Calculating elements of matrix functions using divided differences

Barash, Lev; Güttel, Stefan; Hen, Itay

doi:10.1016/j.cpc.2021.108219

Title: Calculating elements of matrix functions using divided differences

Abstract

In this work, we introduce a method for calculating individual elements of matrix functions. Our technique makes use of a novel series expansion for the action of matrix functions on basis vectors that is memory efficient even for very large matrices. We showcase our approach by calculating the matrix elements of the exponential of a transverse-field Ising model and evaluating quantum transition amplitudes for large many-body Hamiltonians of sizes up to 2⁶⁴ x 2⁶⁴ on a single workstation. We also discuss the application of the method to matrix inverses. We relate and compare our method to the state-of-the-art and demonstrate its advantages. We also discuss practical applications of our method.

Authors:

^[1]; Güttel, Stefan ^[2]; Hen, Itay ^[3]

Russian Academy of Sciences (RAS), Moscow (Russian Federation). L.D. Landau Inst. for Theoretical Physics
Univ. of Manchester (United Kingdom)
Univ. of Southern California, Los Angeles, CA (United States)

Publication Date:: 2021-11-11

Research Org.:: Univ. of Southern California, Los Angeles, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Basic Energy Sciences (BES); Russian Ministry of Science and Higher Education; Engineering and Physical Sciences Research Council (EPSRC)

OSTI Identifier:: 1830658

Grant/Contract Number:: SC0020280; 0029-2019-0003; EP/N510129/1

Resource Type:: Accepted Manuscript

Journal Name:: Computer Physics Communications

Additional Journal Information:: Journal Volume: 271; Journal ID: ISSN 0010-4655

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING

Citation Formats


                    Barash, Lev, Güttel, Stefan, and Hen, Itay. Calculating elements of matrix functions using divided differences.  United States: N. p., 2021. 
Web.  doi:10.1016/j.cpc.2021.108219.

Copy to clipboard


                    Barash, Lev, Güttel, Stefan, & Hen, Itay. Calculating elements of matrix functions using divided differences.  United States.  https://doi.org/10.1016/j.cpc.2021.108219

Copy to clipboard


                    Barash, Lev, Güttel, Stefan, and Hen, Itay. Thu .  
"Calculating elements of matrix functions using divided differences".  United States.  https://doi.org/10.1016/j.cpc.2021.108219.  https://www.osti.gov/servlets/purl/1830658.

Copy to clipboard


                    
@article{osti_1830658,

  title        = {Calculating elements of matrix functions using divided differences},

  author       = {Barash, Lev and Güttel, Stefan and Hen, Itay},

  abstractNote = {In this work, we introduce a method for calculating individual elements of matrix functions. Our technique makes use of a novel series expansion for the action of matrix functions on basis vectors that is memory efficient even for very large matrices. We showcase our approach by calculating the matrix elements of the exponential of a transverse-field Ising model and evaluating quantum transition amplitudes for large many-body Hamiltonians of sizes up to 264 x 264 on a single workstation. We also discuss the application of the method to matrix inverses. We relate and compare our method to the state-of-the-art and demonstrate its advantages. We also discuss practical applications of our method.},

  doi          = {10.1016/j.cpc.2021.108219},

  journal      = {Computer Physics Communications},

  number       = ,

  volume       = 271,

  place        = {United States},

  year         = {2021},

  month        = {11}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1016/j.cpc.2021.108219

Other availability

Search WorldCat to find libraries that may hold this journal

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Off-diagonal series expansion for quantum partition functions
journal, May 2018

Hen, Itay
Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, Issue 5
DOI: 10.1088/1742-5468/aabbe4

Topical Issue Applied and Numerical Linear Algebra (2/2)
journal, November 2020

Güttel, Stefan; Liesen, Jörg
GAMM-Mitteilungen, Vol. 43, Issue 4
DOI: 10.1002/gamm.202000021

Off-diagonal expansion quantum Monte Carlo
journal, December 2017

Albash, Tameem; Wagenbreth, Gene; Hen, Itay
Physical Review E, Vol. 96, Issue 6
DOI: 10.1103/PhysRevE.96.063309

