ExponentialUtilities.jl: High-Performance Matrix Exponentiation and Products

ExponentialUtilities is a package of utility functions for matrix functions of exponential type, including functionality for the matrix exponential and phi-functions. The tools are used by the exponential integrators in OrdinaryDiffEq.

Installation

To install ExponentialUtilities.jl, use the Julia package manager:

using Pkg
Pkg.add("ExponentialUtilities")

Example

using ExponentialUtilities

A = rand(2, 2)
exponential!(A)

v = rand(2);
t = rand();
expv(t, A, v)

Matrix-phi-vector product

The main functionality of ExponentialUtilities is the computation of matrix-phi-vector products. The phi functions are defined as

ϕ_0(z) = exp(z)
ϕ_(k+1)(z) = (ϕ_k(z) - 1) / z

In exponential algorithms, products in the form of ϕ_m(tA)b are frequently encountered. Instead of computing the matrix function first and then computing the matrix-vector product, the common alternative is to construct a Krylov subspaceK_m(A,b) and then approximate the matrix-phi-vector product.

`expv` and `phiv`

expv(t,A,b;kwargs) -> exp(tA)b
phiv(t,A,b,k;kwargs) -> [ϕ_0(tA)b ϕ_1(tA)b ... ϕ_k(tA)b][, errest]

For phiv, allϕ_m(tA)b products up to order k are returned as a matrix. This is because it's more economical to produce all the results at once than individually. A second output is returned if errest=true in kwargs. The error estimate is given for the second-to-last product, using the last product as an estimator. If correct=true, then ϕ_0 through ϕ_(k-1) are updated using the last Arnoldi vector. The correction algorithm is described in [1].

You can adjust how the Krylov subspace is constructed by setting various keyword arguments. See the Arnoldi iteration section for more details.

Contributing

Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to SciML.
See the SciML Style Guide for common coding practices and other style decisions.
There are a few community forums:
- The #diffeq-bridged and #sciml-bridged channels in the Julia Slack
- The #diffeq-bridged and #sciml-bridged channels in the Julia Zulip
- On the Julia Discourse forums
- See also SciML Community page

Reproducibility

The documentation of this SciML package was built using these direct dependencies,

Status `~/work/ExponentialUtilities.jl/ExponentialUtilities.jl/docs/Project.toml`
  [e30172f5] Documenter v1.14.1
  [d4d017d3] ExponentialUtilities v1.27.0 `~/work/ExponentialUtilities.jl/ExponentialUtilities.jl`

and using this machine and Julia version.

Julia Version 1.11.6
Commit 9615af0f269 (2025-07-09 12:58 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)

A more complete overview of all dependencies and their versions is also provided.

Status `~/work/ExponentialUtilities.jl/ExponentialUtilities.jl/docs/Manifest.toml`
  [a4c015fc] ANSIColoredPrinters v0.0.1
  [1520ce14] AbstractTrees v0.4.5
  [79e6a3ab] Adapt v4.3.0
  [4fba245c] ArrayInterface v7.19.0
  [944b1d66] CodecZlib v0.7.8
  [ffbed154] DocStringExtensions v0.9.5
  [e30172f5] Documenter v1.14.1
  [d4d017d3] ExponentialUtilities v1.27.0 `~/work/ExponentialUtilities.jl/ExponentialUtilities.jl`
  [46192b85] GPUArraysCore v0.2.0
  [c145ed77] GenericSchur v0.5.5
  [d7ba0133] Git v1.4.0
  [b5f81e59] IOCapture v0.2.5
  [692b3bcd] JLLWrappers v1.7.1
  [682c06a0] JSON v0.21.4
  [0e77f7df] LazilyInitializedFields v1.3.0
  [d0879d2d] MarkdownAST v0.1.2
  [69de0a69] Parsers v2.8.3
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.4.3
  [2792f1a3] RegistryInstances v0.1.0
  [ae029012] Requires v1.3.1
  [3bb67fe8] TranscodingStreams v0.11.3
  [2e619515] Expat_jll v2.6.5+0
  [f8c6e375] Git_jll v2.50.1+0
  [94ce4f54] Libiconv_jll v1.18.0+0
  [9bd350c2] OpenSSH_jll v10.0.1+0
  [458c3c95] OpenSSL_jll v3.5.1+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.