Acceleration property for the columns of the E-algorithm
Let \((S_ n)_{n\in N}\) be a sequence of complex numbers that satisfies the following relation: \((1)\;S_ n=S+a_ 1g_ 1(n)+a_ 2g_ 2(n)+\dots + a_ kg_ k(n),\) \(n\in N\), where the \(g_ i\) are given sequences. The authors consider the problem of computing the value of \(S\) (which is the limit of \((S_ n)\) if it converges or its antilimit if it diverges). For this, we write relation (1) for the index varying from n to \(n+k\): \((2) S_{n+i}=S+a_ 1g_ 1(n+i)+a_ 2g_ 2(n+i)+\dots+a_ kg_ k(n+i)\), \(i=0,\dots,k\). If the above system (2) is nonsingular then the solution \(S\) is given by the formula \[ S={|\text{col}(\tilde S_ n)\text{col}(\tilde g_ 1)\dots\text{col}(\tilde g_ k)|\over |\text{col}(\tilde 1)\text{col}(\tilde g_ 1)\dots\text{col}(\tilde g_ k)|}\tag{3} \] where we denoted \(\text{col}(\tilde S_ n)=\text{col}(S_ n,S_{n+1},\dots,S_{n+k})\), \(\text{col}(\tilde g_ i)=\text{col}(g_ i(n),g_ i(n+1),\dots,g_ i(n+k))\), \((i=1,\dots,k)\) and \(\text{col}(\tilde 1)=\text{col}(1,\dots,1)\). To simplify the notation, we identify each of these two determinants with its first row. The formula is thus shortened to \[ S={| S_ ng_ 1(n)\dots g_ k(n)|\over | 1 g_ 1(n)\dots g_ k(n)|}. \] If the sequence \((S_ n)\) has not exactly form (1) then the value of \(S\) given by (3) depends on the indices \(n\) and \(k\). Let denote \[ E_ k(S_ n)={| S_ ng_ 1(n)\dots g_ k(n)|\over | 1 g_ 1(n) \dots g_ k(n)|}.\tag{4} \] The \(E\)-algorithm is a recursive algorithm which enables us to compute the numbers \(E_ k(S_ n)\) for all \(k\) and \(n\) without computing the determinants occurring in (4). A convergence acceleration result for the \(E\)-algorithm is proved for sequences such that the error has an asymptotic expansion on a scale of comparison for which a determinantal relation holds. Four examples of sequences \((g_ i(n))\) satisfying this determinantal relation are given. The acceleration result is also generalized to the vector case.
scientific article; zbMATH DE number 95471