On the roots of the orthogonal polynomials and residual polynomials associated with a conjugate gradient method

In this paper we explore two sets of polynomials, the orthogonal polynomi-als and the residual polynomials, associated with a preconditioned conjugate gradient iteration for the solution of the linear system Ax = b. In the context of preconditioning by the matrix C, we show that the roots of the orthogonal polynomials, also known as generalized Ritz values, are the eigenvalues of an orthogonal section of the matrix CA while the roots of the residual polynomi-als, also known as pseudo-Ritz values (or roots of kernel polynomials), are the reciprocals of the eigenvalues of an orthogonal section of the matrix (CA) ?1. When CA is self-adjoint positive deenite, this distinction is minimal, but for the indeenite or non-self-adjoint case this distinction becomes important. We use these two sets of roots to form possibly non-convex regions in the complex plane that describe the spectrum of CA.