swgillan <swgillan@gmail.com> wrote in message <51a9e61a47f44b758c090538d1323b82@x16g2000prn.googlegroups.com>...
> On Dec 11, 4:15=A0pm, "Benp P" <lightat...@hotmail.com> wrote:
> > Hi, I need to solve the equation
> >
> > x'Px =3D v
> >
> > where x is an (n by 1) vector of unknowns and P is a (n by n) square matr=
> ix (v is a scalar).
> >
> > Any help would be great!
> >
> > Thanks,
> > Ben.
>
> Not sure of this would help, but can you perform a cholesky
> factorization on P? P =3D LL' where L is a lower triangular.
>
> x'Px =3D v
> x'LL'x =3D v
> let L'x =3D y
> y'y =3D v
>
> if you can find y, you can just do simple back substition to get x.
>
> Not sure if that helps or not.
Its an idea, although the OP has never stated that
P is positive definite, a HUGEly important factor
in the existence of a Cholesky factor.
Essentially, IF P is symmetric and positive definite,
then X'*P*X = v is the equation of a hyperellipsoid.
All the Cholesky factor does is to transform the
ellipsoid into a hypersphere, still in n dimensions.
Having then turned the problem into Y'*Y = v,
you still do not have a solution for y. Remember,
the OP has asked for a way to recover the unknown.
All that we know is that y lies on a hypersphere
of known radius.
If P is a general matrix, this will fail, although one
might choose another form to factorize P. Perhaps
an LDL' or UDU' form might be chosen. Note that
D need not always be a diagonal matrix in these
forms.
John
