A note on the relationship between the conjugate gradient method and polynomials orthogonal over the spectrum of a linear operator

The relationship between the conjugate gradient method and polynomials orthogonal over the spectrum of a linear operator is discussed. It is shown that as a byproduct of the conjugate gradient construction, two sets of polynomials are generated which are orthogonal with respect to a positive real measure over the spectrum of a self-adjoint operator. The roots of these polynomials can, under certain circumstances, be used to track the eigenvalues or singular values of the relevant operator.