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A linear-algebraic theory called 'multiple Arnoldi method' is presented andrealizes large-scale (order-N) electronic structure calculation withgeneralized eigen-value equations. A set of linear equations, in the form of(zS-H) x = b, are solved simultaneously with multiple Krylov subspaces. Themethod is implemented in a simulation package ELSES (http://www.elses.jp) withtight-binding-form Hamiltonians. A finite-temperature molecular dynamicssimulation is carried out for metallic and insulating materials. A calculationwith $10^7$ atoms was realized by a workstation. The parallel efficiency isshown upto 1,024 CPU cores.

An order-N electronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system