On the convergence of some interval-arithmetic modifications of Newton’s method. (English) Zbl 0536.65026

This interesting paper studies an interval version of Newton’s method for solving systems of n nonlinear equations, in which the auxiliary linear interval equation is solved by Gauss elimination. Main result: The method converges (or shows no existence of a solution) if the spectral radius of the matrix \(| I-IGA({\mathcal A})\cdot {\mathcal A}|\) is less than one, where \({\mathcal A}=f'(x^ 0)\) and IG\(A({\mathcal A})\) is the inverse of \({\mathcal A}\) computed by applying Gauss elimination to the columns of \({\mathcal A}\). Unfortunately the test is unpractical for band matrices, requiring e.g. \(O(n^ 2)\) operations for tridiagonal \({\mathcal A}\).
Reviewer: A.Neumaier


65H10 Numerical computation of solutions to systems of equations
65G30 Interval and finite arithmetic
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