Two Iterative Methods for Solving Linear Interval Systems. (8th October 2018) - Siahlooei, Esmaeil; Shahzadeh Fazeli, Seyed Abolfazl
Abstract : Conjugate gradient is an iterative method that solves a linear systemA x = b, whereA is a positive definite matrix. We present this new iterative method for solving linear interval systemsA ̃ x ̃ = b ̃, whereA ̃ is 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 experiments show that the new interval modified conjugate gradient method minimizes the norm of the difference ofA ̃ x ̃ andb ̃ at every step while the norm is sufficiently small. In addition, we present another iterative method that solvesA ̃ x ̃ = b ̃, whereA ̃ is a diagonally dominant interval matrix. This method, using the idea of steepest descent, finds exact solutionx ̃ for linear interval systems, whereA ̃ x ̃ = b ̃ ; we present a proof that indicates that this iterative method is convergent. Also, our numerical experiments illustrate the efficiency of the proposed methods.