About The Nature Of Saddle Point Problems

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering, such as in the Incompressible flow mechanics fields, the famous Navier-Stokes equations are the partial differential equations with constraints. Typically the constraints represent some basic conservation law(conservation of momentum and mass conservation), Discretization of the equations leads to the saddle point problems. Because saddle point equations can be derived as equilibrium conditions for a physical system, they are sometimes called equilibrium equations; but aother popular name for saddle point systems, especially in the optimization literature, is'KKT system'(from the Karush–Kuhn–Tucker first-order optimality conditions). Therefore , the research of the linear systems of saddle point type has important meaning , but due to their indefiniteness and often poor spectral properties, so significant challenge for solver developers.In this thesis, our main work includes in the following:In the first part , we will conclude the origin of the saddle point problem and its basic conception and classification;In the second part , according to analyze the characters of the general saddle point problem, we get the new sufficient condition that saddle point matrix will have positive spectrum and may be diagonalizable;In the third part , based on the introduction of the precondition, we mainly study the spectral properties of the preconditioned matrices. And we estimite the upper bound and the lower bound of the real part of the eigenvalue much more accurately.