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The Lanczos Approximation for the Γ-Function with Complex Coefficients [article]

William Rea
2020 arXiv   pre-print
We examined the properties of the coefficients of the approximation of the Γ-function with complex values of the free parameter together with the convergence properties of the approximation when using  ...  We report that for fixed real parts of the free parameter that using complex coefficients both increases the computational cost of the Lanczos approximation while drecreasing the accuracy.  ...  Numerical Evaluation It is obvious that using a Lanczos approximation with complex r increases the computational cost of calculating the Γ-function when the same number of coefficients are used.  ... 
arXiv:2005.10449v1 fatcat:o47ztwcthjaovetbpeserdxs7u

The Gamma Function via Interpolation [article]

Matthew F Causley
2021 arXiv   pre-print
A new computational framework for evaluation of the gamma function Γ(z) over the complex plane is developed.  ...  The resulting approximations are competitive with Stirling's formula in terms of overall efficiency.  ...  The author is especially indebted for the suggestion to examine the AAA algorithm.  ... 
arXiv:2104.00697v2 fatcat:jqsl54cvhfhcfi6hdlwotdbqya

Computation of matrix gamma function [article]

Joao R. Cardoso, Amir Sadeghi
2018 arXiv   pre-print
Well-known techniques for the scalar gamma function, such as Lanczos and Spouge methods, are carefully extended to the matrix case.  ...  Bounds for truncation errors and other bounds related with the matrix gamma function will be discussed as well.  ...  If the matrix A has any eigenvalue with non positive real parts, Lanczos formula (4.2) may not be suitable for approximating Γ(A).  ... 
arXiv:1806.10554v1 fatcat:2w7xlxq4zbb63m4jooo7emcefu

Krylov complexity in conformal field theory [article]

Anatoly Dymarsky, Michael Smolkin
2021 arXiv   pre-print
Krylov complexity, or K-complexity for short, has recently emerged as a new probe of chaos in quantum systems.  ...  We find that the bound on OTOC provided by Krylov complexity reduces to bound on chaos of Maldacena, Shenker, and Stanford.  ...  With this we proceed to evaluate the moments (35) and Lanczos coefficients. The latter are shown in Fig. 9 for both approximations superimposed with the asymptotic fit (21) .  ... 
arXiv:2104.09514v1 fatcat:gk7npabpj5corexuvpbikrfube

Random Matrix Theory for Complexity Growth and Black Hole Interiors [article]

Arjun Kar, Lampros Lamprou, Moshe Rozali, James Sully
2021 arXiv   pre-print
Our main tool for establishing this connection is a "complexity renormalization group" framework we develop that allows us to study the effective operator dynamics for different timescales by "integrating  ...  This is a refined, "microcanonical" version of K-complexity that applies to theories with infinite or continuous spectra (including quantum field theories), and in the holographic theories we study exhibits  ...  For a measure with non-compact support and asymptotic fall-off µ(ω) ∼ ω→∞ exp (−|ω/ω 0 | γ ) (A.3) one can show that the Lanczos coefficients grow asymptotically as b n ∼ n→∞ ω 0 n 1/γ .  ... 
arXiv:2106.02046v3 fatcat:hw4cebjnobf7tbhicyncsqpdlq

A Universal Operator Growth Hypothesis [article]

Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, Ehud Altman
2019 arXiv   pre-print
The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's functions grow linearly with rate α in generic systems, with an extra logarithmic correction  ...  Finally we use the hypothesis in conjunction with the recursion method to develop a technique for computing diffusion constants.  ...  (a) The Lanczos coefficients for q = 0.15 are fit to (50) with α = 0.35 and η = 1.74.  ... 
arXiv:1812.08657v4 fatcat:ene7qcmrh5fylfmllman2qoxei

Gauss-Christoffel quadrature for inverse regression: applications to computer experiments [article]

Andrew Glaws, Paul G. Constantine
2017 arXiv   pre-print
Therefore, the proposed approach is most appropriate for deterministic functions with fewer than ten independent inputs.  ...  We call the new algorithms Lanczos-Stieltjes inverse regression (LSIR) and Lanczos-Stieltjes average variance estimation (LSAVE) due to their connection with Stieltjes' method---and Lanczos' related discretization  ...  That is, γ N,k ≈ γ N ≈ γ. (51) By taking more Lanczos iterations, we improve the leftmost approximation, and for k = N , we have γ N,N = γ N .  ... 
arXiv:1710.01372v2 fatcat:pgrloms3j5cqnlccxpcdaoavvy

Lanczos methods for UV-suppressed fermions

A. Boriçi
2003 Nuclear Physics B - Proceedings Supplements  
In this talk I indroduce lattice fermions with suppressed cutoff modes. Then I present Lanczos based methods for stochastic evaluation of the fermion determinant.  ...  I would like to thank Philippe de Forcrand, Alan Irving and Tony Kennedy for discussions related to this talk.  ...  For Wilson (W) and overlap (o) fermions as the input theory one has Γ 5 = γ 5 . For staggered fermions Γ 5 is a diagonal matrix with entries +1/ − 1 on even/odd lattice sites.  ... 
doi:10.1016/s0920-5632(03)01739-0 fatcat:3npmorb235gzpj5vkdfcyb7api

