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Merge pull request #269 from JuliaReach/schillic/bibliography
Use `DocumenterCitations` for bibliography
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‎docs/Project.toml

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[deps]
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Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
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DocumenterCitations = "daee34ce-89f3-4625-b898-19384cb65244"
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IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253"
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[compat]
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Documenter = "1"
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DocumenterCitations = "1.3"
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IntervalArithmetic = "0.21, =0.21" # v0.22 changed the output format

‎docs/make.jl

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using Documenter, IntervalMatrices
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using Documenter, IntervalMatrices, DocumenterCitations
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DocMeta.setdocmeta!(IntervalMatrices, :DocTestSetup, :(using IntervalMatrices);
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recursive=true)
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bib = CitationBibliography(joinpath(@__DIR__, "src", "refs.bib"); style=:alpha)
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makedocs(; sitename="IntervalMatrices.jl",
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modules=[IntervalMatrices],
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format=Documenter.HTML(; prettyurls=get(ENV, "CI", nothing) == "true",
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assets=["assets/aligned.css"], sidebar_sitename=true),
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assets=["assets/aligned.css", "assets/citations.css"],
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sidebar_sitename=true),
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pagesonly=true,
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plugins=[bib],
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pages=["Home" => "index.md",
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"Library" => Any["Types" => "lib/types.md",
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"Methods" => "lib/methods.md"],
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"About" => "about.md",
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"References" => "references.md"])
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"Bibliography" => "bibliography.md"])
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deploydocs(; repo="github.com/JuliaReach/IntervalMatrices.jl.git",
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push_preview=true)

‎docs/src/assets/citations.css

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.citation dl {
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display: grid;
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grid-template-columns: max-content auto; }
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.citation dt {
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grid-column-start: 1; }
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.citation dd {
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grid-column-start: 2;
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margin-bottom: 0.75em; }
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.citation ul {
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padding: 0 0 2.25em 0;
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margin: 0;
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list-style: none !important;}
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.citation ul li {
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text-indent: -2.25em;
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margin: 0.33em 0.5em 0.5em 2.25em;}
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.citation ol li {
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padding-left:0.75em;}

‎docs/src/bibliography.md

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# Bibliography
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```@bibliography
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```

‎docs/src/index.md

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@@ -19,7 +19,7 @@ See the section [Library Outline](@ref) below for details.
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An application of interval matrices is to find the set of states reachable by
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a dynamical system whose coefficients are uncertain. The library
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[ReachabilityAnalysis.jl](http://github.com/JuliaReach/ReachabilityAnalysis.jl)
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implements algorithms that use interval matrices [^3] [^4].
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implements algorithms that use interval matrices [AlthoffKS11, Liou66](@cite).
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## Installing
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@@ -100,9 +100,10 @@ relevant definitions and algebraic rules that apply.
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However, the straightforward application of the rules of interval arithmetic does
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not always give the exact result: in general it only gives an overapproximation
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[^1] [^2]. To illustrate, suppose that we are interested in the quadratic term
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``At + \frac{1}{2}A^2 t^2``, which corresponds to the Taylor-series expansion at order two of
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``e^{At} - I``. Then, at ``t = 1.0``,
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[AlthoffSB08, KoshelevaKMN05](@cite). To illustrate, suppose that we
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are interested in the quadratic term ``At + \frac{1}{2}A^2 t^2``, which
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corresponds to the Taylor-series expansion at order two of ``e^{At} - I``.
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Then, at ``t = 1.0``,
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```jldoctest quickstart
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julia> A + 1/2 * A^2
@@ -138,20 +139,3 @@ Pages = [
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]
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Depth = 2
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```
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## References
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[^1]: Althoff, Matthias, Olaf Stursberg, and Martin Buss. *Reachability analysis
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of nonlinear systems with uncertain parameters using conservative linearization.*
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2008 47th IEEE Conference on Decision and Control. IEEE, 2008.
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[^2]: Kosheleva, Olga, et al. *Computing the cube of an interval matrix is NP-hard.*
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Proceedings of the 2005 ACM symposium on Applied computing. ACM, 2005.
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[^3]: Althoff, Matthias, Bruce H. Krogh, and Olaf Stursberg. *Analyzing reachability
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of linear dynamic systems with parametric uncertainties.*
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Modeling, Design, and Simulation of Systems with Uncertainties.
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Springer, Berlin, Heidelberg, 2011. 69-94.
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[^4]: Liou, M. L. *A novel method of evaluating transient response.*
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Proceedings of the IEEE 54.1 (1966): 20-23.

