Towards acceleration of Rump's fast and parallel circular interval arithmetic for enclosing solution of non linear system of equations

Abstract

It is often the practice to express measurements of experimental problems in terms of uncertainties either due to contamination of measuring instruments or inaccurate measurements in the experimental models. In this note as an attempt at solving a system of nonlinear equation by accelerating the convergence of Rump's fast and parallel interval arithmetic incorporating where in, the Carstensen and Petkovic circular arithmetic for enlarging a disk to be inverted in the complex plane. The problem of excess widths in the midpoint-radius matrix and midpoint –radius vector multiplication is taken into account by using the procedure of Ceberio and Kreinovich for fast multiplication of two interval matrices (or interval matrix and interval vector) whose entries are expressed in terms of midpoint-radius matrix. We used Interval Gaussian Elimination algorithm and Interval Gauss-Siedel iterative method as our basic tools with Newtonian steps, some significant gains over that of Rump’s method were achieved. A stopping criterion for a Newton's step is given in terms of defect measurement instead of the error. AMS SUBJECT CLASSIFICATION (2000): 65 G20, 65G30, 65G40. Key words: Rump's interval operation, zeros of nonlinear system of equation, Carstensen and Petkovic circular interval arithmetic for disk inversion.