Other options

In the future I intend to try other methods of accelerating the convergence of the CG method. The most common way of speeding up the CG algorithm is to use a preconditioning matrix. This is an operator, M, that is in some way approximately equivalent to the operator A but has an inverse that is easy to compute, ( find M such that $AM^{-1} \approx I$). All that is required of AM^-1 is that it has a better condition number than A. There are two forms for the operator that I intend to experiment with, the split form of the 3-D operator and an incomplete Cholesky decomposition of A.

The Cholesky decomposition of A is closely related to the work done by Cole (1991) on Given's rotation of the 3-D operator. The incomplete Cholesky decomposition is an approximate decomposition that has the same sparsity structure as the original operator and is thus cheap to compute and apply.

The preconditioned conjugate gradient method involves solving the equation Mz = r at every iteration of the algorithm. However, the operator M does not change between iterations so that though solving Mz=r may be expensive (e.g., by using the split form of the operator), the decomposed operator can be reused for every iteration.