β Scribed by Ma, Yue; Zhi, Lihong
No coin nor oath required. For personal study only.
The problem of computing a representation for a real polynomial as a sum of minimum number of squares of polynomials can be casted as finding a symmetric positive semidefinite real matrix of minimum rank subject to linear equality constraints. In this paper, we propose algorithms for solving the minimum-rank Gram matrix completion problem, and show the convergence of these algorithms. Our methods are based on the fixed point continuation method. We also use the Barzilai-Borwein technique and a specific linear combination of two previous iterates to accelerate the convergence of modified fixed point continuation algorithms. We demonstrate the effectiveness of our algorithms for computing approximate and exact rational sum of squares decompositions of polynomials with rational coefficients.
π SIMILAR VOLUMES
Let I β K[x 1 ,...,x n ] be a 0-dimensional ideal of degree D where K is a field. It is well-known that obtaining efficient algorithms for change of ordering of GrΓΆbner bases of I is crucial in polynomial system solving. Through the algorithm FGLM, this task is classically tackled by linear algebra