1 research outputs found
Symbolical Index Reduction and Completion Rules for Importing Tensor Index Notation into Programming Languages
In mathematics, many notations have been invented for the concise
representation of mathematical formulae. Tensor index notation is one of such
notations and has been playing a crucial role in describing formulae in
mathematical physics. This paper shows a programming language that can deal
with symbolical tensor indices by introducing a set of tensor index rules that
is compatible with two types of parameters, i.e., scalar and tensor parameters.
When a tensor parameter obtains a tensor as an argument, the function treats
the tensor argument as a whole. In contrast, when a scalar parameter obtains a
tensor as an argument, the function is applied to each component of the tensor.
On a language with scalar and tensor parameters, we can design a set of index
reduction rules that allows users to use tensor index notation for arbitrary
user-defined functions without requiring additional description. Furthermore,
we can also design index completion rules that allow users to define the
operators concisely for differential forms such as the wedge product, exterior
derivative, and Hodge star operator. In our proposal, all these tensor
operators are user-defined functions and can be passed as arguments of
high-order functions.Comment: 13 pages. arXiv admin note: substantial text overlap with
arXiv:1702.0634