  The present paper gives a review of our recent progress and latest results
for novel linear-algebraic algorithms and its application to large-scale
quantum material simulations or electronic structure calculations. The
algorithms are Krylov-subspace (iterative) solvers for generalized shifted
linear equations, in the form of (zS-H)x=b,in stead of conventional generalized
eigen-value equation. The method was implemented in our order-$N$ calculation
code ELSES (http://www.elses.jp/) with modelled systems based on ab initio
calculations. The code realized one-hundred-million-atom, or 100-nm-scale,
quantum material simulations on the K computer in a high parallel efficiency
with up to all the built-in processor cores. The present paper also explains
several methodological aspects, such as use of XML files and 'novice' mode for
general users. A sparse matrix data library in our real problems
(http://www.elses.jp/matrix/) was prepared. Internal eigen-value problem is
discussed as a general need from the quantum material simulation. The present
study is a interdisciplinary one and is sometimes called
'Application-Algorithm-Architecture co-design'. The co-design will play a
crucial role in exa-scale scientific computations.
