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A collection of functionality around rooted trees to generate order conditions for Runge-Kutta methods in Julia for differential equations and scientific machine learning (SciML)

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RootedTrees

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A collection of functionality around rooted trees to generate order conditions for Runge-Kutta methods in Julia. This package also provides basic functionality for BSeries.jl.

API Documentation

The API of RootedTrees.jl is documented in the following. Additional information on each function is available in their docstrings and in the online documentation.

Construction

RootedTrees are represented using level sequences, i.e., AbstractVectors containing the distances of the nodes from the root, see

  • Beyer, Terry, and Sandra Mitchell Hedetniemi. "Constant time generation of rooted trees". SIAM Journal on Computing 9.4 (1980): 706-712. DOI: 10.1137/0209055

RootedTrees can be constructed from their level sequence using

julia> t = rootedtree([1, 2, 3, 2])
RootedTree{Int64}: [1, 2, 3, 2]

In the notation of Butcher (Numerical Methods for ODEs, 2016), this tree can be written as [[τ] τ] or (τ ∘ τ) ∘ (τ ∘ τ), where is the non-associative Butcher product of RootedTrees, which is also implemented.

To get the representation of a RootedTree introduced by Butcher, use butcher_representation:

julia> t = rootedtree([1, 2, 3, 4, 3, 3, 2, 2, 2, 2, 2])
RootedTree{Int64}: [1, 2, 3, 4, 3, 3, 2, 2, 2, 2, 2]

julia> butcher_representation(t)
"[[[τ]τ²]τ⁵]"

There are also some simple plot recipes for Plots.jl. Thus, you can visualize a rooted tree t using plot(t) when using Plots.

Additionally, there is an un-exported function RootedTrees.latexify that can generate LaTeX code for a rooted tree t based on the LaTeX package forest. The relevant code that needs to be included in the preamble can be obtained from the docstring of RootedTrees.latexify (type ? and RootedTrees.latexify in the Julia REPL). The same format is used when you are using Latexify and their function latexify, see Latexify.jl.

Iteration over RootedTrees

A RootedTreeIterator(order::Integer) can be used to iterate efficiently over all RootedTrees of a given order.

Be careful that the iterator is stateful for efficiency reasons, so you might need to use copy appropriately, e.g.,

julia> map(identity, RootedTreeIterator(4))
4-element Array{RootedTrees.RootedTree{Int64,Array{Int64,1}},1}:
 RootedTree{Int64}: [1, 2, 2, 2]
 RootedTree{Int64}: [1, 2, 2, 2]
 RootedTree{Int64}: [1, 2, 2, 2]
 RootedTree{Int64}: [1, 2, 2, 2]

julia> map(copy, RootedTreeIterator(4))
4-element Array{RootedTrees.RootedTree{Int64,Array{Int64,1}},1}:
 RootedTree{Int64}: [1, 2, 3, 4]
 RootedTree{Int64}: [1, 2, 3, 3]
 RootedTree{Int64}: [1, 2, 3, 2]
 RootedTree{Int64}: [1, 2, 2, 2]

Functions on Trees

The usual functions on RootedTrees are implemented, cf. Butcher (Numerical Methods for ODEs, 2016).

  • order(t::RootedTree): The order of a RootedTree, i.e., the length of its level sequence.
  • σ(t::RootedTree) or symmetry(t): The symmetry σ of a rooted tree, i.e., the order of the group of automorphisms on a particular labelling (of the vertices) of t.
  • γ(t::RootedTree) or density(t): The density γ(t) of a rooted tree, i.e., the product over all vertices of t of the order of the subtree rooted at that vertex.
  • α(t::RootedTree): The number of monotonic labelings of t not equivalent under the symmetry group.
  • β(t::RootedTree): The total number of labelings of t not equivalent under the symmetry group.

Additionally, functions on trees connected to Runge-Kutta methods are implemented.

  • elementary_weight(t, A, b, c): Compute the elementary weight Φ(t) of t::RootedTree for the Butcher coefficients A, b, c of a Runge-Kutta method.
  • derivative_weight(t, A, b, c): Compute the derivative weight (ΦᵢD)(t) of t for the Butcher coefficients A, b, c of a Runge-Kutta method.
  • residual_order_condition(t, A, b, c): The residual of the order condition (Φ(t) - 1/γ(t)) / σ(t) with elementary weight Φ(t), density γ(t), and symmetry σ(t) of the rooted tree t for the Runge-Kutta method with Butcher coefficients A, b, c.

Brief Changelog

  • v2.0: Rooted trees are considered up to isomorphisms introduced by shifting each coefficient of their level sequence by the same number.

Referencing

If you use RootedTrees.jl for your research, please cite the paper

@online{ketcheson2021computing,
  title={Computing with {B}-series},
  author={Ketcheson, David I and Ranocha, Hendrik},
  year={2021},
  month={11},
  eprint={2111.11680},
  eprinttype={arXiv},
  eprintclass={math.NA}
}

In addition, you can also refer to RootedTrees.jl directly as

@misc{ranocha2019rootedtrees,
  title={{RootedTrees.jl}: {A} collection of functionality around rooted trees
         to generate order conditions for {R}unge-{K}utta methods in {J}ulia
         for differential equations and scientific machine learning ({SciM}L)},
  author={Ranocha, Hendrik and contributors},
  year={2019},
  month={05},
  howpublished={\url{https://github.com/SciML/RootedTrees.jl}},
  doi={10.5281/zenodo.5534590}
}

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A collection of functionality around rooted trees to generate order conditions for Runge-Kutta methods in Julia for differential equations and scientific machine learning (SciML)

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