ERROR-MINIMIZING KRYLOV SUBSPACE METHODS

Iterative methods for the solution of linear systems are usually contr olled by the observation of the norm of the residual. In reality, the error should be controlled, but the error is not available. The residu als and the errors are connected by the condition number of the system matrix. If the system is well conditioned, the decrease of the errors is closely connected to the decrease of the residuals. For these case s, Krylov subspace methods that minimize the residuals in the Euclidea n norm or in the energy norm are powerful solution techniques. If the system is ill conditioned, the residuals can decrease while the errors increase. For these systems, arising even from very simple and common ly used differential equations, iterative methods that minimize the re siduals may require a large number of iterations to reduce the errors. The user may be misled to stop the iteration too early by small resid uals. Two families of error-minimizing Krylov subspace methods are pro posed to overcome these difficulties. Each of them is suited for diffe rent problem types. Principles for the design of generalized cg method s that minimize the error are derived from the geometric convergence b ehavior of generalized cg methods. These methods use the transposed sy stem matrix multiplied by the system matrix as the iteration matrix. B y this technique a fast convergence is achieved for matrices with clus tered singular values and scattered eigenvalues. A class of Krylov sub space methods minimizing the error by using the simple transposed matr ix as the iteration matrix is proposed. Various realization possibilit ies are inherent in these generalized minimum error methods. The metho ds are analyzed theoretically. Common and related properties with gene ralized conjugate gradient methods are presented. These techniques sho uld be preferred if the eigenvalues are more clustered than the singul ar values. The first promising tests for one distinct method are prese nted.