Positive definiteness of forms: Numerical identification

The question about positive definiteness or semidefiniteness of quadratic forms (or, more generally, polynomial homogeneous forms of an even degree) arises in numerous fields of mathematics and its applications. This is certainly the case for optimization theory, including calculus of variations and optimal control. Effective methods intended to obtain a reliable answer to this question for a given form are of doubtless theoretical and practical interest. For that purpose, we propose to use the traditional unconstrained optimization technique, namely, the steepest descent and the conjugate gradient methods. The effectiveness of this approach is justified by theoretical analysis and computational experiments.