‎docs/src/references.md

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‎docs/src/refs.bib

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@article{Liou66,
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title={A novel method of evaluating transient response},
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author={Liou, ML},
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journal={Proceedings of the {IEEE}},
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volume={54},
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number={1},
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pages={20--23},
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year={1966},
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publisher={IEEE}
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}
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@article{Rump99,
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title={Fast and parallel interval arithmetic},
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author={Rump, Siegfried M},
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journal={{BIT} Numerical Mathematics},
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volume={39},
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number={3},
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pages={534--554},
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year={1999},
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publisher={Springer},
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url={https://www.tuhh.de/ti3/paper/rump/Ru99b.pdf}
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}
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@inproceedings{KoshelevaKMN05,
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author = {Olga Kosheleva and
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Vladik Kreinovich and
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G{\"{u}}nter Mayer and
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Hung T. Nguyen},
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editor = {Hisham Haddad and
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Lorie M. Liebrock and
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Andrea Omicini and
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Roger L. Wainwright},
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title = {Computing the cube of an interval matrix is NP-Hard},
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booktitle = {Symposium on Applied Computing ({SAC})},
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pages = {1449--1453},
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publisher = {{ACM}},
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year = {2005},
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url = {https://doi.org/10.1145/1066677.1067007},
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doi = {10.1145/1066677.1067007}
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}
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@inproceedings{AlthoffSB07,
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author = {Matthias Althoff and
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Olaf Stursberg and
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Martin Buss},
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title = {Reachability analysis of linear systems with uncertain parameters
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and inputs},
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booktitle = {Conference on Decision and Control ({CDC})},
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pages = {726--732},
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publisher = {{IEEE}},
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year = {2007},
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url = {https://doi.org/10.1109/CDC.2007.4434084},
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doi = {10.1109/CDC.2007.4434084}
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}
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@inproceedings{AlthoffSB08,
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author = {Matthias Althoff and
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Olaf Stursberg and
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Martin Buss},
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title = {Reachability analysis of nonlinear systems with uncertain parameters
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using conservative linearization},
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booktitle = {Conference on Decision and Control ({CDC})},
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pages = {4042--4048},
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publisher = {{IEEE}},
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year = {2008},
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url = {https://doi.org/10.1109/CDC.2008.4738704},
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doi = {10.1109/CDC.2008.4738704}
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}
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@phdthesis{Althoff10,
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title={Reachability analysis and its application to the safety assessment of autonomous cars},
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author={Althoff, Matthias},
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year={2010},
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school={Technische Universit{\"a}t M{\"u}nchen}
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}
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@article{Rump10,
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author = {Siegfried M. Rump},
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title = {Verification methods: Rigorous results using floating-point arithmetic},
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journal = {Acta Numerica},
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volume = {19},
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pages = {287--449},
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year = {2010},
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url = {https://www.tuhh.de/ti3/paper/rump/Ru10.pdf},
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doi = {10.1017/S096249291000005X}
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}
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@article{AlthoffKS11,
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title={Analyzing reachability of linear dynamic systems with parametric uncertainties},
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author={Althoff, Matthias and Krogh, Bruce H and Stursberg, Olaf},
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journal={Modeling, Design, and Simulation of Systems with Uncertainties},
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pages={69--94},
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year={2011},
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publisher={Springer}
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}
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@article{GoldsztejnN14,
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author = {Alexandre Goldsztejn and
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Arnold Neumaier},
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title = {On the Exponentiation of Interval Matrices},
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journal = {Reliable Computing},
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volume = {20},
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pages = {53--72},
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year = {2014},
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url = {http://interval.louisiana.edu/reliable-computing-journal/volume-20/reliable-computing-20-pp-053-072.pdf}
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}

‎src/correction_matrices.jl

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### Algorithm
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See Theorem 3 in [1].
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[1] M. Althoff, O. Stursberg, M. Buss. Reachability Analysis of Linear Systems
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with Uncertain Parameters and Inputs. CDC 2007.
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See [AlthoffSB07; Theorem 3](@citet).
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"""
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function correction_hull(A::IntervalMatrix{T}, t, p) where {T}
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F = _exp_remainder(A, t, p)
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### Algorithm
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See Proposition 3.4 in [1].
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[1] M. Althoff. Reachability analysis and its application to the safety
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assessment of autonomous cars. 2010.
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See [Althoff10; Proposition 3.4](@citet).
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"""
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function input_correction(A::IntervalMatrix{T}, t, p) where {T}
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n = checksquare(A)

