AbstractMany popular solution methods for large discrete ill‐posed problems are based on Tikhonov regularization and compute a partial Lanczos bidiagonalization of the matrix. The computational effort required by these methods is not reduced significantly when the matrix of the discrete ill‐posed problem, rather than being a general nonsymmetric matrix, is symmetric and possibly indefinite. This paper describes new methods, based on partial Lanczos tridiagonalization of the matrix, that exploit symmetry. Computed examples illustrate that one of these methods can require significantly less computational work than available structure‐ignoring schemes. Copyright © 2004 John Wiley & Sons, Ltd.