Ill-conditioned linear systems arise in many applications, for example, in the solution of integral equations, and in the solution of non-linear programming problems. In many application of linear algebra, the need arises to find a good approximation matrix (bx) to a vector (x) satisfying an approximating equation Ax b with ill-conditioned matrix (A), given matrix (b). Straightforward the computed solution (bx) is usually meaningless approximation to ( x ) due to the error in the righthand side ( b ) and the severe ill-conditioning of the matrix ( A).In order to avoid this difficulty, one typically replaces the linear systems Ax = b, by a nearby system that is less sensitive to the error in (b) and considers the computed solution of the latter system an approximation of (x). This replacement is known as regularization. This work examines various regularization methods for computing stable solution to ill-conditioned linear systems.

Sheima M. E. Abueldahab is a Lecturer at the faculty of Engineering, University of Khartoum. In 2002 she got a BSc, first honor degree in Mathematics from the Univ. of Khartoum. In 2006 she got a MSc in Industrial and Computational Mathematical Science from Univ. of Khartoum. In 2007 she got a PhD in Mathematics from the Western Cape University.