On an iterative method for solving the least squares problem of rank-deficient systems

Abstract

In this paper, we first show an iterative method for finding the least squares (LS) solution to the inconsistent system Ax=b, where A is an m×n matrix of rank r. The method is an iteration scheme for consistent system of linear equations M=bˆ which is an augmented system associated with Ax=b. It denotes that under some conditions, the sequence, ₀, ₁, ₂, … , converges to the LS solution of the system Ax=b for every initial vector ₀, where (M+γ E)_i=γ E_i−1+bˆ, for i=1, 2, … . In our numerical test, we propose to find E without using decomposition methods. The improved timings are shown with matrices of substantial size.

Keywords:

• iterative method
• index of a matrix
• iterative refinement
• Jordan canonical form

2013 AMS Subject Classification:

• 65F10

Acknowledgement

The research is supported by Scientific Computing Key Laboratory of Shanghai University and the National Natural Science Foundation of China under grant 11271084.