  The Lanczos algorithm has proven itself to be a valuable matrix eigensolver
for problems with large dimensions, up to hundreds of millions or even tens of
billions. The computational cost of using any Lanczos algorithm is dominated by
the number of sparse matrix-vector multiplications until suitable convergence
is reached. Block Lanczos replaces sparse matrix-vector multiplication with
sparse matrix-matrix multiplication, which is more efficient, but for a
randomly chosen starting block (or pivot), more multiplications are required to
reach convergence. We find that a bootstrapped pivot block, that is, an initial
block constructed from approximate eigenvectors computed in a truncated space,
leads to a dramatically reduced number of multiplications, significantly
outperforming both standard vector Lanczos and block Lanczos with a random
pivot. A key condition for speed-up is that the pivot block have a non-trivial
overlap with the final converged vectors. We implement this approach in a
configuration-interaction code for nuclear structure, and find a reduction in
time-to-solution by a factor of two or more, up to a factor of ten.
