This paper addresses the residual scaling techniques (coarse-grid-correction optimization techniques) in multigrid methods. We survey recent developments in this area and prove the equivalence of the over-weighted residual technique and the over-correction technique. This leads to the proof of mathematical equivalence of the pre-scaling and post-scaling acceleration techniques. Two theorems have been proved to unify the concept of the residual scaling techniques. These theoretical results clear the way for developing efficient pre-scaling acceleration techniques for practical applications. Those practical pre-scaling acceleration techniques are discussed in a companion paper: Residual scaling techniques, II: practical applications.
Key words and phrases: Multigrid method, residual scaling techniques, heuristic residual analysis.