Network Properties Revealed through Matrix Functions
journal, January 2010

Estrada, Ernesto; Higham, Desmond J.
SIAM Review, Vol. 52, Issue 4
DOI: 10.1137/090761070

The Ising model with a transverse field. II. Ground state properties
journal, October 1971

Pfeuty, P.; Elliott, R. J.
Journal of Physics C: Solid State Physics, Vol. 4, Issue 15
DOI: 10.1088/0022-3719/4/15/024

Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection
journal, August 2013

Güttel, Stefan
GAMM-Mitteilungen, Vol. 36, Issue 1
DOI: 10.1002/gamm.201310002

Evaluating Matrix Functions by Resummations on Graphs: The Method of Path-Sums
journal, January 2013

Giscard, P. -L.; Thwaite, S. J.; Jaksch, D.
SIAM Journal on Matrix Analysis and Applications, Vol. 34, Issue 2
DOI: 10.1137/120862880

Ising model in a transverse field. I. Basic theory
journal, August 1973

Stinchcombe, R. B.
Journal of Physics C: Solid State Physics, Vol. 6, Issue 15
DOI: 10.1088/0022-3719/6/15/009

Calculating the divided differences of the exponential function by addition and removal of inputs
journal, September 2020

Gupta, Lalit; Barash, Lev; Hen, Itay
Computer Physics Communications, Vol. 254
DOI: 10.1016/j.cpc.2020.107385

Some divided difference inequalities for n-convex functions
journal, June 1985

Farwig, R.; Zwick, D.
Journal of Mathematical Analysis and Applications, Vol. 108, Issue 2
DOI: 10.1016/0022-247X(85)90036-8

Pracniques: further remarks on reducing truncation errors
journal, January 1965

Kahan, W.
Communications of the ACM, Vol. 8, Issue 1
DOI: 10.1145/363707.363723

The Accuracy of Floating Point Summation
journal, July 1993

Higham, Nicholas J.
SIAM Journal on Scientific Computing, Vol. 14, Issue 4
DOI: 10.1137/0914050

Two polynomial methods of calculating functions of symmetric matrices
journal, January 1989

Druskin, V. L.; Knizhnerman, L. A.
USSR Computational Mathematics and Mathematical Physics, Vol. 29, Issue 6
DOI: 10.1016/S0041-5553(89)80020-5

Similar Records in DOE PAGES and OSTI.GOV collections:

Permutation matrix representation quantum Monte Carlo

Journal Article Gupta, Lalit ; Albash, Tameem ; Hen, Itay - Journal of Statistical Mechanics

We present a quantum Monte Carlo algorithm for the simulation of general quantum and classical many-body models within a single unifying framework. The algorithm builds on a power series expansion of the quantum partition function in its off-diagonal terms and is both parameter-free and Trotter error-free. In our approach, the quantum dimension consists of products of elements of a permutation group. As such, it allows for the study of a very wide variety of models on an equal footing. To demonstrate the utility of our technique, we use it to clarify the emergence of the sign problem in the simulationsmore »« less
Cited by 3
https://doi.org/10.1088/1742-5468/ab9e64

Full Text Available
An integral-free representation of the Dyson series using divided differences

Journal Article Kalev, Amir ; Hen, Itay - New Journal of Physics

The Dyson series is an infinite sum of multi-dimensional time-ordered integrals, which serves as a formal representation of the quantum time-evolution operator in the interaction-picture. Using the mathematical tool of divided differences, we introduce an alternative representation for the series that is entirely free from both time ordering and integrals. In this new formalism, the Dyson expansion is given as a sum of efficiently-computable divided differences of the exponential function, considerably simplifying the calculation of the Dyson expansion terms, while also allowing for time-dependent perturbation calculations to be performed directly in the Schrödinger-picture. Here, we showcase the utility of thismore »« less
https://doi.org/10.1088/1367-2630/ac2dae
Acceleration of Commercial Low-Rank Matrix Solver via SLATE

Technical Report Faircloth, Daniel L.

