Solving symmetric arrowhead linear systems by approximate inverses

A new class of approximate inverse arrowhead matrix techniques based on the concept of sparse approximate Choleski-type factorization procedures is introduced for computing fast explicit approximate inverses. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of arrowhead symmetric linear systems. Theoretical results on the rate of convergence of the explicit preconditioned conjugate gradient scheme are given and estimates of the computational complexity required to reduce the L-infinity - norm of the error by a factor c is presented. Application of the proposed method on a linear system is discussed and numerical results are given.