Error analysis of QR decompositions by Givens transformations

The error analysis for computing the QR decomposition by Givens transformations was given originally by Wilkinson for n×n square matrices, and later by Gentleman for n×p (pn) tall thin matrices. The derivations were sufficiently messy that results were quoted by analogy to the derivation of a specific case. A certain lemma makes possible a much simpler derivation, which incidentally substantially tightens the bound. Moreover, it applies to variants of the method other than those originally considered, and suggests why observed errors are even less than this new bound.