TensorOperations.jl

This is also how it works in Julia, where macros digest notation for einsum-like operations before compile-time. In fact the linked file's explanatory comment: (einsum (A i j) (B i k) (C k j)) results in the the updates A[i,j] = \\sum_k B[i,k]C[k,j], which is equivalent to matrix multiplication. very nearly contains the syntax used by all the Julia packages (where @ marks a macro), which is @tensor A[i,j] = B[i,k] * C[k,j] (using https://github.com/Jutho/TensorOperations.jl, but see also OMEinsum, Finch, and my Tullio, TensorCast.)

Ask and ye shall receive (more than ye bargained for): https://github.com/mcabbott/Tullio.jl https://github.com/Jutho/TensorOperations.jl https://github.com/under-Peter/OMEinsum.jl Tullio.jl is the most powerful and interesting to me of the above libraries, but they all have cool strengths.

> It's not JUST google and apple, I'm sure they have enough cash and expertise and will to create enough momentum to attract more domain experts in other areas. Maybe, but I'm doubtful. Scientific domain experts flock to languages like Python, Julia, Matlab, R, etc. because they're interactive and allow them to quickly iterate on ideas, query data, produce plots, etc. Swift is not much of an interactive language and is not built around that kind of repl driven experience. > In particular, do you have another example aside from DifferentialEquations.jl ? Sure, here's a smattering of high quality packages made by and for research scientists: https://github.com/JuliaApproximation/ApproxFun.jl https://github.com/BioJulia https://github.com/JuliaDiffEq/ModelingToolkit.jl https://github.com/crstnbr/MonteCarlo.jl https://gith

Matrix chain multiplication can be seen a special case of tensor contraction, and the https://github.com/Jutho/TensorOperations.jl package has an implementation which tries to optimize the contraction order (macro @tensoropt).

There are a bunch of packages exploring ideas around labelled dimensions, although we are still far from the vision of caring as little as we do about registers: * https://github.com/JuliaArrays/AxisArrays.jl * https://github.com/davidavdav/NamedArrays.jl * https://github.com/invenia/NamedDims.jl * https://github.com/rafaqz/DimensionalData.jl * https://github.com/ITensor/ITensors.jl While these don't pass dimension labels along in the type, there are also a number of Einsum-like packages which should work on any storage order, and could be taught to interface with such types: * https://github.com/Jutho/TensorOperations.jl * https://github.com/under-Peter/OMEinsum.jl * https://github.com/mcabbott/TensorCast.jl

Julia has some really cool libraries for index notation. I've made extensive use of TensorOperations.jl [1] and TensorCast.jl [2]. I find that the capabilities of these two complement each-other really well since TensorOperations.jl is a bit more restricted in the sort of operations it can do, but it does all sorts of really powerful analysis to find the optimal way of doing a given contraction whereas TensorCast.jl is super flexible, permissive and expressive but doesn't always generate an optimal contraction for big high dimensional contractions. I'm excited to see what comes out of Andreas' summer work on his implementation of Einsum! There's definitely room for more cool ideas in this space. In particular, I think baking in AD functionality is a cool approach. [1] https://github.com/Jutho/TensorOperations.jl [2] https://github.com/mc

The julia DSL for this is closer to what you want: "M_ij = A_ik * B_kj" reads @einsum M[i,j] := A[i,k] * B[k,j] Or faster with @tensor from https://github.com/Jutho/TensorOperations.jl & the same notation. I'm not aware of any attempt to distinguish co-vectors; you can have g = Diagonal([-1,1,1,1]) and insert A[i,k]g[k,k′]B[k′,j] and this is handled efficiently.

In Julia it is easy: https://github.com/Jutho/TensorOperations.jl In some fields, e.g. condensed matter physics, you need you need to deal with contracting multidimensional tensors all the time, and with out Einstein notation it would be difficult. The simples example is so-called matrix product state (http://en.wikipedia.org/wiki/Matrix_product_state), which is similar to Markov process. But for more interesting scenarios (e.g. 2d) one needs to use at least 4 dim tensors.