The correct formulation of the physical problem and the poorly conditioned matrix

Abstract

A new approach to the development of methods for solving ill-posed problems is considered, consisting in the fact that it is proposed to formulate the problem correctly, and not to solve the incorrect problem. Correct formulation is performed at the stage of compiling equations describing the physical object. It should be noted that the stage of compiling equations in modern literature is undeservedly ignored. By the example of calculating the electrical circuit, it is shown that the poor conditioning of the SLAE matrix depends on the characteristics of a particular electrical circuit, namely, the range of variation of the parameters of the components of the electrical circuit, as well as the generally accepted criterion for determining the incorrectness of a problem by poor conditioning of the matrix or proximity of the matrix to degeneracy there is a necessary, but not enough. It is determined that the mechanism for the occurrence of an incorrect formulation of a description of a linear electric circuit consists in the unsuccessful choice of variables when compiling a description of an electric circuit. It is proposed, with the correct formulation of the problem, to take into account the parameters of the components of a particular electric circuit when choosing variables of SLAEs at the stage of preparing the equations. It is shown that in the method of nodal potentials and its modifications it is impossible to realize a purposeful choice of variables in the preparation of equations. The requirements to the method of correctly formulating a description of a linear electric circuit are formulated. The calculation of the model example by the method in which the formulated requirements are realized confirmed the fact that the problem is correctly formulated, and the solution is stable even in the case of a poorly conditioned matrix.