Can we check the positive definiteness of a multivariable polynomial using MATLAB functions? Of course in literature, there are few complex methods that use tensor algebra. However, before going to such methods if you have some simple suggestions that will be appreciated. In a stability analysis of systems, It is usually required to check that whether the obtained Lyapunov function is positive or not for x>0. For example, if we obtained the following Lyapunov function with these parameters:
a1 = 0.689 a6 = 0.368 a7 = -0.238 a8=-0.336
That particular example is in a homogeneous form. But if the obtained polynomial is in a nonhomogeneous form then what will be the way to proceed? If we have more than two variables (e.g x1, x2, x3, x4,...)? Please correct me if I am wrong in statements.
For polynomials of degree 2, if you can write V(x) as x'*A*x (in MATLAB notation, assuming x is a column vector) and if A is a matrix with non negative (positive) eigenvalues you know that V is non negative (positive).
Right, this can be. And of course, for the higher degree, we can represent in this form as given in this paper.
See Parrilo and Sturmfels work, in particular
Minimizing polynomial functions. Algorithmic and quantitative real algebraic geometry (Piscataway, NJ, 2001), 83–99, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 60, Amer. Math. Soc., Providence, RI, 2003.
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