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Stability of the Lanczos Method for Matrix Function Approximation [article]

Cameron Musco, Christopher Musco, Aaron Sidford
2017 arXiv   pre-print
The ubiquitous Lanczos method can approximate f(A)x for any symmetric n × n matrix A, vector x, and function f.  ...  Our proof extends work of Druskin and Knizhnerman [DK91], leveraging the stability of the classic Chebyshev recurrence to bound the stability of any polynomial approximating f(x).  ...  We would also like to thank Michael Cohen for the initial idea behind the lower bound proof and Jon Kelner and Richard Peng for a number of helpful conversations.  ... 
arXiv:1708.07788v1 fatcat:pr43viwpwbhr7neo5odrcsfs3e

Stability of the Lanczos Method for Matrix Function Approximation [chapter]

Cameron Musco, Christopher Musco, Aaron Sidford
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
Theoretically elegant and ubiquitous in practice, the Lanczos method can approximate f (A)x for any symmetric matrix A ∈ R n×n , vector x ∈ R n , and function f .  ...  Our proof extends work of Druskin and Knizhnerman [11] , leveraging the stability of the classic Chebyshev recurrence to bound the stability of any polynomial approximating f (x).  ...  We would also like to thank Michael Cohen for the initial idea behind the lower bound proof and Jon Kelner and Richard Peng for a number of helpful conversations.  ... 
doi:10.1137/1.9781611975031.105 dblp:conf/soda/MuscoMS18 fatcat:kfbtdq5aprhihgi3stwnvzjom4

Efficient computation of electromagnetic wave fields on unbounded domains using stability-corrected wave functions and Krylov subspace projection methods

V. Druskin, R. F. Remis, M. Zaslavsky, J. T. Zimmerling
2015 2015 International Conference on Electromagnetics in Advanced Applications (ICEAA)  
We show that dominant open resonant modes can be determined via the Lanczos algorithm and illustrate the performance of our technique through a number of numerical examples for two-and three-dimensional  ...  The extension to infinity is modeled via an optimal complex scaling method and we show that stable time-domain reduced-order models can be efficiently computed via a stability-correction procedure in conjunction  ...  Furthermore, since the stability-corrected wave function is a nonentire function of the Maxwell system matrix, we expect that rational Krylov methods may converge much faster than standard Krylov methods  ... 
doi:10.1109/iceaa.2015.7297066 fatcat:5pxvqjfc2zgahgvvqv2u3iouyu

Numerical stability of Lanczos methods

Eamonn Cahill, Alan Irving, Christopher Johnston, James Sexton
2000 Nuclear Physics B - Proceedings Supplements  
The Lanczos algorithm for matrix tridiagonalisation suffers from strong numerical instability in finite precision arithmetic when applied to evaluate matrix eigenvalues.  ...  A recent application of the Lanczos algorithm proposed by Bai, Fahey and Golub allows quadrature evaluation of inner products of the form ψ^† g(A) ψ.  ...  A random Gaussian fermion vector, ψ, was generated, and the BFG method applied to evaluate ψ † · log(M † κ M κ ) · ψ for Wilson hopping parameter κ = .1650 as a function of the order m of the Lanczos tridiagonal  ... 
doi:10.1016/s0920-5632(00)91816-4 fatcat:7pl42godtnhndkit2o7dzu7aji

The FDTD method and its relation to Fibonacci polynomials

R. F. Remis
2011 2011 International Conference on Electromagnetics in Advanced Applications  
In addition, we compare FDTD with the Spectral Lanczos Decomposition method (SLDM) and show that to capture the evolution of the fields in time, SLDM adjust itself to the spectrum of the system matrix,  ...  In this paper we show that the Finite-Difference Time-Domain method (FDTD method) follows the recurrence relation for Fibonacci polynomials.  ...  Acknowledgments This work was financially supported by the Dutch Technology Foundation (STW) through the MICES project. This support is gratefully acknowledged.  ... 
doi:10.1109/iceaa.2011.6046303 fatcat:m2jnl7blxbgh5ikyefcu3x2b4u

Partition of unity interpolation using stable kernel-based techniques

R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione, G. Santin
2017 Applied Numerical Mathematics  
The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights.  ...  Such technique, taking advantage of the local scheme, leads to a significant benefit in terms of stability, especially for flat kernels.  ...  The second and fifth authors are partially supported by the funds of the University of Padova, project CPDA124755 "Multivariate approximation with application to image reconstruction".  ... 
doi:10.1016/j.apnum.2016.07.005 fatcat:vxdlszjkkrcxznvpolzbrh57vi

A modified Lanczos algorithm for the computation of transient electromagnetic wavefields

R.F. Remis, P.M. van den Berg
1997 IEEE transactions on microwave theory and techniques  
Some theoretical aspects of the method are highlighted and numerical results showing the performance of the method for two-dimensional (2-D) configurations are given.  ...  The method utilizes a modified Lanczos scheme, where a socalled reduced model is constructed. A discretization of the time variable is then superfluous.  ...  A disadvantage of the FDTD method is that the time step is limited by the Courant-Friedrichs-Lewy stability condition [1] .  ... 
doi:10.1109/22.643751 fatcat:yqqgkfahuzatniptrudegmnlmu

