Tensor software explained

Tensor software is a class of mathematical software designed for manipulation and calculation with tensors.

Standalone software

• SPLATT^[1] is an open source software package for high-performance sparse tensor factorization. SPLATT ships a stand-alone executable, C/C++ library, and Octave/MATLAB API.
• Cadabra^[2] is a computer algebra system (CAS) designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor polynomial simplification including multi-term symmetries, fermions and anti-commuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many more. The input format is a subset of TeX. Both a command-line and a graphical interface are available.
• Tela^[3] is a software package similar to MATLAB and GNU Octave, but designed specifically for tensors.

Software for use with Mathematica

• Tensor^[4] is a tensor package written for the Mathematica system. It provides many functions relevant for General Relativity calculations in general Riemann–Cartan geometries.
• Ricci^[5] is a system for Mathematica 2.x and later for doing basic tensor analysis, available for free.
• TTC^[6] Tools of Tensor Calculus is a Mathematica package for doing tensor and exterior calculus on differentiable manifolds.
• EDC and RGTC,^[7] "Exterior Differential Calculus" and "Riemannian Geometry & Tensor Calculus," are free Mathematica packages for tensor calculus especially designed but not only for general relativity.
• Tensorial^[8] "Tensorial 4.0" is a general purpose tensor calculus package for Mathematica.
• xAct:^[9] Efficient Tensor Computer Algebra for Mathematica. xAct is a collection of packages for fast manipulation of tensor expressions.
• GREAT^[10] is a free package for Mathematica that computes the Christoffel connection and the basic tensors of General Relativity from a given metric tensor.
• Atlas 2 for Mathematica^[11] is a powerful Mathematica toolbox which allows to do a wide range of modern differential geometry calculations
• GRTensorM^[12] is a computer algebra package for performing calculations in the general area of differential geometry.
• MathGR^[13] is a package to manipulate tensor and GR calculations with either abstract or explicit indices, simplify tensors with permutational symmetries, decompose tensors from abstract indices to partially or completely explicit indices and convert partial derivatives into total derivatives.
• TensoriaCalc^[14] is a tensor calculus package written for Mathematica 9 and higher, aimed at providing user-friendly functionality and a smooth consistency with the Mathematica language itself. As of January 2015, given a metric and the coordinates used, TensoriaCalc can compute Christoffel symbols, the Riemann curvature tensor, and Ricci tensor/scalar; it allows for user-defined tensors and is able to perform basic operations such as taking the covariant derivatives of tensors. TensoriaCalc is continuously under development due to time constraints faced by its developer.
• OGRe^[15] is a modern free and open-source Mathematica package for tensor calculus, released in 2021 for Mathematica 12.0 and later. It is designed to be both powerful and user-friendly, and is especially suitable for general relativity. OGRe allows performing arbitrarily complicated tensor operations, and automatically transforms between index configurations and coordinate systems behind the scenes as needed for each operation.

Software for use with Maple

• GRTensorII^[16] is a computer algebra package for performing calculations in the general area of differential geometry.
• Atlas 2 for Maple^[17] is a modern differential geometry for Maple.
• DifferentialGeometry^[18] is a package which performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, General Relativity, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. It is included with Maple.
• Physics^[19] is a package developed as part of Maple, which implements symbolic computations with most of the objects used in mathematical physics. It includes objects from general relativity (tensors, metrics, covariant derivatives, tetrads etc.), quantum mechanics (Kets, Bras, commutators, noncommutative variables) etc.

Software for use with Matlab

• Tensorlab^[20] is a MATLAB toolbox for multilinear algebra and structured data fusion.
• Tensor Toolbox^[21] Multilinear algebra MATLAB software.
• MPCA and MPCA+LDA^[22] Multilinear subspace learning software: Multilinear principal component analysis.
• UMPCA^[23] Multilinear subspace learning software: Uncorrelated multilinear principal component analysis.
• UMLDA^[24] Multilinear subspace learning software: Uncorrelated multilinear discriminant analysis.

Software for use with Maxima

Maxima^[25] is a free open source general purpose computer algebra system which includes several packages for tensor algebra calculations in its core distribution.It is particularly useful for calculations with abstract tensors, i.e., when one wishes to do calculations without defining all components of the tensor explicitly. It comes with three tensor packages:^[26]

• itensor for abstract (indicial) tensor manipulation,
• ctensor for component-defined tensors, and
• atensor for algebraic tensor manipulation.

