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

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

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


    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)
    Michel PETITJEAN, Nov 25, 2009
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  2. 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
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