‎src/exponential.jl

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@@ -67,8 +67,7 @@ such that `[exp(A*t)]_{ij} ⊆ m_{ij}`.
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### Algorithm
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See Theorem 1 in *Reachability Analysis of Linear Systems with Uncertain
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Parameters and Inputs* by M. Althoff, O. Stursberg, M. Buss.
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See [AlthoffSB07; Theorem 1](@citet).
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"""
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function exp_overapproximation(A::IntervalMatrix{T}, t, p) where {T}
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n = checksquare(A)
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return S + E
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end
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# Implementation of Prop. 1 in Althoff, Matthias, Bruce H. Krogh, and Olaf Stursberg.
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# "Analyzing reachability of linear dynamic systems with parametric uncertainties."
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# Modeling, Design, and Simulation of Systems with Uncertainties. Springer, Berlin, Heidelberg, 2011. 69-94.
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# Implementation of [AlthoffKS11; Prop. 1](@citet).
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function _exp_remainder(A::IntervalMatrix{T}, t, p; n=checksquare(A)) where {T}
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C = max.(abs.(inf(A)), abs.(sup(A)))
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# compute Q = I + Ct + (Ct)^2/2! + ... + (Ct)^p/p!
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# Estimates the sum of the series in the matrix exponential. See Theorem 1
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# in [1] Althoff, Matthias, Olaf Stursberg, and Martin Buss.
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# Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization.
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# 2008 47th IEEE Conference on Decision and Control. IEEE, 2008.
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# Estimates the sum of the series in the matrix exponential. See
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# [AlthoffSB08; Theorem 1](@citet).
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function _exp_remainder_series(A::IntervalMatrix{T}, t, p; n=checksquare(A)) where {T}
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nA = opnorm(A, Inf)
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c = nA * t / (p + 2)
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### Algorithm
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See Theorem 2 in *Reachability Analysis of Linear Systems with Uncertain
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Parameters and Inputs* by M. Althoff, O. Stursberg, M. Buss.
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See [AlthoffSB07; Theorem 2](@citet).
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"""
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function exp_underapproximation(A::IntervalMatrix{T}, t, p) where {T}
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@assert p > 1 "the order $p < 2 is not supported"
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### Algorithm
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This a variation of the algorithm in [1, Section 6]. If ``A = (aᵢⱼ)`` and
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This a variation of the algorithm in [KoshelevaKMN05; Section 6](@citet). If ``A = (aᵢⱼ)`` and
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``B := αA + βA^2 = (bᵢⱼ)``, the idea is to compute each ``bᵢⱼ`` by factoring
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out repeated expressions (thus the term *single-use expressions*).
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```math
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bᵢⱼ = β\\sum_\\{k, k ≠ i, k ≠ j} a_{ik} a_{kj} + (α + βa_{ii} + βa_{jj}) a_{ij}.
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```
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[1] Kosheleva, Kreinovich, Mayer, Nguyen. Computing the cube of an interval
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matrix is NP-hard. SAC 2005.
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"""
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function quadratic_expansion(A::IntervalMatrix, α::Real, β::Real)
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B = similar(A)
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### Algorithm
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We use the algorithm in [1, Section 4.3], which first scales `A` by factor
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We use the algorithm in [GoldsztejnN14; Section 4.3](@citet), which first scales `A` by factor
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``2^{-l}``, computes the matrix exponential for the scaled matrix, and then
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squares the result ``l`` times.
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```math
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\\exp(A * 2^{-l})^{2^l}
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```
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[1] Goldsztejn, Alexandre, Arnold Neumaier. "On the exponentiation of interval
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matrices". Reliable Computing. 2014.
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"""
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function scale_and_square(A::IntervalMatrix{T}, l::Integer, t, p;
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validate::Bool=true) where {T}
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### Algorithm
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We use the algorithm in [1, Section 4.2].
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[1] Goldsztejn, Alexandre, Arnold Neumaier. "On the exponentiation of interval
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matrices". Reliable Computing. 2014.
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We use the algorithm in [GoldsztejnN14; Section 4.2](@citet).
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"""
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function horner(A::IntervalMatrix{T}, K::Integer;
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validate::Bool=true) where {T}
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H = Iₙ + A / i * H
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end
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# remainder; ref [1] uses a less precise computation here
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# remainder; ref [GoldsztejnN14] uses a less precise computation here
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R = _exp_remainder(A, one(T), K)
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return H + R

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