  An extremely scalable linear-algebraic algorithm was developed for quantum
material simulation (electronic state calculation) with 10$^8$ atoms or
100-nm-scale materials. The mathematical foundation is generalized shifted
linear equations ((zB - A) x = b), instead of conventional generalized
eigenvalue equations. The method has a highly parallelizable mathematical
structure. The fundamental theory is mathematical and is applicable also to
other scientific fields. The benchmark shows an extreme strong scaling and a
qualified time-to-solution on the full system of the K computer. The method was
demonstrated in a real material research for ultra-flexible (organic) devices,
key devices of next-generation IoT products. The present paper shows that an
innovative scalable algorithm for a real research can appear by the co-design
among application, algorithm and architecture.

Materials Science;Computational Physics