The authors present a method for building efficient preconditioners for the solution of shifted linear systems of the form \((A+\alpha I) x_{\alpha} = b\), where \(A\) is symmetric and positive definite, \(I\) is the identity and \(\alpha > 0\). In this respect they derive an update technique for an \(LDL^T\) factorization of \(A\) that modifies only the nonzero entries of the \(L\) factor while leaving \(D\) unchanged.