Software for use with R

• Tensor is an R package for basic tensor operations.
• rTensor provides several tensor decomposition approaches.
• nnTensor provides several non-negative tensor decomposition approaches.
• ttTensor provides several tensor-train decomposition approaches.
• tensorBF is an R package for Bayesian Tensor decomposition.
• MTF^[27] Bayesian Multi-Tensor Factorization for data fusion and Bayesian versions of Tensor PCA and Tensor CCA. Software: MTF.

Software for use with Python

• TensorLy^[28] provides several tensor decomposition approaches.

Software for use with Julia

• TensorDecompositions.jl provides several tensor decomposition approaches.
• TensorToolbox.jl provides several tensor decomposition approaches. This follows the functionality of MATLAB Tensor toolbox and Hierarchical Tucker Toolbox.
• ITensors.jl^[29] is a library for rapidly creating correct and efficient tensor network algorithms. This is the Julia version of ITensor, not a wrapper around the C++ version but full implementations by Julia language.
• TensorOperations.jl provides fast tensor operations using a convenient Einstein index notation.
• TensorRules.jl provides a macro, which enable us to use automatic differentiation (AD) libraries (e.g., Zygote.jl, ForwardDiff.jl) with @tensor and @tensoropt macros in TensorOperations.jl.

Software for use with SageMath

• SageManifolds: tensor calculus on smooth manifolds; all SageManifolds^[30] code is included in SageMath since version 7.5; it allows for computations in various vector frames and coordinate charts, the manifold not being required to be parallelizable.

Software for use with Java

• ND4J: N-dimensional arrays for the JVM is a Java library for basic tensor operations and scientific computing.
• Tensor: computation for regular or unstructured multi-dimensional tensors. Scalar entries are either in numeric or exact precision. API inspired by Mathematica. Java 8 library in with no external dependencies.

Libraries

• Redberry^[31] is an open source computer algebra system designed for symbolic tensor manipulation. Redberry provides common tools for expression manipulation, generalized on tensorial objects, as well as tensor-specific features: indices symmetries, LaTeX-style input, natural dummy indices handling, multiple index types etc. The HEP package includes tools for Feynman diagrams calculation: Dirac and SU(N) algebra, Levi-Civita simplifications, tools for calculation of one-loop counterterms etc. Redberry is written in Java and provides extensive Groovy-based programming language.
• libxm^[32] is a lightweight distributed-parallel tensor library written in C.
• FTensor^[33] is a high performance tensor library written in C++.
• TL^[34] is a multi-threaded tensor library implemented in C++ used in Dynare++. The library allows for folded/unfolded, dense/sparse tensor representations, general ranks (symmetries). The library implements Faa Di Bruno formula and is adaptive to available memory. Dynare++ is a standalone package solving higher order Taylor approximations to equilibria of non-linear stochastic models with rational expectations.
• vmmlib^[35] is a C++ linear algebra library that supports 3-way tensors, emphasizing computation and manipulation of several tensor decompositions.
• Spartns^[36] is a Sparse Tensor framework for Common Lisp.
• FAstMat^[37] is a thread-safe general tensor algebra library written in C++ and specially designed for FEM/FVM/BEM/FDM element/edge wise computations.
• Cyclops Tensor Framework ^[38] is a distributed memory library for efficient decomposition of tensors of arbitrary type and parallel MPI+OpenMP execution of tensor contractions/functions.
• TiledArray^[39] is a scalable, block-sparse tensor library that is designed to aid in rapid composition of high-performance algebraic tensor equation. It is designed to scale from a single multicore computer to a massively-parallel, distributed-memory system.
• libtensor ^[40] is a set of performance linear tensor algebra routines for large tensors found in post-Hartree–Fock methods in quantum chemistry.
• ITensor ^[41] features automatic contraction of matching tensor indices. It is written in C++ and has higher-level features for quantum physics algorithms based on tensor networks.
• Fastor ^[42] is a high performance C++ tensor algebra library that supports tensors of any arbitrary dimensions and all their possible contraction and permutation thereof. It employs compile-time graph search optimisations to find the optimal contraction sequence between arbitrary number of tensors in a network. It has high level domain specific features for solving nonlinear multiphysics problem using FEM.
• Xerus ^[43] is a C++ tensor algebra library for tensors of arbitrary dimensions and tensor decomposition into general tensor networks (focusing on matrix product states). It offers Einstein notation like syntax and optimizes the contraction order of any network of tensors at runtime so that dimensions need not be fixed at compile-time.

