Path: news.mathworks.com!not-for-mail From: "John D'Errico" <woodchips@rochester.rr.com> Newsgroups: comp.soft-sys.matlab Subject: Re: Solving x'Px = v Date: Fri, 12 Dec 2008 10:00:05 +0000 (UTC) Organization: John D'Errico (1-3LEW5R) Lines: 48 Message-ID: <ghtcn5$764$1@fred.mathworks.com> References: <ghsae8$nlk$1@fred.mathworks.com> <51a9e61a-47f4-4b75-8c09-0538d1323b82@x16g2000prn.googlegroups.com> Reply-To: "John D'Errico" <woodchips@rochester.rr.com> NNTP-Posting-Host: webapp-03-blr.mathworks.com Content-Type: text/plain; charset="ISO-8859-1" Content-Transfer-Encoding: 8bit X-Trace: fred.mathworks.com 1229076005 7364 172.30.248.38 (12 Dec 2008 10:00:05 GMT) X-Complaints-To: news@mathworks.com NNTP-Posting-Date: Fri, 12 Dec 2008 10:00:05 +0000 (UTC) X-Newsreader: MATLAB Central Newsreader 869215 Xref: news.mathworks.com comp.soft-sys.matlab:506503 swgillan <swgillan@gmail.com> wrote in message <51a9e61a-47f4-4b75-8c09-0538d1323b82@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 hyper-ellipsoid. All the Cholesky factor does is to transform the ellipsoid into a hyper-sphere, 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 hyper-sphere 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