A massively parallel order-N electronic structure theory was developed by interdisciplinary research between physics, applied mathematics, and computer science. (1) A high parallel efficiency with ten-million-atom nanomaterials was realized on the K computer with up to 98,304 processor cores. The mathematical foundation is a novel linear algebraic algorithm for generalized shifted linear equations. The calculation was carried out using our code ''ELSES'' (www.elses.jp) with modelled (tight-binding-form) systems based on ab initio calculations. (2) A postcalculation analysis method, called the -orbital crystalline orbital Hamiltonian population ( -COHP) method, is presented, since this method is ideal for huge electronic structure data distributed among massive nodes. The analysis method is demonstrated in an sp 2 -sp 3 nanocomposite carbon solid, with the original visualization software ''VisBAR''. The present research indicates the general aspects of computational physics with current or next-generation supercomputers.KEYWORDS: order-N electronic structure theory, numerical linear algebra, massive parallel computation, sp 2 -sp 3 nanocomposite carbon solid, crystalline orbital Hamiltonian populationA common issue in current computational physics is the theory for large calculations with modern massively parallel supercomputers, such as the K computer. Large calculations should accompany a large-data analysis theory, as a postcalculation procedure, so as to obtain a physical conclusion from huge numerical data distributed among massive computer nodes. Interdisciplinary research studies between physics, applied mathematics and computer science are thus crucial and such a study is sometimes called ''application-algorithm-architecture co-design''.The present paper is devoted to theories, for both largescale calculation and large-data analysis, as order-N electronic structure theories suitable for modern massively parallel computers.Order-N electronic structure theories are those in which the computational time is proportional to the number of atoms in the system N and are promising for large-scale calculation. The reference list for the order-N theories can be found, for example, in Ref. 1. Recently, several novel linear algebraic algorithms, with Krylov subspace, have been developed for the order-N theory, i.e., the generalized shifted conjugate-orthogonal conjugate-gradient method, [1][2][3] generalized Lanczos method, 1) generalized Arnoldi method, 1) Arnoldi ðM; W; GÞ method, 4) multiple Arnoldi method, 5) and generalized shifted quasi-minimal-residual method. 3) Their common foundation is the ''generalized shifted linear equation'', or the set of linear equationswhere z is a (complex) energy value and H and S denote the Hamiltonian and overlap matrices in the linear-combinationof-atomic-orbital (LCAO) representation, respectively. The matrices are sparse real-symmetric M Â M matrices, and S is positive definite. The vector b is the input and the vector x is the solution vector. Equation (1) is solved, iterativ...
