Some Convergence Properties of the Conjugate Gradient Method in Hilbert Space

A rate of convergence of the conjugate gradient method for minimizing the convex quadratic functionals in Hilbert space is investigated. Daniel’s approach via spectral analysis is developed and the theory of moments is applied. Main results: the investigation of conjugate gradient method properties is replaced by the investigation of some approximation problems; a rate of convergence is characterized by a Jacobi matrix closely related to the Hessian of the functional. Examples have been constructed when the convergence of the conjugate gradient method is only linear.