# cubic form and eigenvalue for matrix equation in R3 with cubic term

Discussion in 'Numerical Analysis' started by Michel PETITJEAN, Nov 25, 2009.

1. ### Michel PETITJEANGuest

sci.math.num-analysis
cubic form and eigenvalue for matrix equation in R3 with cubic term

Hello

I have two problems:

(1) I am looking for an unknown unit vector u in R3 satisfying to:
V u + ( Q1 u u' Q1 u + Q2 u u' Q2 u + Q3 u u' Q3 u ) + L u = 0
where u' is the transposed of u
V, Q1, Q2, Q3 are known symmetric 3,3 matrices, V is positive definite,
and L is an eigenvalue to be computed.
Apart Newton methods, is there some numerical method devised
for this class of equations involving a trilinear term T(u,u,u) ?

(2) I am looking for an unknown unit vector v satisfying to:
v'Nv = 0 and K(v,v,v)=0
where N is a known diagonal matrix
(neither positive definite nor negative definite)
and K is a known cubic form
(symmetric in its three indices, of course).
From v'v=1 and v'Nv=0 only a single unknown remains.
Is there some better way to operate ?

Apart Bezout theorem, is there more to say about the number of roots
in these two problems ?

Thanks for pointers and suggestions.

Michel Petitjean,
CEA/DSV/iBiTec-S/SB2SM (CNRS URA 2096),
91191 Gif-sur-Yvette Cedex, France.
Phone: +331 6908 4006 / Fax: +331 6908 4007
E-mail: , (preferred)
http://petitjeanmichel.free.fr/itoweb.petitjean.html

Michel PETITJEAN, Nov 25, 2009

2. ### Helmut JarauschGuest

Just a random idea (might turn out silly)

Let rho_k = u' Q_k u and put A=(V+rho_1*Q1+rho_2*Q2+rho_3*Q3)
Try to minimize the constrained problem

min u' (A-Lu)^2 u / (u' u) subject to u' Q_k u = rho_k for 1 <= k <= 3
where one minimizes over (u,L,rho_1,rho_2,rho_3)

--
Helmut Jarausch

Lehrstuhl fuer Numerische Mathematik
RWTH - Aachen University
D 52056 Aachen, Germany

Helmut Jarausch, Nov 26, 2009