I've to solve the following nonlinear matrix equation, and I was wondering if there is any bigger class where to fit such an equation so that I can either:
- Obtain a closed-form solution (ideal)
- Use appropriate algorithms to solve it numerically.
Assume that $V\in\mathbb{R}^{1\times k}$, $B_i\in\mathbb{R}^{k\times 1}$, $T_i\in\mathbb{R}^{k \times k}$ with $i=1,\dots,k$, and finally $L\in\mathbb{R}^{1\times k}$. The equation reads:
$$V - \sum_{i=1}^{k} VB_iVT_i = L$$
And it has to be solved for $V$ (all the other matrices are assumed to be known). Any help or suggestion will be extremely appreciated! Thanks!!!
We are dealing with a system of $k$ equations of second degree in $k$ unknowns $V=(v_i)$. Using the Grobner basis software of Maple, several tries, with various values of $k$, seem to show that:
If we randomly choose (generical case) the $(B_i),(T_i),L$, then we obtain $2^k$ complex solutions, that is, according to the Bezout theorem, the maximum of isolated solutions for a $k-k$ system of degree $2$. For example, when $k=5$, we obtain $32$ complex solutions; curiously, when (for $k=5$ again) we consider the system $V-\sum_{i=1}^{4} VB_iVT_i-L=0$, then we obtain only $31$ solutions.
There is no closed form for the solutions: for example, for $n=3$, Maple gives the result in the form of a triangular decomposition $P_8(v_1)=0,v_2=Q_2(v_1),\cdots,v_n=Q_n(v_1)$ where $P_8$ is a polynomial of degree $8$ and the $Q_i$ are polynomial of degree $7$. The Galois group of $P_8$ is $S_8$ and, consequently, is not solvable. Note that, of the eight roots of $P_8$, only two are real.
The problem with the Grobner's method is that it works badly for not so large values of $k$ as $k=8$. On the other hand, pure Grobner gives all complex solutions; using special methods for the real case, we can obtain only the real solutions. Yet, this method is even slower than the previous one.
So it is better to turn to numerical methods.