Please, send me references and links to anything related to Tensor Computer Algebra.

Other packages in tensor calculus

$: Commercial,
A: Abstract calculus,
C: Component calculus.

• For Wolfram Language & Mathematica:
  • MathTensor (AC$)
    • L. Parker and S.M. Christensen, MathTensor, a system for doing tensor analysis by computer, Addison-Wesley, Reading, MA 1994.
    • Post on the motivation and development of MathTensor.
  • Indicial Tensor Package Using Notebook Interface
  • Tensors in Physics, Cartan (C$)
  • Tensorial ($), with an older free version Tensorial 3.0.
  • MathSymbolica (AC$)
  • Ricci [Lee] (AC)
  • Tools of Tensor Calculus (AC)
  • MathGR (AC). Contact the author to get the code.
    • Yi Wang, MathGR: a tensor and GR computation package to keep it simple, arxiv:1306.1295
  • EinS (A)
    • S.A. Klioner, EinS: a Mathematica package for computations with indexed objects, gr-qc/0011012.
  • GRTensorM (C)
  • RGTC (C)
  • Tetrad, Ricci [Aguirregabiria] (C)
  • diffgeo (C)
  • grassmann (C)
  • Tensors (C)
  • AVF (C)
  • Atlas2 Community (C$)
  • TensorTools in Kranc
  • CCGRG (C)
  • VEST: abstract vector calculus simplification in Mathematica
  • OGRe: An Object-Oriented General Relativity Package for Mathematica
  • SimpleTensor -- a user-friendly Mathematica package for elementary tensor and differential-geometric calculations
  • Gravitas: Computational General Relativity in the Wolfram Language:
    • Symbolic and Analytic Computation
    • ADM Formalism and Numerical Relativity
• For Maple:
  • DifferentialGeometry, built-in (C)
  • Vector, built-in (C)
  • GRTensorII (C)
  • Atlas (C$)
  • Riemann (C), Riegeom (A), Canon (A):
    • R. Portugal and S. Sautú, Applications of Maple to general relativity, Comp. Phys. Commun. 105 (1997) 233-253.
    • R. Portugal, The Riegeom package: abstract tensor calculation, Comp. Phys. Commun. 126 (2000) 261-268.
    • L.R.U. Manssur and R. Portugal, Comp. Phys. Commun. 157 (2004) 173-180.
  • NP, NPSpinor, NPTools:
    • S.R. Czapor and R.G. McLenaghan, Gen. Rel. Grav. 19 (1987) 623.
    • S.R. Czapor, R.G. McLenaghan and J. Carminati, Gen. Rel. Grav. 24 (1992) 911.
    • S. Cyganowski and J. Carminati, Comp. Phys. Commun. 115 (1998) 200-214.
  • GHP, GHPII:
    • J. Carminati and K.T. Vu, GHP: A Maple Package for Perfoming Calculations in the Geroch-Held-Penrose Formalism, Gen. Rel. Grav. 33 (2001) 295.
    • K.T. Vu and J. Carminati, The GHP II Package with Applications, Gen. Rel. Grav. 35 (2003) 263.
• For Reduce:
  • Redten (C)
  • GRG (C)
  • EXCALC, doc (A)
  • RicciR (A)
    • J. Kadlecsik, Ricci calculus package in REDUCE, Comp. Phys. Commun. 93 (1996), 265-282.
  • ATENSOR (A)
    • V.A. Ilyin and A.P. Kryukov, ATENSOR - REDUCE program for tensor simplification, Comp. Phys. Commun. 96 (1996) 36-52.
• For R:
  • TensorA
• For Maxima (the successor of Macsyma and its tensor packages):
  • itensor (A)
  • ctensor (C)
  • atensor (algebras)
    • V. Toth, Tensor manipulation in GPL Maxima, arXiv:cs/0503073.
• Sheep and derivates:
  Two relevant documents:
  • M.A.H. MacCallum, Standard Sheep Letter (1994).
  • J. Skea, Lecture Notes on Sheep (1994).
  • The On-Line Invariant Classification Database
  • Sheep (C)
  • Classi (C)
  • Stensor (AC)
• For Sage:
  • SageManifolds (C)
• In sympy:
  • diffgeom (C)
• Standalone
  • Cadabra (A)
  • Redberry (A)
  • OrtoCartan, doc (C)
  • LambdaTensor , doc (C)
  • ITensor (C)
• See also Tela, the Tensor Language.

Reviews on Tensor Computer Algebra

• M.A.H. MacCallum, in Proc. 4th Canadian Conference on General Relativity and Relativistic Astrophysics, G. Kunstatter, D.E. Vincent, J.G. Williams eds. (World Scientific, Singapore 1992).
• D. Hartley, in Relativity and scientific computing. Computer algebra, numerics, visualization, ed. F.W. Hehl, R.A. Puntigam and H. Ruder (Springer, Berlin 1996).
• M.A.H. MacCallum, Computer algebra and applications in relativity and gravity, in Recent Developments in Gravitation and Mathematical Physics: Proceedings of the First Mexican School on Gravitation and Mathematical Physics, ed. A. Macias, T. Matos, O. Obregon and H. Quevedo (World Scientific, Singapore 1996), pp. 3-41.
• gr-qc/9804068: J. Socorro, A. Macias, F. W. Hehl, Computer algebra in gravity: Programs for (non-)Riemannian spacetimes. I.
• gr-qc/0105094: C. Heinicke, F. W. Hehl, Computer algebra in gravity.
• M.A.H. MacCallum, Computer Algebra in General Relativity, Int. J. Mod. Phys. A 17 (2002) 2707-2710.
• gr-qc/0301076: S. Husa, C. Lechner, Computer algebra applications for Numerical Relativity.
• M.A.H. MacCallum, Computer algebra in gravity research, Living Reviews in Relativity 21, 6 (2018)

Tensor canonicalization

• R. Portugal, Algorithmic simplification of tensor expressions, J. Phys. A 32 (1999) 7779-7789.
• gr-qc/9803023: R. Portugal, An algorithm to simplify tensor expressions, Comp. Phys. Commun. 115 (1998) 215-230.
• math-ph/0107031: R. Portugal, B. F. Svaiter, Group-theoretic Approach for Symbolic Tensor Manipulation: I. Free Indices.
• math-ph/0107032: L. R. U. Manssur, R. Portugal, Group-theoretic Approach for Symbolic Tensor Manipulation: II. Dummy Indices.
• gr-qc/9809022: A. Balfagon, X. Jaen, Simplifying Tensor Polynomials with Indices.
• gr-qc/9912062: X. Jaen, A. Balfagon, Nondimensional Simplification of Tensor Polynomials with Indices.
• arXiv:0803.0862 [cs.SC]: J. M. Martín-García, xPerm: fast index canonicalization for tensor computer algebra.
• arXiv:1702.08114 [cs.CS]: B. E. Niehoff, Faster Tensor Canonicalization.
• arXiv:2208.11946 [cs.CS]: D. Price, K. Peeters, M. Zamaklar, Hiding canonicalisation in tensor computer algebra.