Perturbation of the least squares problem
Let \(\bar A=A+\delta A\), \(\bar b=b+\delta b\) where A is a matrix, b is a vector, and \(\delta\) A, \(\delta\) b are perturbations in A and b. Let \(x_{LS}\) be the minimum norm least squares solution of \(Ax=b\), and \(\bar x_{LS}\) the minimum norm least squares solution of \(\bar Ax=\bar b\). This paper shows that there exists a least squares solution \(\hat x\) to \(Ax=b\) for which \(\| \hat x-\bar x_{LS}\|\) satisfies an error estimate in terms of \(\{\| A\|,\| b\|,\| \delta A\|,\| \delta b\|,\| \bar x_{LS}\|,\| \hat x\|,\| b-Ax_{LS}\| \}\).
scientific article; zbMATH DE number 4168842