Numeric tensor framework: Exploiting and extending Einstein notation

The numeric tensor (NT) framework addresses and unifies a growing body of work on high-dimensional algebra and software for technical computing. Its NT algebra exploits and extends Einstein notation, offering unmatched capabilities, including N-dimensional operators, associativity, commutativity, entrywise products, and linear invertibility. High-performance C++ and MATLAB NT software allows practitioners to directly program with NT algebra. The advantages of NT algebra are highlighted using the example of canonical-polyadic (CP) tensor decomposition. Corresponding dense benchmarks demonstrate that the NT software matches or surpasses leading competitors, i.e., the MATLAB Tensor Toolbox, NumPy, and Blitz++, while supporting a more general set of arithmetic operations.