🚀 go-pugleaf

a correction is applied below:

Jon wrote:
> 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
correct above to,
T=[(AA^T-I)^T(AA^T-I)]^-1(AA^T-I)^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
> 

Jon wrote 

> > 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,

> > 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)

REVOLUTIONARY !!!

Why don't you tell C.F. Gauss or E. Galois; they would love to hear
that you have easily solved a problem they have shown to be
principally not solvable!!!

RMoebs1@compuserve.de (Peter Pan) wrote in message news:<3cc72195.0407251355.26ad613d@posting.google.com>...
> Jon wrote 
> 
> > > 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,
>  
> > > 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)
> 
> REVOLUTIONARY !!!
> 
> Why don't you tell C.F. Gauss or E. Galois; they would love to hear
> that you have easily solved a problem they have shown to be
> principally not solvable!!!

Jon is evidently still using the same wrong procedure. He doesn't come up
with 2, 3, or 4 as roots of the (cubic!) polynomial he gives. (He cheated
by rounding.)
     -- Christopher Heckman

Jon <jon8338@peoplepc.com> wrote in message news:<MSHMc.12907$f4.5254@newsread3.news.atl.earthlink.net>...
> a correction is applied below:
> 
> Jon wrote:
> > 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
> > 

Alright Jon, its time for some strong criticism.  Take a look at this:
t^3 - 2t^2 - 2t + 4 = 0
t = 2, sqrt(2), -sqrt(2)

> > (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 = (1, -2, -2)

> > 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])
> > 

I have decided to try this polynomial with two sets of vectors one is
marked with primes and the other is not.

u[1] = (1/3, -2/3, -2/3)
u[2] = (2/3, -1/3, 2/3)
u[3] = (2/3, 2/3, -1/3)

u'[1] = (1/3, -2/3, -2/3)
u'[2] = (2/sqrt(5), 1/sqrt(5), 0)
u'[3] = (2/sqrt(5), 0, 1/sqrt(5))

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

Q = -(4)/3 * (1/3, -2/3, -2/3) = (-4/9, 8/9, 8/9)
Q' = -(4)/3 * (1/3, -2/3, -2/3) = (-4/9, 8/9, 8/9)

> >     .212  -.249       1 0 0
> > A = .691  -.640    I= 0 1 0
> >     .691   .717       0 0 1
> > 
    [ 2/3  2/3]
A = [-1/3  2/3]
    [ 2/3 -1/3]

     [2/sqrt(5) 2/sqrt(5)]
A' = [1/sqrt(5) 0        ]
     [0         1/sqrt(5)]

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

For the unprimed set of vectors, the expression T = (AA^T -I)^-1 Q
does not exist since (AA^T -I) is not invertible in this case.  This
is an EXTREME problem.  You can't even guarantee the equations
generated to be solvable!

For the primed set of vectors, T' = (4/9, -8/9, -8/9).  Now Jon, since
you did some rounding in your example, how would you round these to
the correct values of 2, 1.41421356237309504880168872421, and
-1.41421356237309504880168872421?

> > 
> > 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
> correct above to,
> T=[(AA^T-I)^T(AA^T-I)]^-1(AA^T-I)^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   
> > 
This has to be the worst, YOUR ERROR VALUES ARE UP TO HALF A MILLION
TIMES LARGER THAN THE T VALUES!!!!!!!!!!  When an engineer would see
those kinds of error values, he would scream!

> > 
> > Jon Giffen
> > jon8338@peoplepc.com
> >

brianscsmith@yahoo.com (Brian Smith) wrote in message news:<12f59340.0407260451.3bbd755@posting.google.com>...
> Jon <jon8338@peoplepc.com> wrote in message news:<MSHMc.12907$f4.5254@newsread3.news.atl.earthlink.net>...
> > a correction is applied below:
> > 
> > Jon wrote:
> > > 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
> > > 
> 
> Alright Jon, its time for some strong criticism. 

I've found it's not worth it telling him about this. I'm posting replies
to his, mainly to let other people know his methods don't work.
     -- Christopher Heckman

Jon <jon8338@peoplepc.com> wrote 

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

OK, Jon, now put your money where your mouth (and your webside) is:

Solve 

t^5 + t^2 + 1 = 0

Iff you can do that, I'll have a closer look at your 'theory'.

RMoebs1@compuserve.de (Peter Pan) wrote in message news:<3cc72195.0407271357.62c798a9@posting.google.com>...
> Jon <jon8338@peoplepc.com> wrote 
> 
> > > APPLICATION OF
> > > http://mypeoplepc.com/members/jon8338/polynomial/id7.html
> > > 
> > > EXAMPLE
> 
> OK, Jon, now put your money where your mouth (and your webside) is:
> 
> Solve 
> 
> t^5 + t^2 + 1 = 0
> 
> Iff you can do that, I'll have a closer look at your 'theory'.

Forget it. His method can't even solve a factored cubic ((t-2)(t-3)(t-4)=0,
his own example) correctly.
     -- Christopher Heckman