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.
Software for use with Maple
GRTensorII[15] is a computer algebra package for performing calculations in the general area of differential geometry.
Atlas 2 for Maple[16] is a modern differential geometry for Maple.
DifferentialGeometry[17] 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[18] 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, noncomutative variables) etc.
Software for use with Matlab
Tensorlab[19] is a MATLAB toolbox for multilinear algebra and structured data fusion.
Tensor Toolbox[20] Multilinear algebra MATLAB software.
MPCA and MPCA+LDA[21] Multilinear subspace learning software: Multilinear principal component analysis.
UMPCA[22] Multilinear subspace learning software: Uncorrelated multilinear principal component analysis.
UMLDA[23] Multilinear subspace learning software: Uncorrelated multilinear discriminant analysis.
Software for use with Maxima
Maxima[24] 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:[25]
itensor for abstract (indicial) tensor manipulation,
ctensor for component-defined tensors, and
atensor for algebraic tensor manipulation.
Software for use with R
Tensor[26] is an R package for basic tensor operations.
rTensor[27] provides several tensor decomposition approaches.
tensorBF[28] is an R package for Bayesian Tensor decomposition.
MTF[29] Bayesian Multi-Tensor Factorization for data fusion and Bayesian versions of Tensor PCA and Tensor CCA. Software: MTF.
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 [31] 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[32] 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[33] is a lightweight distributed-parallel tensor library written in C.
FTensor[34] is a high performance tensor library written in C++.
TL[35] 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[36] is a C++ linear algebra library that supports 3-way tensors, emphasizing computation and manipulation of several tensor decompositions.
Spartns[37] is a Sparse Tensor framework for Common Lisp.
FAstMat[38] 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 [39] is a distributed memory library for efficient decomposition of tensors of arbitrary type and parallel MPI+OpenMP execution of tensor contractions/functions.
TiledArray[40] 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 [41] is a set of performance linear tensor algebra routines for large tensors found in post-Hartree–Fock methods in quantum chemistry.
ITensor [42] 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 [43] 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 [44] 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.
References
"SPLATT - Parallel Sparse Tensor Decomposition | Karypis Lab".
"Cadabra: A field-theory motivated approach to computer algebra".
http://www.ava.fmi.fi/prog/tela.html Tela
"Archived copy". Archived from the original on 2009-05-31. Retrieved 2010-01-01. Tensors in Physics
http://www.math.washington.edu/~lee/Ricci/ Ricci
https://sites.google.com/view/xjaen-ttc/home TTC
http://www.inp.demokritos.gr/~sbonano/RGTC/ EDC and RGTC
http://home.comcast.net/~djmpark/TensorialPage.html Tensorial
http://www.xact.es/ xAct
http://library.wolfram.com/infocenter/MathSource/4781/ GREAT
http://digi-area.com/Mathematica/atlas/ atlas 2 for Mathematica
http://grtensor.phy.queensu.ca/ GRTensorII
https://github.com/tririver/MathGR; https://arXiv.org/abs/1306.1295 MathGR
http://www.stargazing.net/yizen/Tensoria.html TensoriaCalc
http://grtensor.phy.queensu.ca/ GRTensorII
http://digi-area.com/Maple/atlas/ atlas 2 for Maple
"Utah State University Differential Geometry Software Project | Utah State University Research | DigitalCommons@USU".
"Overview of the Physics Package - Maple Programming Help".
"Tensorlab | A Matlab package for tensor computations".
http://sandia.gov/~tgkolda/TensorToolbox Tensor Toolbox
http://www.mathworks.com/matlabcentral/fileexchange/26168 MPCA and MPCA+LDA
UMPCA
http://www.mathworks.com/matlabcentral/fileexchange/35782 UMLDA
http://maxima.sourceforge.net/ Maxima on sourceforge.net
http://maxima.sourceforge.net/docs/manual/en/maxima_27.html#SEC90 The itensor Package Guide
Rougier, Jonathan (2012-05-05), tensor: Tensor product of arrays, retrieved 2016-12-10
Wells, James Li and Jacob Bien and Martin (2015-12-15), rTensor: Tools for Tensor Analysis and Decomposition, retrieved 2016-12-10
Khan, Suleiman A.; Ammad-ud-din, Muhammad (2016-12-29), tensorBF: Bayesian Tensor Factorization, retrieved 2017-01-07
Khan, Suleiman A.; Leppäaho, Eemeli; Kaski, Samuel (2016-06-10). "Bayesian multi-tensor factorization". Machine Learning. 105 (2): 233–253. arXiv:1412.4679. doi:10.1007/s10994-016-5563-y. ISSN 0885-6125.
http://sagemanifolds.obspm.fr/ SageManifolds
Gibson, Adam (2017-11-11), ND4J: N-dimensional arrays for the JVM, retrieved 2016-12-10
Homepage: http://redberry.cc . Redberry project at Bitbucket: http://bitbucket.org/redberry/redberry
https://github.com/ilyak/libxm libxm
http://www.wlandry.net/Projects/FTensor FTensor
http://www.cepremap.cnrs.fr/juillard/mambo/download/manual/dynare++/tl.pdf TL
https://vmml.github.com/vmmlib/ vmmlib
http://aleph0.info/spartns/ Spartns
http://www.cimec.org.ar/petscfem FastMat
"Fork of cyclops-community/ctf repository updated haphazardly, previously this was main repo location: Solomonik/ctf". 2018-10-31.
"A massively-parallel, block-sparse tensor framework written in C++: ValeevGroup/tiledarray". 2019-02-03.
"Downloads - iOpenShell".
"ITensor - Intelligent Tensor Library".
"A light-weight high performance SIMD based tensor algebra framework in C++11/14/17: Romeric/Fastor". 2019-01-11.
"Xerus Documentation - Xerus".
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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