Conjugate Gradient Algorithms Using Multiple Recursions

Much is already known about when a conjugate gradient method can be implemented with short recursions for the direction vectors. The work done in 1984 by Faber and Manteuuel 3] gave necessary and suucient conditions on the iteration matrix A, in order for a conjugate gradient method to be implemented with a single recursion of a certain form. However, this form does not take into account all possible recursions. This became evident when Jagels and Reichel 5, ?] used an algorithm of Gragg for unitary matrices 4] to demonstrate that the class of matrices for which a practical conjugate gradient algorithm exists can be extended to include unitary and shifted unitary matrices. The implementation uses short double recursions for the direction vectors. This motivates the study of multiple recursion algorithms. In this talk, we show that the conjugate gradient method for unitary and shifted unitary matrices can be implemented using a single short term recursion of a special type called an (`; m) recursion with`; m 1. We then examine the class of matrices for which a conjugate gradient method can be carried out using a general (`; m) recursion. This class includes the class of normal matrices with rational degree (`; m) as well as low rank perturbations of these matrices. Under some circumstances, an (`; m) recursion can break down. We also show that any (`; m) recursion can be reformulated as m short recursions that will not break down. Necessary and suucient conditions have been established for`; m 2. These results