Gauss quadrature based finite temperature Lanczos method

The finite temperature Lanczos method(FTLM),which is an exact diagonalization method intensively used in quantum many-body calculations,is formulated in the framework of orthogonal polynomials and Gauss quadrature.The main idea is to reduce finite temperature static and dynamic quantities into weighted summations related to one-and twodimensional Gauss quadratures.Then lower order Gauss quadrature,which is generated from Lanczos iteration,can be applied to approximate the initial weighted summation.This framework fills the conceptual gap between FTLM and kernel polynomial method,and makes it easy to apply orthogonal polynomial techniques in the FTLM calculation.

Generalized Lanczos method for systematic optimization of tensor network states

We propose a generalized Lanczos method to generate the many-body basis states of quantum lattice models using tensor-network states （TNS）. The ground-state wave function is represented as a linear superposition composed from a set of TNS generated by Lanczos iteration. This method improves significantly the accuracy of the tensor-network algorithm and provides an effective way to enlarge the maximal bond dimension of TNS. The ground state such obtained contains significantly more entanglement than each individual TNS, reproducing correctly the logarithmic size dependence of the entanglement entropy in a critical system. The method can be generalized to non-Hamiltonian systems and to the calculation of low-lying excited states, dynamical correlation functions, and other physical properties of strongly correlated systems.

ON CONVERGENCE PROPERTY OF THE LANCZOS METHOD FOR SOLVING A COMPLEX SHIFTED HERMITIAN LINEAR SYSTEM

We discuss the convergence property of the Lanczos method for solving a complex shifted Hermitian linear system （αI ＋ H）x = f. By showing the colinear coefficient of two system＇s residuals, our convergence analysis reveals that under the condition Re（α） ＋ λmin（H） 〉 0, the method converges faster than that for the real shifted Hermitian linear system （Re（α）I ＋ H）x = f. Numerical experiments verify such convergence property.

VIBRATION MODE ANALYSIS OF MULTILAYERED COMPOSITE PLATES AND SHELLS USING NATURAL APPROACH

Argyris'natural approach is employed to analyze vibranon mode of multilayered composite plates and shells.The shells can be either symmetric or unsymmetric.The spectral transformation Lanczos method with selective or fully orthogonalization is used to solve the eigenvalue problem of pencil(K,M).Some problems on shift,which is essential for the success of this method, are discussed.A few numerical examples, including composite square plates and conical shells,are presented. The results show that the method in this paper is efficient and reliable for vibration mode analysis.

A Nonmonotone Hybrid Method of Conjugate Gradient and Lanczos-type for Solving Nonlinear Systems

In this paper,we construct a new algorithm which combines the conjugate gradient and Lanczos methods for solving nonlinear systems.The iterative direction can be obtained by solving a quadratic model via conjugate gradient and Lanczos methods.Using the backtracking line search,we will find an acceptable trial step size along this direction which makes the objective function nonmonotonically decreasing and makes the norm of the step size monotonically increasing.Global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions.Finally,we present some numerical results to illustrate the effectiveness of the proposed algorithm.

PRECONDITIONING BLOCK LANCZOS ALGORITHM FOR SOLVING SYMMETRIC EIGENVALUE PROBLEMS

A preconditioned iterative method for computing a few eigenpairs of large sparse symmetric matrices is presented in this paper. The proposed method which combines the preconditioning techniques with the efficiency of block Lanczos algorithm is suitable for determination of the extreme eigenvalues as well as their multiplicities. The global convergence and the asymptotically quadratic convergence of the new method are also demonstrated. [ABSTRACT FROM AUTHOR]

Error bounds of Lanczos approach for trust-region subproblem

Because of its vital role of the trust-region subproblem （TRS） in various applications, for example, in optimization and in ill-posed problems, there are several factorization-free algorithms for solving the large-scale sparse TRS. The truncated Lanczos approach proposed by N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint [SIAM J. Optim., 1999, 9： 504-525] is a natural extension of the classical Lanczos method for the symmetric linear system and eigenvalue problem and, indeed follows the classical Rayleigh-Ritz procedure for eigenvalue computations. It consists of 1） projecting the original TRS to the Krylov subspa^es to yield smaller size TRS＇s and then 2） solving the resulted TRS＇s to get the approximates of the original TRS. This paper presents a posterior error bounds for both the global optimal value and the optimal solution between the original TRS and their projected counterparts. Our error bounds mainly rely on the factors from the Lanczos process as well as the data of the original TRS and, could be helpful in designing certain stopping criteria for the truncated Lanczos approach.

A Spectral Time-Domain Method for Computational Electrodynamics

Ever since its introduction by Kane Yee over forty years ago,the finitedifference time-domain(FDTD)method has been a widely-used technique for solving the time-dependent Maxwell’s equations that has also inspired many other methods.This paper presents an alternative approach to these equations in the case of spatially-varying electric permittivity and/or magnetic permeability,based on Krylov subspace spectral(KSS)methods.These methods have previously been applied to the variable-coefficient heat equation and wave equation,and have demonstrated high-order accuracy,as well as stability characteristic of implicit timestepping schemes,even though KSS methods are explicit.KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Golub and Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral,rather than physical,domain.We show how they can be generalized to coupled systems of equations,such as Maxwell’s equations,by choosing appropriate basis functions that,while induced by this coupling,still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields.We also discuss the application of block KSS methods to problems involving non-self-adjoint spatial differential operators,which requires a generalization of the block Lanczos algorithm of Golub and Underwood to unsymmetric matrices.

Zero Temperature Numerical Studies of Multiband Lattice Models of Strongly Correlated Electrons

Relative to single-band models,multiband models of strongly interacting electron systems are of growing interest because of their wider range of novel phenomena and their closer match to the electronic structure of real materials.In this brief review we discuss the physics of three multiband models(the three-band Hubbard,the periodic Anderson,and the Falicov-Kimball models)that was obtained by numerical simulations at zero temperature.We first give heuristic descriptions of the three principal numerical methods(the Lanczos,the density matrix renormalization group,and the constrainedpath Monte Carlo methods).We then present generalized versions of the models and discuss the measurables most often associated with them.Finally,we summarize the results of their ground state numerical studies.While each model was developed to study specific phenomena,unexpected phenomena,usually of a subtle quantum mechanical nature,are often exhibited.Just as often,the predictions of the numerical simulations differ from those of mean-field theories.