The Davidson Method

Crouzeix, Michel; Philippe, Bernard; Sadkane, Miloud

doi:10.1137/0915004

Cited by 167 publications

(164 citation statements)

References 7 publications

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“…To overcome this problem, we develop an iterative diagonalization scheme using a plane-wave basis set. Because the TC Hamiltonian is non-Hermitian owing to the similarity transformation, some standard methods such as the conjugate gradient (CG) method [39] do not work and so we adopted the block-Davidson algorithm [40,41]. Whereas the block-Davidson algorithm has been successfully applied to other conventional methods such as DFT, some modifications described below are necessary to adopt it to the TC method.…”

Section: Block-davidson Algorithm For Plane-wave Calculationmentioning

confidence: 99%

Iterative diagonalization of the non-Hermitian transcorrelated Hamiltonian using a plane-wave basis set: Application to sp-electron systems with deep core states

Ochi

Yamamoto

Arita

et al. 2016

The Journal of Chemical Physics

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We develop an iterative diagonalization scheme in solving a one-body self-consistent-field equation in the transcorrelated (TC) method using a plane-wave basis set. Non-Hermiticity in the TC method is well handled with a block-Davidson algorithm. We verify the required computational cost is efficiently reduced by our algorithm. In addition, we apply our plane-wave-basis TC calculation to some simple sp-electron systems with deep core states to elucidate an impact of the pseudopotential approximation to the calculated band structures. We find a position of the deep valence bands is improved by an explicit inclusion of core states, but an overall band structure is consistent with a regular setup that includes core states into the pseudopotentials. This study offers an important understanding for the future application of the TC method to strongly correlated solids.

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Section: Block-davidson Algorithm For Plane-wave Calculationmentioning

confidence: 99%

Iterative diagonalization of the non-Hermitian transcorrelated Hamiltonian using a plane-wave basis set: Application to sp-electron systems with deep core states

Ochi

Yamamoto

Arita

et al. 2016

The Journal of Chemical Physics

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“…If X is also positive definite, then it is a feasible interior point. In this case, Slater's theorem applies (see, e.g., [16]) and strong duality holds; that is, the minimum value of the primal problem (4) is equal to the maximum value of the Lagrange dual problem (5).…”

Section: Note That the Adjoint Of This Operator Ismentioning

confidence: 99%

“…Subspace methods have been use to compute a few of the extreme eigenvalues and the corresponding eigenvectors of a large, sparse, symmetric matrices, such as the Lanczos method, which is based on Krylov subspaces [14], and the Davidson method which uses Rayleigh matrices [6]. Generalized Davidson [19] and JacobiDavidson type algorithms [20] have been introduced, and theoretical studies have been done in [5,12]. Davidson type subspace methods have been applied to solve graph partitioning problem [9,10] and extended eigenproblem [13].…”

Section: Introductionmentioning

confidence: 99%

A Subspace Semidefinite Programming for Spectral Graph Partitioning

Oliveira

Stewart

Soma

2002

Lecture Notes in Computer Science

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Abstract.A semidefinite program (SDP) is an optimization problem over n × n symmetric matrices where a linear function of the entries is to be minimized subject to linear equality constraints, and the condition that the unknown matrix is positive semidefinite. Standard techniques for solving SDP's require O(n 3 ) operations per iteration. We introduce subspace algorithms that greatly reduce the cost os solving large-scale SDP's. We apply these algorithms to SDP approximations of graph partitioning problems. We numerically compare our new algorithm with a standard semidefinite programming algorithm and show that our subspace algorithm performs better.

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“…Since it may be expensive to work with (A-iJ 1)-1 (i.e., to solve equations as (A-{} I)z = r exactly) in each step, one might prefer suitable approximations K of A -{)I or of A -µI (preconditioners; cf. [5,8,17,18,20]). Then, in JD, the search subspace is expanded by some appropriate linear combination of K-1 (A -7'J I) u and K-1 u while the Shift-and-Invert Arnoldi, properly adapted, will expand by K-1 u only.…”

Section: ( E) Solve the Correction Equation (Approximately)mentioning

confidence: 99%

Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems

et al. 1996

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Abstract.In this paper we will show how the Jacobi-Davidson iterative method can be used to solve generalized eigenproblems. Similar ideas as for the standard eigenproblem are used, but the projections, that are required to reduce the given problem to a small manageable size, need more attention. We show that by proper choices for the projection operators quadratic convergence can be achieved. The advantage of our approach is that none of the involved operators needs to be inverted. It turns out that similar projections can be used for the iterative approximation of selected eigenvalues and eigenvectors of polynomial eigenvalue equations. This approach has already been used with great success for the solution of quadratic eigenproblems associated with acoustic problems.

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The Davidson Method

Cited by 167 publications

References 7 publications

Iterative diagonalization of the non-Hermitian transcorrelated Hamiltonian using a plane-wave basis set: Application to sp-electron systems with deep core states

Iterative diagonalization of the non-Hermitian transcorrelated Hamiltonian using a plane-wave basis set: Application to sp-electron systems with deep core states

A Subspace Semidefinite Programming for Spectral Graph Partitioning

Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems

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