Performing accurate simulations of large- and multi-scale electromagnetics problems has far-reaching implications. The same physics governs applications as diverse as rare earth content reduction in DC motor design; wireless propagation in dense urban environments and novel antenna design, both key issues in the pending 5G communications boom; electromagnetic sounding for subsurface exo-terrestrial interrogation; burgeoning RF-based medical imaging modalities; and exploration of intra- and inter-cellular quasistatic and RF signaling. With such diversity of simulation problems, a scalable, high performance computational electromagnetics (CEM) software package will provide state-of-the-art design and analysis capability to commercial, medical, and government users. IERUS Technologies has developedmore »« less
Calculating the divided differences of the exponential function by addition and removal of inputs

Journal Article Gupta, Lalit ; Barash, Lev ; Hen, Itay - Computer Physics Communications

We introduce a method for calculating the divided differences of the exponential function by means of addition and removal of items from the input list to the function. Our technique exploits a new identity related to divided differences recently derived by F. Zivcovich. We show that upon adding an item to or removing an item from the input list of an already evaluated exponential, the re-evaluation of the divided differences can be done with only O(sn) floating point operations and O(sn) bytes of memory, where [z₀,...,z_n] are the inputs and s ∝ max_i,j |z_i – z_j |. We demonstrate ourmore »« less
Cited by 5
https://doi.org/10.1016/j.cpc.2020.107385

Full Text Available
THE DENSITY MATRIX IN MANY-ELECTRON QUANTUM MECHANICS. I. GENERALIZED PRODUCT FUNCTIONS. FACTORIZATION AND PHYSICAL INTERPRETATION OF THE DENSITY MATRICES

Journal Article McWeeny, R - Proc. Roy. Soc. (London)

Many-electron wave functions are usually constructed from antisymmetrized products of one-electron orbitals (determinants) and energy calculations are based on the matrix element expressions due to Slater (1931). The orbitals in such a product are replaced by "group functions," each describing any number of electrons, and the necessary generalization of Slater's results is carried out. It is first necessary to develop the density matrix theory of N- particle systems and to show that for systems described by "generalized product functions" the density matrices of the whole system may be expressed in terms of those of the component electron groups. The matrixmore »« less
https://doi.org/10.1098/rspa.1959.0191

Similar Records

Title: Calculating elements of matrix functions using divided differences

Abstract

Citation Formats

Off-diagonal series expansion for quantum partition functions journal, May 2018

Topical Issue Applied and Numerical Linear Algebra (2/2) journal, November 2020

Off-diagonal expansion quantum Monte Carlo journal, December 2017

Network Properties Revealed through Matrix Functions journal, January 2010

The Ising model with a transverse field. II. Ground state properties journal, October 1971

Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection journal, August 2013

Evaluating Matrix Functions by Resummations on Graphs: The Method of Path-Sums journal, January 2013

Ising model in a transverse field. I. Basic theory journal, August 1973

Calculating the divided differences of the exponential function by addition and removal of inputs journal, September 2020

Some divided difference inequalities for n-convex functions journal, June 1985

Pracniques: further remarks on reducing truncation errors journal, January 1965

The Accuracy of Floating Point Summation journal, July 1993

Two polynomial methods of calculating functions of symmetric matrices journal, January 1989

Off-diagonal series expansion for quantum partition functions
journal, May 2018

Topical Issue Applied and Numerical Linear Algebra (2/2)
journal, November 2020

Off-diagonal expansion quantum Monte Carlo
journal, December 2017

Network Properties Revealed through Matrix Functions
journal, January 2010

The Ising model with a transverse field. II. Ground state properties
journal, October 1971

Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection
journal, August 2013

Evaluating Matrix Functions by Resummations on Graphs: The Method of Path-Sums
journal, January 2013

Ising model in a transverse field. I. Basic theory
journal, August 1973

Calculating the divided differences of the exponential function by addition and removal of inputs
journal, September 2020

Some divided difference inequalities for n-convex functions
journal, June 1985

Pracniques: further remarks on reducing truncation errors
journal, January 1965

The Accuracy of Floating Point Summation
journal, July 1993

Two polynomial methods of calculating functions of symmetric matrices
journal, January 1989