Both methods are fourth-order accurate in time.

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As it is not necessary to use equal spacing between nodes, this node can be chosen to lie between the existing nodes for the given coefficient. With the use of Newton interpolation to construct the approximation of each component from the chosen nodes, the additional node can be accounted for efficiently to obtain the second approximation. The time step can then be adjusted according to the size of the error estimate as in standard adaptive methods. Other spatial discretizations The main idea behind KSS methods, that higher-order accuracy in time can be obtained by compo- nentwise approximation, is not limited to the enhancement of spectral methods that employ Fourier basis functions.

## The Lanczos algorithm – Number Crunch

Then, f. The first expression in 35 can be computed analytically if the members of the basis fuj gN j D1 can be simply expressed in terms of j , as in Fourier spectral methods. The other two expressions in 35 are readily obtained from bases for Krylov subspaces generated by v. If A is sparse, then each block recursion coefficient, across all components, can be represented as the sum of a sparse matrix and a low-rank matrix [27].

## The Lanczos Method: Evolution and Application

Therefore, it is worthwhile to explore the adaptation of KSS methods to other spatial discretiza- tions for which the recursion coefficients can be computed efficiently. The temporal order of accuracy achieved in the case of Fourier spectral methods applies to such discretizations, as only the measures in the Riemann—Stieltjes integrals are changing, not the integrands. This is demon- strated in [27], in which block KSS methods are adapted to spatial discretization via finite elements. Future work will consist of asymptotic analysis of the recursion coefficients arising from these discretizations in order to develop effective algorithms for prescribing quadrature nodes.

Application to nonlinear PDE There are several exponential integrators that can be used for solving stiff systems of nonlinear ODE, such as those proposed in [28, 31—36]. These methods involve the approximation of products of matrix functions and vectors, where the functions include the matrix exponential. A comparison of several of these methods with implicit and explicit integrators is given in [37]. The one-node block KSS method, which is first-order accurate in time for diffusion equations, has already been successfully applied to nonlinear diffusion equations from image processing [29, 30].

To achieve higher-order accuracy, an approach such as that described in [37], in which the KSS method is used to approximate the product of a function of the Jacobian with a vector, will be investigated. Over the last few decades, much advancement has been made in the approximation of quantities of the form f.

Techniques for com- puting f. On the other hand, techniques for computing the bilinear form uT f. Because they allow individual attention to be paid to each compo- nent of f.

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KSS methods represent one approach of exploiting this flexibil- ity, and their use of distinct block Gaussian quadrature rules for each component makes them an attractive choice for various problems in terms of their accuracy. However, it has been established in this paper that the use of block Gaussian rules is neither necessary nor sufficient for maximizing accuracy, and alternative approaches based on prescribing quadrature nodes, whether based on asymptotic block Lanczos iteration or not, cannot only achieve comparable or greater accuracy than block Gaussian rules but can also do so with much greater effi- ciency.

Therefore, it is worthwhile to continue the exploration of such componentwise approaches to the solution of stiff systems of ODE, both linear and nonlinear, through the approximation of matrix functions. Hochbruck M, Lubich C. A Gautschi-type method for oscillatory second-order differential equations. Numerische Mathematik ; — On Krylov subspace approximations to the matrix exponential operator. Moret I, Novati P. RD-rational approximation of the matrix exponential operator. BIT Numerical Mathematics ; — Preconditioning Lanczos approximations to the matrix exponential.

Lambers JV. Derivation of high-order spectral methods for time-dependent PDE using modified moments. Electronic Transactions on Numerical Analysis ; — Enhancement of Krylov subspace spectral methods by block Lanczos iteration. Krylov subspace methods for variable-coefficient initial-boundary value problems. PhD Thesis, Stanford University, Krylov subspace spectral methods for variable-coefficient initial-boundary value problems. Analysis of 1-D wave propagation in inhomogeneous media. Numerical Functional Analysis and Optimization ; — An explicit, stable, high-order spectral method for the wave equation based on block Gaussian quadra- ture.

Implicitly defined high-order operator splittings for parabolic and hyperbolic variable-coefficient PDE using modified moments. International Journal of Computational Science ; — Numerical Algorithms ; — A multigrid block krylov subspace spectral method for variable-coefficient elliptic PDE. Practical implementation of Krylov subspace spectral methods. Journal of Scientific Computing ; — Spectral methods for time-dependent variable-coefficient PDE based on block Gaussian quadrature.

A spectral time-domain method for computational electrodynamics.

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Advances in Applied Mathemat- ics and Mechanics ; 1 6 — Golub GH, Meurant G. Matrices, moments and quadrature. Bounds for the error of linear systems of equations using the theory of moments. Journal of Mathematical Analysis and Applications ; — Golub GH. Some modified matrix eigenvalue problems. SIAM Review ; — Bounds for matrix moments. Rocky Mountain Journal of Mathematics ; — Davis P, Rabinowitz P.

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## Quantum Imaginary Time Evolution, Quantum Lanczos, and Quantum Thermal Averaging

Mathematics of Computation ; — Orthogonal polynomials—constructive theory and applications. Golub GH, Welsch J. Calculation of Gauss quadrature rules. Mathematics of Computation ; — Atkinson K. An Introduction to Numerical Analysis, 2nd Ed. Golub GH, Underwood R.

The block Lanczos method for computing eigenvalues. Academic Press: New York, ; — Solution of time-dependent PDE through component-wise approximation of matrix functions. Exponential integrators for large systems of differential equations. A nonlinear nonlocal diffusion model for color image denoising. In preparation. Guidotti P, Longo K. Two enhanced fourth order diffusion models for image denoising.

Journal of Mathematical Imaging and Vision ; — Caliari M, Ostermann A. Implementation of exponential Rosenbrock-type integrators. Applied Numerical Mathe- matics ; — A method for exponential propagation of large systems of stiff nonlinear differential equations.

Hochbruck M, Ostermann A. Exponential Runge—Kutta methods for parabolic problems. Applied Numerical Mathematics ; — A class of explicit exponential general linear methods. Tokman M. Journal of Computational and Applied Mathematics ; — Loffeld J, Tokman M.

Related Papers. By James Lambers. Download pdf. Remember me on this computer.

According to It then follows from the above statements that the matrix UM is zero everywhere except on the diagonal and the two codiagonals that are the subdiagonals and the superdiagonals whose. Search all titles. Search all titles Search all collections. Your Account Logout. By Dzevad Belkic. Edition 1st Edition. First Published Ncond Type : Int positive integer Description : The number of conduction electrons not used in grand canonical ensemble.

TPQ method The seed of the random generator is given by this parameter and the random vector is used as the initial vector. LanczosTarget Type : Int positive integer Description : An integer giving the target of the eigenvalue for judging the convergence of the Lanczos method. Description : An integer giving the interval steps of calculating the correlation functions. Description : An integer giving the interval steps of output the wave function.