$l_2 $-Solutions to Underdetermined Linear Systems

Abstract

Various algorithms for obtaining minimal $l_2 $-solutions to consistent linear systems of equations $Ax = y$ are discussed. These algorithms are based upon factorizations of matrices, A, with full row rank as $A = LDU$, where L is unit lower triangular, D is diagonal and U is either upper unit trapezoidal or such that rows of U are orthogonal. Included are well-known techniques and also a new method which requires, in general, fewer multiplications whenever the numbers of rows and columns of A satisfy certain relations.