drudge

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Drudge

Drudge is a symbolic algebra system built on top of the SymPy_ library with its primary focus on tensorial and non-commutative algebras. It is motivated by complex symbolic problems in quantum chemistry and many-body theory, but it could be useful for any tedious and error-prone symbolic manipulation and simplification in any problem with indexed quantities, symbolic summations, and non-commutative algebra.

Built on the generic algorithm for the canonicalization of combinatorial objects, like strings and graphs, in libcanon_, drudge is able to find a canonical form for mathematical expressions with tensors with symmetries and symbolic summations. For instance, for a 4th-order tensor u with symmetry

.. math::

u_{abcd} = -u_{bacd} = -u_{abdc} = u_{badc}

expression like

.. math::

\sum_{cd} u_{acbd} \rho_{dc} - \sum_{cd} u_{cabd} \rho_{dc}
+ \sum_{cd} u_{cdbc} \rho_{cd}

can be automatically simplified into a single term like,

.. math::

3 \sum_{cd} u_{acbd} \rho_{dc}

despite the initial different placement of the indices to the symmetric u tensor and different naming of the dummy indices for summations.

In addition to the full consideration of the combinatorial properties of symmetric tensors and summations during the simplification, drudge also offers a general system for handling non-commutative algebraic systems. Currently, drudge directly supports the CCR and CAR algebra_ for treating fermions and bosons in many-body theory, general Clifford algebras, and su(2) algebra in its Cartan-Killing basis. Other non-commutative algebraic systems should be able to be added with ease.

Based on the symbolic results from drudge, a companion package gristmill_ is able to automatically optimize and generate numerical code. For computations with heavy dependence on tensor contraction and sums of tensor contractions, substantial optimization could be given.

Drudge is developed by Jinmo Zhao and Prof Gustavo E Scuseria at Rice University, and was supported as part of the Center for the Computational Design of Functional Layered Materials, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award DE-SC0012575.

.. _SymPy: http://www.sympy.org .. _libcanon: https://github.com/tschijnmo/libcanon .. _CCR and CAR algebra: https://en.wikipedia.org/wiki/CCR_and_CAR_algebras .. _Clifford algebras: https://en.wikipedia.org/wiki/Clifford_algebra .. _su(2) algebra: https://en.m.wikipedia.org/wiki/Special_unitary_group#Lie_Algebra .. _gristmill: https://github.com/tschijnmo/gristmill