ON THE BREAKDOWNS OF THE GALERKIN AND LEAST-SQUARES METHODS
摘要
The Galerkin and least-squares methods are two classes of the most popular Krylovsubspace methOds for solving large linear systems of equations. Unfortunately, both the methodsmay suffer from serious breakdowns of the same type: In a breakdown situation the Galerkinmethod is unable to calculate an approximate solution, while the least-squares method, althoughdoes not really break down, is unsucessful in reducing the norm of its residual. In this paper wefrst establish a unified theorem which gives a relationship between breakdowns in the two meth-ods. We further illustrate theoretically and experimentally that if the coefficient matrix of alienar system is of high defectiveness with the associated eigenvalues less than 1, then the restart-ed Galerkin and least-squares methods will be in great risks of complete breakdowns. It appearsthat our findings may help to understand phenomena observed practically and to derive treat-ments for breakdowns of this type.