TTC: Symbolic tensor calculus with indices

Abstract

In an earlier paper [P. Castellvı́, X. Jaén, and E. Llanta, Comput. Phys. 9, 3 (1995)] we presented the capabilities of TTC (Tools of Tensor Calculus, a Mathematica package, see http://baldufa.upc.es/ttc) when inputs are made in index notation. In that article inputs were merely an effective way to indicate explicit operations to be performed on tensors. Here we describe the way TTC operates with index notation at a symbolic level, that is, when inputs and outputs are expressed in index notation and tensors are not explicitly declared. The operations are performed only at a symbolic level. This is not just an appearance of symbolic computing supplemented with a limited collection of rules. TTC makes algorithm simplifications on symbolic indexed expressions while using the symmetry properties of tensors. © 1998 American Institute of Physics.

KLEIN: a MATHEMATICA Package for Radar Polarimetry Based on Spinor and Tensor Algebra

“…The Mathematica package Klein (written by the second author) demonstrates that spinor and tensor algebra is an effective tool for solving problems in radar polarimetry and that the structure and phenomenology of radar polarimetry can be explained by means of spinors and tensors (Cartan, 1966;Penrose and Rindler, 1984). The design of Klein is in contrast to other systems, like the commercial package MathTensor (Parker and Christiansen, 1994), or, the freely available packages Ricci (Lee, 2000), TCC (Balfagón et al, 2000), and Lucy (Schray, 1985), tailored to the requirements of radar polarimetry.…”

Cadabra: a field-theory motivated symbolic computer algebra system

“…Inheritance of a property is, itself, again implemented as a property (in the example above, the \bar node is declared to have the 2 In order to determine the number of terms in a basis of monomials of tensors, cadabra relies on an external program for the computation of tensor product representations (using the LiE program [16]). 3 "Minimal" here does not necessarily mean that the expression has been reduced to the shortest possible form, which is a problem which to the best of my knowledge remains unresolved. That is, while the algorithm removes dependent terms, as in 2R abcd + 2R bcad + R cabd → R abcd + R bcad (because the third term is found to be expressible as a linear combination of the first two), it does not reduce this further to −R cabd (typical cases are of course more complicated than this example).…”

“…The nested list is just one of the many possible views or representations of the (rather heavily labelled) graph structure representing the expression. While it is perfectly possible to tackle many of the problems mentioned above in a list-based system (as several tensor algebra packages for general purpose computer algebra systems illustrate [2][3][4][5]), this may not be the most elegant, efficient or robust approach (the lack of a system which is able to solve all of the sample problems in Section 3 in an elegant way exemplifies this point). By endowing the core of the computer algebra system with data structures which are more appropriate for the storage of field-theory expressions, it becomes much simpler to write a computer algebra system which can resolve the problems listed above.…”

“…However, the same algorithm can also be used to write a sum in a "minimal" form. 3 That is, by projecting each term using (3) the program can perform the simplification…”

“…it would require tedious work to construct such rules for more general expressions, involving more than just the Riemann or Ricci tensors). An alternative approach is taken in [3,11,14], in which the set of all identities for a particular tensor is used to rewrite a given expression in canonical form. This idea of handling multi-term symmetries using a sum-substitution algorithm goes back to at least [15].…”