Multiparameter descent methods Original Research Article

A descent method for solving a system of linear equations Ax=b consists of the iterations xk+1=xk+λkzk, where zk is a vector and λk a parameter chosen to minimize some functional. In this paper, we will consider multiparameter generalizations of such descent methods, namely iterations of the form xk+1=xk+ZkΛk, where Zk is a matrix and Λk a vector chosen to minimize some functional. Multiparameter generalizations of the conjugate direction method and the Lanczos method will be obtained and their algebraic properties discussed. Multiparameter conjugate and biconjugate gradient algorithms and other Lanczos-type algorithms for implementing this Lanczos method will also be given.