TABLE OF CONTENTS As was seen in Section 6.6, when A is symmetric, the Arnoldi algorithm simplifies to the Lanczos procedure, which is defined through a three-term recurrence. As a consequence, FOM is mathematically equivalent to the CG algorithm in this case. Similarly, the full GMRES algorithm gives rise to the CR algorithm. It is clear that the CG-type algorithms, i.e., algorithms defined through short-term recurrences, are more desirable than those algorithms that require storing entire sequences of vectors, as in the GMRES process. These algorithms require less memory and fewer operations per step.
Therefore, the question is, Is it possible to define algorithms based on optimal Krylov subspace projection that give rise to sequences involving short-term recurrences? An optimal Krylov subspace projection means a technique that minimizes a certain norm of the error, or residual, on the Krylov subspace. Such methods can be defined from the Arnoldi process.
It is sufficient to consider the Arnoldi process. If Arnoldi's algorithm reduces to the s-term incomplete orthogonalization algorithm (Algorithm 6.6 with k ? s), i.e., if h ij = 0 for i < j ? s + 1, then an ( s ? 1)-term recurrence can be defined for updating the iterates, as was done in Section 6.4.2. Conversely, if the solution is updated as x j+1 = x j + ? j p j and p j satisfies a short recurrence, then the residual vectors will...
