We present a technique for building effective and low cost preconditioners for sequences of shifted linear systems $(A + alpha I) x_alpha = b$, where A is symmetric positive definite and $alpha > 0$. This technique updates a preconditioner for A, available in the form of an $LDL^T$ factorization, by modifying only the nonzero entries of the L factor in such a way that the resulting preconditioner mimics the diagonal of the shifted matrix and reproduces its overall behavior. This approach is supported by a theoretical analysis as well as by numerical experiments, showing that it works efficiently for a broad range of values of $alpha$.
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