TriangularSolve.jl
18335
TriangularSolve.jl | 18335 | |
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12 | 471 | |
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6.6 | 6.5 | |
6 days ago | 3 months ago | |
Julia | Jupyter Notebook | |
MIT License | - |
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TriangularSolve.jl
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Why Fortran is easy to learn
> But in the end, it's FORTRAN all the way down. Even in Julia.
That's not true. None of the Julia differential equation solver stack is calling into Fortran anymore. We have our own BLAS tools that outperform OpenBLAS and MKL in the instances we use it for (mostly LU-factorization) and those are all written in pure Julia. See https://github.com/YingboMa/RecursiveFactorization.jl, https://github.com/JuliaSIMD/TriangularSolve.jl, and https://github.com/JuliaLinearAlgebra/Octavian.jl. And this is one part of the DiffEq performance story. The performance of this of course is all validated on https://github.com/SciML/SciMLBenchmarks.jl
18335
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Why Fortran is easy to learn
I would say Fortran is a pretty great language for teaching beginners in numerical analysis courses. The only issue I have with it is that, similar to using C+MPI (which is what I first learned with, well after a bit of Java), the students don't tend to learn how to go "higher level". You teach them how to write a three loop matrix-matrix multiplication, but the next thing you should teach is how to use higher level BLAS tools and why that will outperform the 3-loop form. But Fortran then becomes very cumbersome (`dgemm` etc.) so students continue to write simple loops and simple algorithms where they shouldn't. A first numerical analysis course should teach simple algorithms AND why the simple algorithms are not good, but a lot of instructors and tools fail to emphasize the second part of that statement.
On the other hand, the performance + high level nature of Julia makes it a rather excellent tool for this. In MIT graduate course 18.337 Parallel Computing and Scientific Machine Learning (https://github.com/mitmath/18337) we do precisely that, starting with direct optimization of loops, then moving to linear algebra, ODE solving, and implementing automatic differentiation. I don't think anyone would want to give a homework assignment to implement AD in Fortran, but in Julia you can do that as something shortly after looking at loop performance and SIMD, and that's really something special. Steven Johnson's 18.335 graduate course in Numerical Analysis (https://github.com/mitmath/18335) showcases some similar niceties. I really like this demonstration where it starts from scratch with the 3 loops and shows how SIMD and cache-oblivious algorithms build towards BLAS performance, and why most users should ultimately not be writing such loops (https://nbviewer.org/github/mitmath/18335/blob/master/notes/...) and should instead use the built-in `mul!` in most scenarios. There's very few languages where such "start to finish" demonstrations can really be showcased in a nice clear fashion.
What are some alternatives?
Pkg.jl - Pkg - Package manager for the Julia programming language
GPUCompiler.jl - Reusable compiler infrastructure for Julia GPU backends.
SuiteSparse.jl - Development of SuiteSparse.jl, which ships as part of the Julia standard library.
Octavian.jl - Multi-threaded BLAS-like library that provides pure Julia matrix multiplication
Fortran-code-on-GitHub - Directory of Fortran codes on GitHub, arranged by topic
BLIS.jl - This repo plans to provide a low-level Julia wrapper for BLIS typed interface.
rr - Record and Replay Framework
MPI.jl - MPI wrappers for Julia
SciMLBenchmarks.jl - Scientific machine learning (SciML) benchmarks, AI for science, and (differential) equation solvers. Covers Julia, Python (PyTorch, Jax), MATLAB, R