From Orthogonal Polynomials to Iteration Schemes for Linear Systems: CG and CR Revisited

Large systems of linear equations arise frequently in numerical analysis and are the basis of many models in engineering and other applied sciences. This note provides a study for the solution of Hermitian linear systems. One particular feature which distinguishes this paper from the usual literature on polynomial based iteration methods is its emphasis on the properties of the underlying polynomials rather than more conventional matrix manipulations. In particular, a development and discussion of the properties of orthogonal polynomials leads to unified analysis of the state-of-the-art methods conjugate gradient and conjugate residual, respectively.