  A deflated restarted Lanczos algorithm is given for both solving symmetric
linear equations and computing eigenvalues and eigenvectors. The restarting
limits the storage so that finding eigenvectors is practical. Meanwhile, the
deflating from the presence of the eigenvectors allows the linear equations to
generally have good convergence in spite of the restarting. Some
reorthogonalization is necessary to control roundoff error, and several
approaches are discussed. The eigenvectors generated while solving the linear
equations can be used to help solve systems with multiple right-hand sides.
Experiments are given with large matrices from quantum chromodynamics that have
many right-hand sides.
