Two Iterative Methods for Solving Linear Interval Systems

Conjugate gradient is an iterative method that solves a linear system𝐴𝑥 = 𝑏,where𝐴is a positive definite matrix. We present this new iterative method for solving linear interval systems̃𝐴̃𝑥=̃𝑏,wherẽ𝐴is a diagonally dominant interval matrix, as defined in this paper. Our method is based on conjugate gradient algorithm in the context view of interval numbers. Numerical experiment sshow that the new interval modified conjugate gradient method minimizes the norm of the difference of̃𝐴̃𝑥and̃𝑏at every stepwhile the norm is sufficiently small. In addition, we present another iterative method that solves̃𝐴̃𝑥=̃𝑏,wherẽ𝐴is a diagonally dominant interval matrix. This method, using the idea of steepest descent, finds exact solutioñ𝑥for linear interval systems, wherẽ𝐴̃𝑥=̃𝑏; we present a proof that indicates that this iterative method is convergent. Also, our numerical experiments illustrate the efficiency of the proposed methods