Quasilinear sequence transformations (Q1381377)

\(p>0\) and \(k\geq 0\) are integers. The elements of \(E\equiv\mathbb{R}^p\) are represented as column vectors. \({\mathfrak G}{\mathfrak L}(E)\) is the linear isomorphism group over \(E\) and \(I\) is its unit member. \(E(p,k)\) is \(E\times\cdots\times E\) (\(k+1\) times). A transformation over \(E\) is extended to \(E(p,k)\) by setting \(A(x,y,\dots, z)= (Ax,Ay,\dots, Az)\). The scalar part \({\mathcal S}\{{\mathfrak D}\}\) of any domain \({\mathfrak D}\) in \(E(p,k)\) is that part consisting of elements of the form \((x,\dots, x)\) alone; with \(x\in E\), \(X= (x,\dots, x)\) is written as \(\text{scal}(x)\); red is the reduction operator from \({\mathcal S}\{E(p,k)\}\) to \(E\) defined by setting \(\text{red}(X)= x\). \(f\) is a transformation from \(E(p, k)\) to \(E\); \(\text{dom}(f)\) and \({\mathcal N}(f)\) are its domain and kernel respectively. \(f\) for which \(X+\text{scal}(b)\) is in \(\text{dom}(f)\) whenever \(X\) is in \(\text{dom}(f)\) and \(b\in E\) and for which also \(f(X+ \text{scal}(b))= f(X)+ b\) is said to be translative. \({\mathfrak M}\) being a set of multipliers in \({\mathfrak G}{\mathfrak L}(E)\), a transformation \(f\) for which (a) \(MX\) is in \(\text{dom}(f)\) whenever \(X\) is in \(\text{dom}(f)\) and (b) \(f(MX)= Mf(X)\), both for all \(M\in{\mathfrak M}\), is said to be \({\mathfrak M}\)-homogeneous. An \({\mathfrak M}\)-homogeneous transformation which is also translative is said to be \({\mathfrak M}\)-quasilinear. The transformation \(\varepsilon(n, 2k)\) over \(E(p,2k)\) determined recursively by a constrained variant of the vector \(\varepsilon\)-algorithm in which \(y(\neq 0)\in E\) is fixed, \(\varepsilon(m,- 1)= 0\in E\) \((n<m\leq n+ 2k)\), \(\varepsilon(m, 0)= x(m)\in E\) \((n\leq m\leq n+ 2k)\), \(\varepsilon(m, 2r+1)= \varepsilon(m+1, 2r-1)+ y/\langle y,\Delta\varepsilon(m, 2r)\rangle\) \((0\leq r<k; n\leq m<n+ 2k- 2r)\), \(\varepsilon(m, 2r+2)= \varepsilon(m+1, 2r)+ \Delta\varepsilon(m, 2r)/\langle\Delta \varepsilon(m, 2r+1), \Delta\varepsilon(m, 2r)\rangle\) \((0\leq r<k; n\leq m< n+2k- 2r-1)\) is considered (scalar products are denoted by triangular brackets; \(\Delta\varepsilon(m, 2r)\) is \(\varepsilon(m+ 1,2r)- \varepsilon(m, 2r)\) and so on). It is \({\mathfrak M}\)-quasilinear, \({\mathfrak M}\) being the system of operators \(\lambda I\) with \(\lambda(\neq 0)\in\mathbb{R}\). It is also shown that the transformation \(\varepsilon(n, 2k)\) over \(E(p, 2k)\) determined by the vector \(\varepsilon\)-algorithm itself is \({\mathfrak M}\)-quasilinear where \({\mathfrak M}\) is now the system of operators \(\lambda T\) with \(\lambda(\neq 0)\in\mathbb{R}\) and \(T\) fixed and orthogonal. [This result was given by the reviewer in: Upon a conjecture concerning a method for solving linear equations, and certain other matters, Trans. of the Twelfth Conference of Army Mathematicians, USA War Office (1966).] For a transformation due to Henrici, \({\mathfrak M}\) is \({\mathfrak G}{\mathfrak L}(E)\). Results concerning the structure of quasilinear transformations are derived. If \(f\) is translative (in particular, if \(f\) is quasilinear with respect to some set) \((*)\) \(\text{dom}(f)= {\mathcal N}(f)+{\mathcal S}\{E(p, k)\}\). It follows that translative transformations are characterized by their kernels in the sense that \(f\) and \(g\), both translative over \(E(p,k)\), have equal kernels if and only if they are equal over \(E(p, k)\). The stated decomposition over \(E(p,k)\) is unique: a projection \({\mathcal P}(f)\) from \(\text{dom}(f)\) to \({\mathcal N}(f)\) is defined and in terms of it \(f\) has a composition reduction: \(f=\text{red}\circ (I-{\mathcal P}(f))\). All of the preceding theory is presented in terms of quasilinear transformations, but in fact it holds with regard to transformations that are translative alone. The only point at which quasilinearity comes into play concerns \({\mathfrak M}\)-invariance of \({\mathcal N}(f): M{\mathcal N}(f)\equiv{\mathcal N}(f)\) for all \(M\in{\mathfrak M}\). Convergence acceleration results are given. The transformation \(\varepsilon(n, 2k)\) determined recursively by a constrained variant of the vector \(\varepsilon\)-algorithm is \({\mathfrak M}\)-quasilinear over a group larger than that stated by the author, namely over the Abelian group \({\mathfrak M}\) of matrices \(\lambda I+\mu yy^*\) with \(\lambda\neq 0\), \(\lambda+ \mu\langle y,y\rangle\neq 0\) (\(y^*\) is the transpose of \(y\) and may be taken to be the complex conjugate transpose in an obvious complex extension of the theory). As is easily verified, the effect of replacing \(x(m)\) by \(\{\lambda I+\mu yy^*\}x(m)\) is to replace all \(\varepsilon(m, 2r)\) in the same way and to replace \(\varepsilon(m, 2r+1)\) by \(\varepsilon(m, 2r+1)/(\lambda+ \mu\langle y,y\rangle)\). The exposition is perhaps a little confused concerning the result \((*)\). In the construction of subsequent illustrative examples, the converse of this result is used. No harm is done: in each case \({\mathcal N}(f)\) relates to transformations \(f\) that are translative, and working backwards naturally leads to acceptable examples \(f\). But it is not clear that the converse result holds.