Numerical Stability of Lanczos Methods

E Cahill
2000 Nuclear Physics B - Proceedings Supplements  
The Lanczos algorithm for matrix tridiagonalisation suffers from strong numerical instability in finite precision arithmetic when applied to evaluate matrix eigenvalues.  ...  A recent application of the Lanczos algorithm proposed by Bai, Fahey and Golub allows quadrature evaluation of inner products of the form ψ † · g(A) · ψ.  ...  A random Gaussian fermion vector, ψ, was generated, and the BFG method applied to evaluate ψ † · log(M † κ M κ ) · ψ for Wilson hopping parameter κ = .1650 as a function of the order m of the Lanczos tridiagonal  ... 
doi:10.1016/s0920-5632(00)00436-9 fatcat:x642khr3zvhjrilo6jwfhhdk5q

Krylov subspace methods for the Dirac equation

Randolf Beerwerth, Heiko Bauke
2015 Computer Physics Communications  
As the Lanczos algorithm requires only matrix-vector products and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations.  ...  The unboundedness of the Dirac Hamiltonian does not hinder the applicability of the Lanczos algorithm.  ...  In our numerical test we employed the pseudospectral method with a basis set of 512 basis functions for the spatial discretization of the wave function.  ... 
doi:10.1016/j.cpc.2014.11.008 fatcat:4smnocnwrve75fsz5dcyh6qxla

Fast Estimation of $tr(f(A))$ via Stochastic Lanczos Quadrature

Shashanka Ubaru, Jie Chen, Yousef Saad
2017 SIAM Journal on Matrix Analysis and Applications  
In addition, we present error bounds for other useful tools such as approximating the log-likelihood function in the context of maximum likelihood estimation of Gaussian processes.  ...  We establish multiplicative and additive error bounds for the approximations obtained by this method.  ...  This paper is a study of the method we call the Stochastic Lanczos Quadrature (SLQ) for approximating the trace of functions of large matrices [6, 7, 20] . The method combines three key ingredients.  ... 
doi:10.1137/16m1104974 fatcat:5uvmhc5gevfrjb3inyzluqukxm

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Qianqian Yang, Ian Turner, Timothy Moroney, Fawang Liu
2014 Applied Mathematical Modelling  
The computational heart of this approach is the efficient computation of a matrixfunction-vector product f (A)b, where A is the matrix representation of the Laplacian obtained from the finite volume method  ...  A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation.  ...  Preconditioned Lanczos method The standard Lanczos approximation to the matrix-function-vector product f (A)b, for symmetric A is (see, for example, van der Vorst [30] ): f (A)b ≈ ||b||V m f (T m )e 1  ... 
doi:10.1016/j.apm.2014.02.005 fatcat:dydqjt2i5bamrp4npago4kgccu

A Krylov Stability-Corrected Coordinate-Stretching Method to Simulate Wave Propagation in Unbounded Domains [article]

Vladimir Druskin, Rob Remis
2012 arXiv   pre-print
Pure imaginary stretching functions based on Zolotarev's optimal rational approximation of the square root are implemented leading to perfectly matched layers with a controlled accuracy over a complete  ...  A new Krylov-based solution method via stability-corrected operator exponents is presented which allows us to construct reduced-order models (ROMs) that respect the delicate spectral properties of the  ...  , and preliminary calculations that verified some of the results of this work.  ... 
arXiv:1202.0424v3 fatcat:wpjvyakypve3lgqp2yvaprfkqy

Error bounded Padé approximation via bilinear conformal transformation

Chung-Ping Chen, D. F. Wong
1999 Proceedings of the 36th ACM/IEEE conference on Design automation conference - DAC '99  
All of the existing methods approximate the transfer function directly in the frequency domain and hence can not provide error bounds in the time domain.  ...  Although the stability and precision of model reduction methods have been greatly improved, the following important question has not been answered: "What is the error bound in the time domain?".  ...  H q is a Heisenberg matrix for the Arnoldi algorithm and is a tridiagonal matrix for the Lanczos algorithm.  ... 
doi:10.1145/309847.309850 dblp:conf/dac/ChenW99 fatcat:l45edsgezjfyta7wb2rwnbcxzu

A partial Padé-via-Lanczos method for reduced-order modeling

Zhaojun Bai, Roland W. Freund
2001 Linear Algebra and its Applications  
The classical Lanczos process can be used to efficiently generate Padé approximants of the transfer function of a given single-input single-output time-invariant linear dynamical system.  ...  We present an algorithm for computing partial Padé approximants via suitable rank-1 updates of the tridiagonal matrices generated by the Lanczos process.  ...  The authors are grateful to the referees and the editor for their constructive comments that helped us to improve the presentation of the paper.  ... 
doi:10.1016/s0024-3795(00)00291-3 fatcat:pffwzs5nl5hv7hy6gmb32jucoa

An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem

Peter Benner, Heike Faβbender
1997 Linear Algebra and its Applications  
An implicitly restarted symplectic Lanczos method for the Hamihonian eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis.  ...  The inherent numerical difficulties of the symplectic Lanczos method are addressed by inexpensive implicit restarts.  ...  NUMERICAL PROPERTIES OF THE IMPLICITLY RESTARTED SYMPLECTIC LANCZOS METHOD Stability Issues It is well known that for general Lanczos-like methods the stability of the overall process is improved when  ... 
doi:10.1016/s0024-3795(96)00524-1 fatcat:fg4c53x2ova37ddw5soqikeevi
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