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taylorexponentialmatrix.jl

Computing the matrix exponential with an optimized Taylor polynomial approximation

https://github.com/pnavaro/taylorexponentialmatrix.jl

Science Score: 18.0%

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Keywords

julia-language matrix-exponential matrix-exponentiation taylor-polynomial
Last synced: 6 months ago · JSON representation ·

Repository

Computing the matrix exponential with an optimized Taylor polynomial approximation

Basic Info
  • Host: GitHub
  • Owner: pnavaro
  • License: mit
  • Language: Julia
  • Default Branch: master
  • Homepage:
  • Size: 42 KB
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  • Stars: 1
  • Watchers: 1
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Topics
julia-language matrix-exponential matrix-exponentiation taylor-polynomial
Created about 6 years ago · Last pushed over 2 years ago
Metadata Files
Readme License Citation

README.md

TaylorExponentialMatrix.jl

This is a julia translation of the matlab code available here.

Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation

Philipp Bader (Departament de Matemàtiques, Universitat Jaume I, Castellón, Spain), Sergio Blanes (Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Spain) and Fernando Casas (IMAC and Departament de Matemàtiques, Universitat Jaume I, Castellón, Spain)

```julia julia> using Pkg

julia> pkg" add https://github.com/pnavaro/TaylorExponentialMatrix.jl"

julia> A = rand(5,5) 5×5 Array{Float64,2}: 0.0224285 0.160116 0.504822 0.370332 0.203693 0.861772 0.156394 0.178399 0.645844 0.229411 0.0630692 0.584537 0.358806 0.763173 0.410573 0.320181 0.391341 0.78607 0.619399 0.055634 0.450914 0.0945151 0.277274 0.0576302 0.560325

julia> exp(A) # version from LinearAlgebra 5×5 Array{Float64,2}: 1.45688 0.636229 1.1295 1.13607 0.591914 1.41956 1.77159 1.2015 1.70089 0.734244 0.918259 1.29852 2.42545 2.00871 1.02066 1.04361 1.20838 1.88584 2.99255 0.655514 0.848529 0.454553 0.838875 0.638862 2.02498

julia> using TaylorExponentialMatrix

julia> expm2(A) # Version using Taylor polynomial aproximation (simple algorithm) 5×5 Array{Float64,2}: 1.45688 0.636229 1.1295 1.13607 0.591914 1.41956 1.77159 1.2015 1.70089 0.734244 0.918259 1.29852 2.42545 2.00871 1.02066 1.04361 1.20838 1.88584 2.99255 0.655514 0.848529 0.454553 0.838875 0.638862 2.02498

julia> expm3(A) # Version using Taylor polynomial aproximation (sophisticated algorithm) 5×5 Array{Float64,2}: 1.45688 0.636229 1.1295 1.13607 0.591914 1.41956 1.77159 1.2015 1.70089 0.734244 0.918259 1.29852 2.42545 2.00871 1.02066 1.04361 1.20838 1.88584 2.99255 0.655514 0.848529 0.454553 0.838875 0.638862 2.02498

```

See also

Owner

  • Name: Pierre Navaro
  • Login: pnavaro
  • Kind: user
  • Location: Rennes
  • Company: CNRS

Scientific Software Engineer, Institut de Recherche Mathématique de Rennes, France.

Citation (CITATION.bib) 

@Article{math7121174,
AUTHOR = {Bader, Philipp and Blanes, Sergio and Casas, Fernando},
TITLE = {Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation},
JOURNAL = {Mathematics},
VOLUME = {7},
YEAR = {2019},
NUMBER = {12},
ARTICLE-NUMBER = {1174},
URL = {https://www.mdpi.com/2227-7390/7/12/1174},
ISSN = {2227-7390},
ABSTRACT = {A new way to compute the Taylor polynomial of a matrix exponential is presented which reduces the number of matrix multiplications in comparison with the de-facto standard Paterson-Stockmeyer method for polynomial evaluation. Combined with the scaling and squaring procedure, this reduction is sufficient to make the Taylor method superior in performance to Padé approximants over a range of values of the matrix norms. An efficient adjustment to make the method robust against overscaling is also introduced. Numerical experiments show the superior performance of our method to have a similar accuracy in comparison with state-of-the-art implementations, and thus, it is especially recommended to be used in conjunction with Lie-group and exponential integrators where preservation of geometric properties is at issue.},
DOI = {10.3390/math7121174}
}

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