Homogeneous system of polynomial equations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:56:37Z http://mathoverflow.net/feeds/question/56572 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56572/homogeneous-system-of-polynomial-equations Homogeneous system of polynomial equations Tony 2011-02-24T22:33:53Z 2012-10-16T12:22:00Z <p>Hi all,</p> <p>Previously I asked a question that currently has no satisfactory answer <a href="http://mathoverflow.net/questions/55939/least-sum-squares-given-constraints-on-subcomponents" rel="nofollow">http://mathoverflow.net/questions/55939/least-sum-squares-given-constraints-on-subcomponents</a></p> <p>It comes from an engineering problem. I was thinking to formulate it differently and hope that someone becomes interested and/or know how to solve it.</p> <p>By formulating differently, I will have the following system:</p> <p>$\mathbf{D} \mathbf{R} \mathbf{\theta} = \mathbf{0}_{2N \times 1}$</p> <p>D is a $2N \times 6N$ block diagonal matrix that contains unknowns $\mathbf{D} = diag(\mathbf{x}^T - a_1 \mathbf{z}^T, \mathbf{y}^T - b_1 \mathbf{z}^T, \dots, \mathbf{x}^T - a_N \mathbf{z}^T, \mathbf{x}^T - b_N \mathbf{z}^T)$ where $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are 3x1 orthogonal unit vectors that we need to find (ie., $\mathbf{x}^T \mathbf{x} = 1, \mathbf{x}^T \mathbf{y} = 0, \dots$). $a_i, b_i$ are known parameters.</p> <p>R is a $6N \times M$ matrix that contains only numerical entries (measured and computed). $\mathbf{\theta}$ is a vector of M other unknown parameters.</p> <p>By doing so, I isolated unknowns in two separate matrices (vectors). However, it is still not trivial. </p> <p>I tried to solve this, but there is no obvious way. One way is that I tried to find $\mathbf{x}, \mathbf{y}, \mathbf{z}$ such that $\mathbf{D} \mathbf{R}$ is rank-deficient. I'm not sure if this can be solved, either exactly or in least-squared, in closed-form or numerically. Any idea, discussion is appreciated.</p> <p>edit: I was not clear. If I set determinant of any MxM sub-matrix of $\mathbf{D} \mathbf{R}$ to 0 and express it as a function of elements in $\mathbf{x}, \mathbf{y}, \mathbf{z}$, then I have a number of homogeneous polynomials. That's why the title comes.</p> <p>edit2: M is much smaller than 2N.</p> http://mathoverflow.net/questions/56572/homogeneous-system-of-polynomial-equations/56976#56976 Answer by Tony for Homogeneous system of polynomial equations Tony 2011-03-01T06:58:41Z 2011-03-01T06:58:41Z <p>One suggests that we can have a good prior for $\theta$ (but not for $\mathbf{D}$). Then, he proposed that we fix $\theta$, find $\mathbf{R}$ for least square $|| \mathbf{D} \mathbf{R} \theta||$ given the constraints above; then fix $\mathbf{R}$ and find $\theta$ for least square $|| \mathbf{D} \mathbf{R} \theta||$ given $||\theta|| = 1$. Then keep repeating (fix $\theta$ then $\mathbf{R}$) until they converges.</p> <p>The latter least square is easy, and the former, I believe there is a closed form or numerical solution.</p> <p>But my question is if such numerical scheme works? Is there any proof for it? What is the name of this method? I could not recall this numerical scheme.</p> <p>Please enlighten.</p>