Shifted Conjugate-Orthogonal–Conjugate-Gradient Method and Its Application to Double Orbital Extended Hubbard Model

Abstract

We discuss the shifted conjugate-orthogonal-conjugate-gradient (COCG) method, which can be used to solve a series of linear equations generated by a number of scalar shifts, without performing timeconsuming matrix-vector operations, except at a single reference energy. This is a type of CG method and is robust and has an expression of the accuracy of the calculated result that requires no additional calculation cost. The shifted COCG is useful for calculating the Green's function of a many-electron Hamiltonian, which has a very large dimension. We applied the shifted COCG method to the double orbital extended Hubbard model with twelve electrons on a periodic ffiffi ffi 8 p  ffiffi ffi 8 p site system, with the dimension of the Hamiltonian equal to 64,128,064, and we found that the ground state is an insulator. We also discuss the main points of reducing the amount of memory required to apply COCG algorithm.

LDA+DMFTapproach to resonant inelastic x-ray scattering in correlated materials

“…Their propagation by the resolvent (ω in + E n − H AIM + iΓ) −1 T i |n is computed using conjugate-gradient-based method. Though the RIXS calculation for different photon energies ω in can be parallelized in a straightforward way, one can also adopt the shift and seed-switching techniques in the conjugategradient-based method (for constant Γ case) [46,47], see Appendix. I.…”

Solution of generalized shifted linear systems with complex symmetric matrices

“…This enables us to use the seed switching technique [30] (see also [38] for its application) for the generalized shifted COCG method in a similar manner.…”

Efficient and accurate linear algebraic methods for large-scale electronic structure calculations with nonorthogonal atomic orbitals

“…The seed switching is a very efficient technique to avoid restarting the calculation from the beginning with a new seed energy. 16,25 One chooses a new seed energy σ new s and can continue the calculation without discarding the information of the previous calculation with the old σ s by using the shift property. Figure 3 shows the behavior of the residual norms in the seed-switching process.…”