Notes and References

1. Web site: SPLATT - Parallel Sparse Tensor Decomposition | Karypis Lab.
2. Web site: Cadabra: A field-theory motivated approach to computer algebra.
3. http://www.ava.fmi.fi/prog/tela.html Tela
4. Web site: Cartan: A Mathematica package for tensor analysis . 2010-01-01 . dead . https://web.archive.org/web/20090531083427/http://www.adinfinitum.no/cartan/ . 2009-05-31 . Tensors in Physics
5. http://www.math.washington.edu/~lee/Ricci/ Ricci
6. https://sites.google.com/view/xjaen-ttc/home TTC
7. http://www.inp.demokritos.gr/~sbonano/RGTC/ EDC and RGTC
8. http://home.comcast.net/~djmpark/TensorialPage.html Tensorial
9. http://www.xact.es/ xAct
10. http://library.wolfram.com/infocenter/MathSource/4781/ GREAT
11. http://digi-area.com/Mathematica/atlas/ atlas 2 for Mathematica
12. http://grtensor.phy.queensu.ca/ GRTensorII
13. https://github.com/tririver/MathGR; https://arxiv.org/abs/1306.1295 MathGR
14. http://www.stargazing.net/yizen/Tensoria.html TensoriaCalc
15. https://github.com/bshoshany/OGRe OGRe
16. http://grtensor.phy.queensu.ca/ GRTensorII
17. http://digi-area.com/Maple/atlas/ atlas 2 for Maple
18. Web site: Utah State University Differential Geometry Software Project | Utah State University Research | DigitalCommons@USU.
19. Web site: Overview of the Physics Package - Maple Programming Help.
20. Web site: Tensorlab | A Matlab package for tensor computations.
21. http://sandia.gov/~tgkolda/TensorToolbox Tensor Toolbox
22. http://www.mathworks.com/matlabcentral/fileexchange/26168 MPCA and MPCA+LDA
23. http://www.mathworks.com/matlabcentral/fileexchange/35432 UMPCA
24. http://www.mathworks.com/matlabcentral/fileexchange/35782 UMLDA
25. http://maxima.sourceforge.net/ Maxima on sourceforge.net
26. http://maxima.sourceforge.net/docs/manual/en/maxima_27.html#SEC90 The itensor Package Guide
27. Khan. Suleiman A.. Leppäaho. Eemeli. Kaski. Samuel. 2016-06-10. Bayesian multi-tensor factorization. Machine Learning. en. 105. 2. 233–253. 10.1007/s10994-016-5563-y. 0885-6125. 1412.4679. 11871330.
28. Kossaifi. Jean. Panagakis. Yannis. Anandkumar. Anima. Pantic. Maja. 2019. TensorLy: Tensor Learning in Python. Journal of Machine Learning Research. en. 20. 26. 1–6.
29. Fishman. Matthew. Shite. Steven R.. Stoudenmire. E. Miles. 2020-06-28. The ITensor Software Library for Tensor Network Calculations. Mathematical Software. en. 2007.14822v1.
30. http://sagemanifolds.obspm.fr/ SageManifolds
31. Homepage: http://redberry.cc . Redberry project at Bitbucket: http://bitbucket.org/redberry/redberry
32. https://github.com/ilyak/libxm libxm
33. http://www.wlandry.net/Projects/FTensor FTensor
34. http://www.cepremap.cnrs.fr/juillard/mambo/download/manual/dynare++/tl.pdf TL
35. https://vmml.github.com/vmmlib/ vmmlib
36. http://aleph0.info/spartns/ Spartns
37. http://www.cimec.org.ar/petscfem FastMat
38. Web site: Fork of cyclops-community/ctf repository updated haphazardly, previously this was main repo location: Solomonik/ctf. GitHub. 2018-10-31.
39. Web site: A massively-parallel, block-sparse tensor framework written in C++: ValeevGroup/tiledarray. GitHub. 2019-02-03.
40. Web site: Downloads - iOpenShell.
41. Web site: ITensor - Intelligent Tensor Library.
42. Web site: A light-weight high performance SIMD based tensor algebra framework in C++11/14/17: Romeric/Fastor. GitHub. 2019-01-11.
43. Web site: Xerus Documentation - Xerus.