The main result may be presented in the following terms. \(\mathbb{C}[p]\) is the set of all vectors having \(p\) complex components, \(\mathbb{C}[p]^{k+1}\) is the source space \(\mathbb{C}[p]\times \dots \times\mathbb{C}[p]\) comprising \(k+1\) domains, and \({\mathfrak A}\subseteq \mathbb{C}[p]^{k+1}\). \(X[k]\) is the sequence of vectors \(x(0),\dots,x(k)\) in \(\mathbb{C}[p]^{k+1}\) and, with \(b\in\mathbb{C}[p]\), \(X[k]+b\) is \(x(0) + b,\dots,x(k)+b\). \(x(\omega\mid\upsilon)\) is the component with index \(\upsilon\) in \(x(\omega)\) \((0\leq \omega \leq k\); \(1\leq \upsilon \leq p)\). \(f: {\mathfrak A}\to\mathbb{C}[p]\) being a suitable mapping, \(f(X[k]\mid \tau)\) is the component with index \(\tau\) in \(f(X[k])\). Assuming appropriate differentiability conditions, the derivative summand \({\mathcal D}(\omega)\) is the \(p\times p\) matrix having the element \(\partial f(X[k]\mid \tau)/\partial x(\omega\mid\upsilon)\) in the \(\tau\)th row and \(\upsilon\)th column; the derivative \({\mathcal D}f\) is defined by the sum \({\mathcal D}f(X[k])=\sum{\mathcal D}(\omega)\) \([0\leq \omega\leq k]\); the tensor \({\mathcal D}^ 2f\) is defined as the derivative of \({\mathcal D}f\) in a similar sense. The result in question is (1) the mapping \(F:\mathbb{C}[p]^{k+1}\to\mathbb{C}[p]\) satisfies the relationship \[ {F(X[k]+b)=}F(X[k])+b \] for all \(X[k]\), \(X[k]+b\) in \(\mathfrak A\) if and only if (2) a mapping \(f:{\mathfrak A}\to\mathbb{C}[p]\) exists for which both \({\mathcal D}^ 2f(X[k])\) is the zero trivalent tensor and \(F\) has the reciprocal logarithmic derivative representation \(F(X[k]) = {\mathcal D}f(X[k])^{-1}f(X[k])\). Many methods for accelerating the convergence of sequences of vectors make use of functions having the property just stated. Some of them are considered in detail.