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Could somebody motivate the comparison being made? It is not apriori obvious what Grassman numbers have to do with linear solvers. Thanks.

spacedome 6y ago
They are solving the linear system Ax=b, written in Julia/Matlab usually as A\b, specifically for a fixed size 5x5 matrix. The Grassmann algebras can give a way of representing the matrix and vectors, and a comparison is being made to the representation (and solver) given by StaticArrays.jl, which just uses typical matrices and arrays but optimized for small sizes. No real motivation to use Grassmann algebras if your only concern is generally solving a linear system, other than this method seeming to be faster, which mostly (imo) points to the StaticArrays.jl method needing to be optimized. Edit: This post also does not mention numerical stability of the Grassmann.jl method, which is a real concern in practice, even for such small systems.
- deleted 6y ago
  [deleted]
deltasquared 6y ago
This is a pretty neat trick, they implemented exterior algebra see https://en.wikipedia.org/wiki/Exterior_algebra https://en.wikipedia.org/wiki/Exterior_algebra You see this kind of math in textbooks focused on computer graphics.
contravariant 6y ago
Do you have a Physics background by any chance? Anyway, I believe the module might be named after the mathematician, not the number system called after the same mathematician. Regardless the name Grassman is tied to exterior algebras (and therefore geometric algebras), which are implemented by this library and which can be used to solve matrix equations (although I couldn't recommend using the geometric algebra approach for higher dimensions, unless you can do so symbolically).
- deltasquared 6y ago
  I do have a bit of physics in my background. I read about Grassmann recently in a game programming book. Foundations of game engine development Volume 1 By Eric Lengyel ends with a nice explanation of Grassmann algebra. The road to reality by Roger Penrose also has a treatment of Grassmann algebra in physics context. I agree the library is likely a reference to the mathematician.
  - contravariant 6y ago
    Interesting. Yeah it's nothing too important it's just that I've only ever seen physicists use Grassman numbers, whereas mathematicians tend to work with the exterior algebra directly (which works in much the same way except the wedge operator is used explicitly, IMHO this makes things clearer).
    - DreamScatter 6y ago
      Grassmann algebra and exterior algebra are the same thing, regardless of whether a mathematician or physicist uses them.
      - contravariant 6y ago
        Sure, but the Grassman numbers are a bit of an odd model of Grassman algebra with its own notation. They live in some anonymous vector space that is effectively defined in terms of the Grassman numbers, whereas mathematicians usually have some vector space they're interested in and construct an exterior algebra from its elements. The differences are minor but they're two clearly distinct approaches. Like I said it's nothing terribly important but it's still an interesting difference.
        deltasquared 6y ago
        I agree, the subtle difference between the physics and math mindset is interesting. I wound up on the sidelines as a computer scientist who worked in a physics lab.
6gvONxR4sf7o 6y ago
There's a neat kind of linear algebra (multilinear algebra, technically) that doesn't just deal with vectors (think vectors and planes and volumes), but it's often computationally unweildy. This is a fast library for it that competes with simple matrix code in efficiency.

spacedome 6y ago

They are solving the linear system Ax=b, written in Julia/Matlab usually as A\b, specifically for a fixed size 5x5 matrix. The Grassmann algebras can give a way of representing the matrix and vectors, and a comparison is being made to the representation (and solver) given by StaticArrays.jl, which just uses typical matrices and arrays but optimized for small sizes. No real motivation to use Grassmann algebras if your only concern is generally solving a linear system, other than this method seeming to be faster, which mostly (imo) points to the StaticArrays.jl method needing to be optimized. Edit: This post also does not mention numerical stability of the Grassmann.jl method, which is a real concern in practice, even for such small systems.

deleted 6y ago
[deleted]

deltasquared 6y ago

This is a pretty neat trick, they implemented exterior algebra see https://en.wikipedia.org/wiki/Exterior_algebra https://en.wikipedia.org/wiki/Exterior_algebra You see this kind of math in textbooks focused on computer graphics.

contravariant 6y ago

Do you have a Physics background by any chance? Anyway, I believe the module might be named after the mathematician, not the number system called after the same mathematician. Regardless the name Grassman is tied to exterior algebras (and therefore geometric algebras), which are implemented by this library and which can be used to solve matrix equations (although I couldn't recommend using the geometric algebra approach for higher dimensions, unless you can do so symbolically).

deltasquared 6y ago
I do have a bit of physics in my background. I read about Grassmann recently in a game programming book. Foundations of game engine development Volume 1 By Eric Lengyel ends with a nice explanation of Grassmann algebra. The road to reality by Roger Penrose also has a treatment of Grassmann algebra in physics context. I agree the library is likely a reference to the mathematician.
- contravariant 6y ago
  Interesting. Yeah it's nothing too important it's just that I've only ever seen physicists use Grassman numbers, whereas mathematicians tend to work with the exterior algebra directly (which works in much the same way except the wedge operator is used explicitly, IMHO this makes things clearer).
  - DreamScatter 6y ago
    Grassmann algebra and exterior algebra are the same thing, regardless of whether a mathematician or physicist uses them.
    - contravariant 6y ago
      Sure, but the Grassman numbers are a bit of an odd model of Grassman algebra with its own notation. They live in some anonymous vector space that is effectively defined in terms of the Grassman numbers, whereas mathematicians usually have some vector space they're interested in and construct an exterior algebra from its elements. The differences are minor but they're two clearly distinct approaches. Like I said it's nothing terribly important but it's still an interesting difference.
      - deltasquared 6y ago
        I agree, the subtle difference between the physics and math mindset is interesting. I wound up on the sidelines as a computer scientist who worked in a physics lab.

6gvONxR4sf7o 6y ago

There's a neat kind of linear algebra (multilinear algebra, technically) that doesn't just deal with vectors (think vectors and planes and volumes), but it's often computationally unweildy. This is a fast library for it that competes with simple matrix code in efficiency.