if T is described from two orthogonal 
systems, its linear requirement precipitates.  T remains invariant
despite the coordinate system used to describe it. Consequently, my 
revised web site showing the details,

APPLICATION OF
http://mypeoplepc.com/members/jon8338/polynomial/id7.html

EXAMPLE

(t-2)(t-3)(t-4)=0
(t^2-5t+6)(t-4)=0
       t-4
-------------
t^3-5t^2+ 6t
    -4t^2+20t-24
----------------
t^3-9t^2+26t-24

N=+/-(26,-9,1)
N*U[2]=0 at U[2]=(4,13,13)
               i    j  k
U[3]=NxU[2]= 26   -9  1 =(-130,-334,374)
               4   13 13

N/|N|=
u[1]=( .944,-.327,.036)=(u[11],u[12],u[13])
u[2]=( .212, .691,.691)=(u[21],u[22],u[23])
u[3]=(-.249,-.640,.717)=(u[31],u[32],u[33])

(-a[0]/|N|)u[1]=(.8717)u[1]
Q = +/-(-.823,.285,-.0314)

     .212  -.249       1 0 0
A = .691  -.640    I= 0 1 0
     .691   .717       0 0 1

T = (AA^T -I)^-1 Q   =(1.085,6.552,6.607)=(t,t^2,t^3)

t = 1.085  throw out
t = 2.560~3
t = 1.876~2


EXAMPLE

n=8

T=[((AA^T-I)^-1)^T((AA^T-I)^-1)]^-1((AA^T-I)^-1)^T Q

(Least Squares)

           LEAST SQUARES SOLUTION TO MATRIX				
k   a[k]	               SUMMARY			
0  -30     t^k         |t^k|         t        ERROR	
1    9   2.384E-01   2.384E-01   2.384E-01  -2.731E+01	
2    9   2.031E+01   2.031E+01   4.507E+00   1.590E+06	
3    1  -1.902E+01   1.902E+01   2.669E+00   2.557E+04	
4    3  -4.038E+00   4.038E+00   1.418E+00   2.165E+02	
5    6   4.742E+00   4.742E+00   9.485E-01   1.830E+00	
6    1  -3.753E+01   3.753E+01   1.830E+00   1.416E+03	
7    1   3.320E+01   3.320E+01   1.649E+00   6.553E+02	
8    9   3.010E+01   3.010E+01   1.530E+00   3.787E+02	


Jon Giffen
jon8338@peoplepc.com