Note on generalized Archimedes-Borchardt algorithm
The Newton process as follows \[ x_{n+1}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right)\qquad n\in\mathbb{N}_0,\tag{1} \] where \(a\) is a positive number yields \(x_n\longrightarrow\sqrt{a}\) as \(n\to\infty\). By setting \(y_n=\frac{a}{x_n}\), we see that (1) is equivalent to \[x_{n+1}=A(x_n, y_n) \quad\text{and}\quad y_{n+1}=H(x_n, y_n)\quad (n\in\mathbb{N}_0)\,, \tag{2}\] where \(A\) and \(H\) are the arithmetic and harmonic means, respectively. A generalization of (2) is as follows: \[ x_{n+1}=M(x_n, y_n) \quad\text{and}\quad y_{n+1}=N(x_n, y_n)\quad (n\in\mathbb{N}_0)\,, \] where \(M\) and \(N\) are means on an interval \(I\), and \(M:I^2\longrightarrow I\) is called a bivariate mean on \(I\) such that \[ \min\{x, y\} \le M(x, y) \le \max\{x, y\},\quad x, y\in I\,. \] The mean \(M\) is \textit{strict}, if \[ \min\{x, y\} < M(x, y) < \max\{x, y\},\quad x, y\in I, x\neq y\,, \] and is \textit{symmetric} if \[ M(x, y)=M(y, x),\qquad x, y\in I\,, \] and continuous. The Archimedes-Borchardt algorithm has the following convergence property:\\ If \(x_0, y_0\) are points of an interval \(I\) and \(M, N:I^2\longrightarrow I\) are continuous, strict and symmetric means, then the sequences \((x_n)_{n\in\mathbb{N}}\) and \((y_n)_{n\in\mathbb{N}}\) defined by \[ x_{n+1}=M(x_n, y_n) \quad\text{and}\quad y_{n+1}=N(x_n, y_n)\quad (n\in\mathbb{N}_0)\,, \] converge monotonically to a common limit. In the paper under review, the authors give a generalization of the Archimedes-Borchardt algorithm by using a Gauss iteration process and prove a convergence result extending the one of the above algorithm.
scientific article; zbMATH DE number 7439040
scientific article