3
$\begingroup$

I have a system of $n \times 1$ equations $$ 0 = A\,vec(xx^t) + B x + C $$ where

  • $x$ is a $n \times 1$ vector of unknowns

  • $x^t$ means transpose

  • $vec$ means $xx^t$ has been vectorized so has dimension $n^2 \times 1$

  • $A$ is a known matrix with dimensions $n \times n^2$

  • $B$ is a known matrix with dimensions $n \times n$

  • $C$ is a known vector of dimension $n \times 1$

I can solve these problems using a nonlinear solver. However, I am trying to find out if there are any theoretical results on how to solve this class of problems. I can find a lot of work on solving matrix quadratic equations. However, I can't find anything that has specifically this form, and I cannot figure out if I can rewrite this system in a way that is equivalent to other matrix quadratic problems I come across.

Improve this question
$\endgroup$
2
  • $\begingroup$ I edited this so it was formatted properly, but I think you need to supply more context at the very least. Even then it's not clear this will be on-topic, but I'm not saying myself one way or the other. $\endgroup$
    – David Roberts ♦
    Commented Dec 4, 2016 at 8:06
  • $\begingroup$ Since $\rm x x^{\top}$ is symmetric, wouldn't it make more sense to half-vectorize instead? $\mbox{vech} (\rm x x^{\top})$ has only $\binom{n+1}{2}$ entries. One could use the duplication matrix, too. $\endgroup$
    – Rodrigo de Azevedo
    Commented Dec 4, 2016 at 11:53

1 Answer 1

Reset to default
5
$\begingroup$

Shameless advertisement to a paper of mine: http://www.sciencedirect.com/science/article/pii/S0024379511004484 Quadratic vector equations, in Linear Algebra and its Applications, volume 438, 2013. Arxiv version on https://arxiv.org/abs/1004.1500.

I studied this exact problem in the case in which $A,B,C$ all contain nonnegative entries apart from the diagonal of $B$ which is nonpositive ($-B$ has to be an M-matrix, more precisely).

There are a few matrix equations that can be reduced to this form (for instance, some nonsymmetric Riccati equations or matrix quadratic polynomial equations); in this paper I tried to study them all in a unified fashion and work with the most general hypotheses. I dealt with the existence of a minimal nonnegative solution, and with the convergence of several numerical methods.

In future works, we have developed other algorithms that are tailored to the case in which there are two very close solutions.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .