The acceleration of the convergence of certain approximant sequences (Q2755015)

The subject of this paper is the numerical solution of the abstract equation NEWLINE\[NEWLINEf(x)=\theta_Y,NEWLINE\]NEWLINE where \(f:D\rightarrow Y\), \(D\subset X\) and \((X,\|.\|_X)\), \((Y,\|.\|_Y)\) are two linear normed spaces, while \(\theta_Y\) is the null-element of \(Y\). The author considers a class of iterative processes generating sequences of elements \(\{x_k\}\), \(x_k\in X\) such that \(\|x^*-x_n\|_X\rightarrow 0\), where \(f(x^*)=\theta_Y\). NEWLINENEWLINENEWLINEThe first section of the paper recalls processes published elsewhere by various authors -- all of them being certain generalizations of the classical Newton process. In general, existence of the \(p\)-th Fréchet derivative of \(f\) for \(p\geq 1\), as well as invertibility of \(f'\) is assumed. The classical Newton equation NEWLINE\[NEWLINEf'(x_n)(x_{n+1}-x_n)+f(x_n)=0NEWLINE\]NEWLINE is there replaced by following system of two inequalities of the form NEWLINE\[NEWLINE\|f(x_n)+\sum_{i=1}^p{1\over i!}f^{(i)}(x_n)(x_{n+1}-x_n)^i\|_Y \leq a\|f(x_n)\|_Y^{p+1},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\|f'(x_n)(x_{n+1}-x_n)\|_Y \leq b\|f(x_n)\|_YNEWLINE\]NEWLINE for certain constants \(a\) and \(b\) independent of \(n\). NEWLINENEWLINENEWLINEThe author raises the following problem: ``How we can influence the construction of the sequence \(\{x_n\}\) using an additional sequence \(\{y_n\}\), such that the order of the approximant sequence will grow substantially, and we will need far fewer operations in order to obtain \(y_n\) from \(x_n\) than to obtain \(x_{n+1}\).'' NEWLINENEWLINENEWLINEThe proposed construction is as follows. We are looking for a pair of sequences \(\{x_n\}\), \(\{y_n\}\) satisfying for certain contants \(a\) and \(b\) NEWLINE\[NEWLINEf'(x_n)(x_n-y_n)+f(y_n)=\theta_Y,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\|f(x_n)+\sum_{i=1}^p{1\over i!}f^{(i)}(x_n)(y_n-x_n)^i\|_Y\leq a\|f(x_n)\|_Y^{p+1},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\|f'(x_n)(x_{n+1}-x_n)\|\leq b\|f(x_n)\|_Y.NEWLINE\]NEWLINE A theorem on the convergence and its rate is given for this new process. Examples of several realizations are presented. One of them is as follows NEWLINE\[NEWLINEf'(x_n)(y_n-x_n)+f(x_n)=\theta_Y,NEWLINE\]NEWLINE NEWLINE\[NEWLINEf'(x_n)(x_{n+1}-y_n)+f(y_n)=\theta_Y.NEWLINE\]NEWLINE This is equivalent to J. F. Traub's process which is of the order 3. In order to compute \(y_n\) and \(x_{n+1}\) the same Jacobian \(f'(x_n)\) can be used.