Block Gauss and Anti-Gauss Quadrature with Application to Networks

C. Fenu, D. Martin, L. Reichel, G. Rodriguez
2013 SIAM Journal on Matrix Analysis and Applications  
Approximations of the latter functions are computed by applying a few steps of the nonsymmetric block Lanczos method to A with initial block-vectors V and W .  ...  a few steps of the symmetric block Lanczos method to A with initial block-vector W ∈ R m×k .  ...  The authors would like to thank the referees for comments. LR would like to thank Giuseppe Rodriguez and Sebastiano Seatzu for an enjoyable visit to Cagliari.  ... 
doi:10.1137/120886261 fatcat:maxug7qm4zhftjmfrpnrot7sju

The Lanczos algorithm and complex Gauss quadrature

Stefano Pozza, Miroslav S. Pranić, Zdeněk Strakoš
2018 Electronic Transactions on Numerical Analysis  
Gauss quadrature can be naturally generalized in order to approximate quasi-definite linear functionals, where the interconnections with (formal) orthogonal polynomials, (complex) Jacobi matrices, and  ...  In particular, the existence of the n-weight (complex) Gauss quadrature corresponds to successfully performing the first n steps of the Lanczos algorithm for generating biorthogonal bases of the two associated  ...  We are grateful to Martin Gutknecht and to the anonymous referee of an earlier version of the present work for the many comments and suggestions that has led us to writing this survey.  ... 
doi:10.1553/etna_vol50s1 fatcat:il5nyk6smffwbixadaf7u54t7u

Julia and Mandelbrot Sets of the Gamma Function Using Lanczos Approximation

Richard Eneojo Amobeda
2016 Applied and Computational Mathematics  
To overcome this challenge, the Lanczos approximation of the Gamma function which presents an efficient and easy way to compute algorithms for approximating the Gamma function to an arbitrary precision  ...  Various Julia and Mandelbrot sets associated with the Gamma function are generated using the iterative function f (z) = (z) λ Γ + λ , with different parameter λ − values.  ...  Lanczos Gamma Approximation The Gamma function (z) Γ is defined as z 1 t 0 (z) = t e dt (z) > 0 ∞ − − Γ ℜ ∫ (2) with the fundamental recurrence relation (n 1) = n (n) Γ + Γ (3) The functional relation  ... 
doi:10.11648/j.acm.20160502.16 fatcat:cyzuegeygndjnksz3jea3d5rki

Krylov complexity of many-body localization: Operator localization in Krylov basis [article]

Fabian Ballar Trigueros, Cheng-Ju Lin
2021 arXiv   pre-print
Extrapolating the Lanczos coefficients to the thermodynamic limit, we study the spectral function and also find that the corresponding single-particle problem is localized for both unextrapolated and extrapolated  ...  Lanczos coefficients, resulting in a bounded Krylov complexity in time.  ...  (e) The attempt to fit the low frequency part of the cumulative histogram of Γn in Eq. (13), obtained by taking the differ- spectral function using the nmax = 1000 result with ω γ+1 ence between  ... 
arXiv:2112.04722v1 fatcat:zrt36mxngjbzzkzisy2r2wp22a

Acceleration Techniques for Approximating the Matrix Exponential Operator

M. Popolizio, V. Simoncini
2008 SIAM Journal on Matrix Analysis and Applications  
We show that some recently developed acceleration procedures may be restated as preconditioning techniques for the partial fraction expansion form of an approximating rational function.  ...  Moreover, we provide a performance evaluation among several numerical approaches to approximate the action of the exponential of large matrices.  ...  The authors would like to thank Marlis Hochbruck and Igor Moret for insightful conversations.  ... 
doi:10.1137/060672856 fatcat:mivn6sbzozfhdouo3s3soms2ga

Stopping Criteria for Rational Matrix Functions of Hermitian and Symmetric Matrices

Andreas Frommer, Valeria Simoncini
2008 SIAM Journal on Scientific Computing  
Building upon earlier work by Golub, Meurant, Strakos and Tichy, we derive new a posteriori error bounds for Krylov subspace approximations to f (A)b, the action of a function f of a real symmetric or  ...  Our numerical results include experiments with the matrix exponential, as used in exponential integrators, and with the matrix sign function, as used in lattice QCD simulations, and demonstrate the accuracy  ...  We would like to thank Zdenek Strakoš for his helpful comments on an earlier version of this paper and for suggesting the second matrix in Example 7.2.  ... 
doi:10.1137/070684598 fatcat:tiiaaffj2bcxhhnm7vav4q5ojq

Quantum chaos and the complexity of spread of states [article]

Vijay Balasubramanian, Pawel Caputa, Javier Magan, Qingyue Wu
2022 arXiv   pre-print
We propose a measure of quantum state complexity defined by minimizing the spread of the wave-function over all choices of basis.  ...  Our measure is controlled by the "survival amplitude" for a state to remain unchanged, and can be efficiently computed in theories with discrete spectra.  ...  We are grateful to Alex Streicher for conversations about numerical evaluation of the Lanczos algorithm.  ... 
arXiv:2202.06957v2 fatcat:izptquh2hjex7phnjiryx2dugy
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