| Details: |
Consider a singular or rectangular system of the form
m×n m
Ax=b, A∈R , b∈R . (1)
The singular systems are arises in various branch of Science and Engineering such as statistical models, forecast modelling and partial differential equations. Several iterative methods are proposed for singular system to improve the convergence rate as well better complexity. To deal such singular system, in recent past many re- searchers has considered the splitting theory such as proper regular splitting and proper weak regular splitting. For example, If A = B − C is a proper splitting (if R(A) = R(U) and N(A) = N(U)) of A ∈ Rm×n, then the iterative scheme
xk+1 = B†Cxk + B†b (2)
for (1) converges to A†b, the least squares solution for any initial vector x0 iff ρ(B†C) < 1.
In this talk, we will discuss alternating iterative schemes based on some split- tings and generalized inverses. In addition to these, we will explain a few other approaches such as regularization theory and pre-conditioning techniques to relax some conditions. |