The bordering method as a preconditioning method
The problem of solving a system of linear algebraic equations \[ Ax = b \tag{1} \] where \(A\) is a symmetric positive definite \((n\times n)\)-matrix is considered. Let \(\{ \lambda_{i} , v^{i} \}\) \((i = 1, 2,\dots, n)\) be the eigenvalues and eigenvectors of \(A\), where \(\lambda_{1} \leq \lambda_{2} \leq \cdots \leq \lambda_{n}\). Let the vector \[ x =\sum_{i=1}^{n}x_{i} v^{i} \tag{2} \] be a solution of the system (1). If the matrix \(A\) of the system (1) is an almost degenerate then its eigenvalues are close to zero. In this case methods of approximation (conjugate gradient method, for example) give a big value for some components of the vector of errors. In the article a method of bordering of matrices (usually used for obtaining inverse matrices) is proposed for obtaining an absolutly degenerate matrix for the system (1) and then the conjugate gradient method is becoming more effective.
scientific article; zbMATH DE number 1266872