Comparing the pseudocode for Chebyshev Iteration with the pseudocode for the Conjugate Gradient method shows a high degree of similarity, except that no inner products are computed in Chebyshev Iteration .
Scalars and must be selected so that they define a family of ellipses with common center and foci and which contain the ellipse that encloses the spectrum (or more general, field of values) of and for which the rate of convergence is minimal:
where is the length of the -axis of the ellipse.
We provide code in which it is assumed that the ellipse degenerate to the interval , that is all eigenvalues are real. For code including the adaptive determination of the iteration parameters and the reader is referred to Ashby .
The Chebyshev method has the advantage over GMRES that only short recurrences are used. On the other hand, GMRES is guaranteed to generate the smallest residual over the current search space. The BiCG methods, which also use short recurrences, do not minimize the residual in a suitable norm; however, unlike Chebyshev iteration, they do not require estimation of parameters (the spectrum of ). Finally, GMRES and BiCG may be more effective in practice, because of superlinear convergence behavior , which cannot be expected for Chebyshev.
For symmetric positive definite systems the ``ellipse'' enveloping the spectrum degenerates to the interval on the positive -axis, where and are the smallest and largest eigenvalues of . In circumstances where the computation of inner products is a bottleneck , it may be advantageous to start with CG, compute estimates of the extremal eigenvalues from the CG coefficients, and then after sufficient convergence of these approximations switch to Chebyshev Iteration . A similar strategy may be adopted for a switch from GMRES, or BiCG-type methods, to Chebyshev Iteration.