Comparison of splittings used with the conjugate gradient algorithm

We consider several possible criteria for comparing splittings used with the conjugate gradient algorithm. We present sharp upper bounds for the error at each step of the algorithm and compare several widely used splittings with respect to their effect on these sharp upper bounds. We then consider a more stringent comparison test, and present necessary and sufficient conditions for the error at each step with one splitting to be smaller than that with another, for all pairs of corresponding initial guesses.