Componentwise error estimates for approximate solutions to systems of equations with the aid of Dahlquist constants (Q1822458)

Suppose that, in some set D of \(R^ n\) the mapping \(f: R^ n\to R^ n\) has a fixed point \(x^*\) and satisfies (1) \(\nu\) (f(x)-f(y))\(\leq K\nu (x-y)\), where \(\nu (x)=(| x_ 1|,...,| x_ n|)^ T\), \(K\geq 0\) is a given matrix, and the natural ordering of \(R^ n\) is used. Assume further that a matrix M with non-negative off-diagonal entries exists such that for \(\epsilon >0\) there is a \(\delta >0\) so that \(\nu [(I+hf)(x)-(I+hf)(y)]-\nu (x-y)\leq h(M+\epsilon I)(x-y)\) for x,y\(\in D\) and \(0<h<\delta\). Then it is shown that \(\nu (x^*-x^ 1)\leq (I-M)^{- 1}K\nu (x^ 0-x^ 1)\) for any \(x^ 0\in D\) and \(x^ 1=f(x^ 0)\in D\). This extends an estimate of \textit{G. Söderlind} [BIT 24, 391-393 (1984; Zbl 0554.65039)] to this case of componentwise bounds. Then maps \(f(x)=x-H(x)g(x)\) with suitable matrices H(x) are considered as they arise in iterative methods for solving a given equation (1) \(g(x)=0\). This leads to bounds of approximate solutions of (1).