Error vector choice in direct inversion in the iterative subspace method

Based on Banach's principle, we formally obtain possible choices for an error vector in the direct inversion in the iterative subspace (DIIS) method. These choices not only include all previously proposed error vectors, but also a new type of error vector which is computationally efficient and applicable to much wider range of problems. The error vector analysis also reveals a strong connection between DIIS and damping, thus adding to understanding of the reasons behind DIIS's effect on convergence. We illustrate our conclusions with several examples.