Scalar and Tensor Parameters for Importing Tensor Index Notation including Einstein Summation Notation

TL;DR

This paper introduces scalar and tensor parameters in functional programming to seamlessly incorporate tensor index notation, including Einstein summation, simplifying tensor computations without extra function specifications.

Contribution

The paper presents a novel method for importing tensor index notation into functional programming using scalar and tensor parameters with simplified index rules.

Findings

Enables applying functions to tensors using Einstein notation

No need for extra tensor-handling code in functions

Simplifies tensor computations in functional programming

Abstract

In this paper, we propose a method for importing tensor index notation, including Einstein summation notation, into functional programming. This method involves introducing two types of parameters, i.e, scalar and tensor parameters, and simplified tensor index rules that do not handle expressions that are valid only for the Cartesian coordinate system, in which the index can move up and down freely. An example of such an expression is "c = A_i B_i". As an ordinary function, 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. In this paper, we show that introducing these two types of parameters and our simplified index rules enables us to apply arbitrary user-defined functions to tensor arguments using index notation including Einstein summation notation without requiring an additional description to enable each function to handle tensors.