An efficient LDU algorithm for the minimal least squares solution of linear systems

Abstract

Abstract The minimal least squares solutions is a topic of interest due to the broad range of applications of this problem. Although it can be obtained from other algorithms, such as the Singular Value Decomposition (SVD) or the Complete Orthogonal Decomposition (COD), the use of LDU factorizations has its advantages, namely the computational cost and the low fill-in that can be obtained using this method. If the right and left null-subspaces (which can also be named as Null and Image subspaces, respectively) are to be obtained, the use of these factorizations leads to fundamental subspaces, which are sparse by definition. Here an algorithm that takes advantage of both the Peters–Wilkinson method and Sautter method is presented. This combination allows for a good performance in all cases. The method also optimizes memory use by storing the right null-subspace and the left null-subspace in the factored matrix.