On the Minimality of Squared Error of Solutions to Systems of Equations Transformed to the Best Parameter under Small Homogeneous Perturbations

Solving of systems of nonlinear equations with a scalar parameter is studied. The set of solutions to such systems is a curve in the space of variables of the equation system and the parameter. Its construction is usually carried out using numerical methods and is associated with numerous difficulties arising due to the presence of limiting and essentially singular points on the curve of the set of solutions. To find such curves, the method of solution continuation with respect to a parameter and the best parameterization is used, which allows us to reduce the solution to the Cauchy problem for a system of differential equations of solution continuation. Stability of the solution to perturbations introduced into the continuation system is investigated. For the first time, the previously formulated proposition about the minimality of the squared error of the solution to the continuation system under homogeneous small perturbations of its matrix is completely proved. The theoretical results are illustrated by the example of the numerical construction of Bernoulli’s lemniscate.