kauri 2.1.1

Hopf algebras, B-series, and Runge-Kutta methods on rooted trees

Kauri is a Python package for symbolic and algebraic manipulation of rooted trees, with applications to B-series, Runge-Kutta methods, and backward error analysis. It implements multiple Hopf algebraic structures on both non-planar and planar rooted trees, and provides tools for symbolic computation, visualization, and numerical integration.

Installation

pip install kauri

The base install is lightweight (pure Python, no external dependencies) and provides tree algebra, enumeration, indexing, and Hopf algebraic operations.

For visualization, Runge-Kutta analysis, and B-series (requires matplotlib, numpy, scipy, sympy, tqdm):

pip install kauri[full]

Documentation

Full documentation is available at https://kauri.readthedocs.io

Features

Hopf algebras

Algebra	Non-planar	Planar
Butcher-Connes-Kreimer (BCK)	kauri.bck	kauri.nck
Grossman-Larson (GL)	kauri.gl	kauri.pgl
Calaque-Ebrahimi-Fard-Manchon (CEM)	kauri.cem	--
Munthe-Kaas-Wright (MKW)	--	kauri.mkw

Each algebra provides: coproduct, counit, antipode, map_product, map_power. Additionally, kauri.gl and kauri.pgl provide product, and kauri.mkw provides shuffle_product.

Tree types

Non-planar (commutative)	Planar (noncommutative)
Tree	PlanarTree
Forest	OrderedForest
ForestSum	ForestSum
TensorProductSum	TensorProductSum

Additional modules

• B-series (BSeries) -- symbolic and numerical manipulation of truncated B-series
• Runge-Kutta methods (RK) -- 15+ predefined methods with order verification, composition, and numerical integration
• Commutator-free methods (CFMethod) -- Lie group integrators with planar order theory
• Odd-even decomposition (oddeven, planar_oddeven) -- symmetric splitting of characters
• Map algebra (Map) -- BCK/CEM convolution products, composition, exp/log
• Tree generation -- enumeration of non-planar, planar, and coloured trees
• SVG display -- inline visualization in Jupyter notebooks

Examples

BCK coproduct

import kauri as kr
import kauri.bck as bck

t = kr.Tree([[], [[]]])
cp = bck.coproduct(t)
kr.display(cp)

Labelled BCK antipode

t = kr.Tree([[[3],2],[1],0])
s = bck.antipode(t)
kr.display(s)

Grossman-Larson coproduct

import kauri.gl as gl

t = kr.Tree([[], [[]]])
cp = gl.coproduct(t)
kr.display(cp)

CEM coproduct

import kauri.cem as cem

t = kr.Tree([[], [[]]])
cp = cem.coproduct(t)
kr.display(cp)

NCK coproduct

import kauri.nck as nck

pt = kr.PlanarTree([[], [[]]])
cp = nck.coproduct(pt)
kr.display(cp)

PGL product

import kauri.pgl as pgl

t1 = kr.PlanarTree([[]])
t2 = kr.PlanarTree([[], []])
p = pgl.product(t1, t2)
kr.display(p)

MKW coproduct

import kauri.mkw as mkw

pt = kr.PlanarTree([[], [[]]])
cp = mkw.coproduct(pt)
kr.display(cp)

Trees of order 4

for t in kr.trees_of_order(4):
    kr.display(t)

Runge-Kutta order conditions

t = kr.Tree([[],[]])
print(kr.rk_order_cond(t, s=3, explicit=True))

a10**2*b1 + b2*(a20 + a21)**2 - 1/3

Truncated B-series of RK4

import sympy as sp

y1 = sp.symbols('y1')
y = sp.Matrix([y1])
f = sp.Matrix([y1 ** 2])

m = kr.rk4.elementary_weights_map()
bs = kr.BSeries(y, f, weights=m, order=5)
print(bs.series())

Odd-even decomposition

import kauri.oddeven as oddeven

# The square root of the identity in BCK convolution
sqrt_id = oddeven.id_sqrt

# Verify: sqrt_id * sqrt_id == identity
t = kr.Tree([[],[]])
print((sqrt_id * sqrt_id)(t))  # Same as kr.ident(t)

Modified equations and preprocessing

# Modified equation of a B-series method
phi = kr.rk4.elementary_weights_map()
mod_eq = phi.modified_equation()

# Preprocessed integrator
preprocessed = phi.preprocessed_integrator()

Citation

If you found this library useful in your research, please consider citing:

@misc{shmelev2025ees,
  title={Explicit and Effectively Symmetric Runge-Kutta Methods},
  author={Shmelev, Daniil and Ebrahimi-Fard, Kurusch and Tapia, Nikolas and Salvi, Cristopher},
  journal={arXiv:2507.21006},
  year={2025}
}