A note on conjugate gradient convergence. III
In this third paper of a series [Part I: ibid. 76, No. 2, 209-230 (1997; Zbl 0905.65047); Part II: ibid. 85, No. 4, 655-683 (2000; reviewed above)], the authors collect the previous results. They show that for some right-hand side (r.h.s.) vectors, one iteration will bring a certain improvement independent of the size of the matrix -- derived from the discretization of a two-point boundary value problem. For other r.h.s. vectors the improvement tends to zero as the size of the system is increased. In the end it is concluded that if the r.h.s. vector projects only on a few eigenvectors, especially at the ends of the spectrum, conjugate gradients brings quick improvements, whereas if the r.h.s. vector projects equally onto many eigenvectors, conjugate gradients can be very slow -- a fact that is already well known.
scientific article; zbMATH DE number 1491750