Vec : a library for fixed-length lists and low-dimensional linear algebra

Scott E. Dillard <[email protected]>

http://github.com/sedillard/Vec

--- Synopsis ---

Vectors are represented by lists with type-encoded lengths. The constructor is
@:.@, which acts like a cons both at the value and type levels, with @()@
taking the place of nil. So @x:.y:.z:.()@ is a 3d vector. The library provides
a set of common list-like functions (map, fold, etc) for working with vectors.
Built up from these functions are a small but useful set of linear algebra
operations: matrix multiplication, determinants, solving linear systems,
inverting matrices.


--- Design Principles --- 

  * Simplicity : Beyond the initial complexities of type-level lists and
    numbers, I've tried to keep the API simple. There is no vector-space
    class, nor a complicated hierarchy of linear/affine/projective
    transformations. These can be added on top of the library easily.

  * Purity : The library is written in the functional style. For most
    functions this does not hinder performance at all, but some I am still
    working on (Gaussian elimination) so if this library is a bottleneck you
    can easily drop down to C. 

  * Low Dimension : Although the dimensionality is limited only by what GHC
    will handle, the library is meant for 2,3 and 4 dimensions. For general
    linear algebra, check out the excellent hmatrix library and blas bindings.

    
In pursuit of to the above goal of simplicity, vectors and matrices are
instances of Num and Fractional.  All arithmetic is done component-wise and
literals construct uniform vectors and matrices. There are many interesting
projects aiming to overhaul Haskell's number classes, but for now the type of
(*) is 'a -> a -> a' so that's what we're working with. It is easy to
incorporate this library into a more mathematically consistent class hierarchy
(provided you can design one.) 

The rule is simple : 
  If the method is unary, it's a map. 
  If it's binary, it's a zipWith.


-- Performance ----

(:.) is strict in both arguments, but it is also polymorphic, so at runtime
vectors will be realized as linked lists, albeit with less pattern matching.
However the library provides packed representations for 2,3 and 4d vectors of
Ints, Floats and Doubles. 'Vec3F x y z' constructs a packed vector of unboxed
Floats. Functions 'pack' and 'unpack' convert between packed and unpacked
types. When vector operations are bracketed by 'pack' and 'unpack', GHC can
unfold them into very efficient code. The 'Storable' instances for vectors also
generate fast code.  Without optimizations, the code falls back into
linked-list mode. The optimizations depend on inlining, so you may need to
increase your unfolding threshold in certain situations.


-- GHC Extensions ---

This library makes heavy use of functional dependencies. I have tried to
tweak things so that they "just work." However, every now and then you will
get incomprehensible error messages, usually about how this isn't an
instance of that. These are how type errors typically manifest, so first
double check to make sure you aren't trying to mix vectors of different
dimension or component types. If you still get these errors, manual type
annotations usually make them go away.


--- Related Work ---

See previous work by David Menendez,
  http://haskell.org/pipermail/haskell/2005-May/015815.html

and of course Oleg Kiselyov,
  http://okmij.org/ftp/papers/number-parameterized-types.pdf

Other vector and linear algebra packages :

vector-space, by Conal Elliott : 
  http://hackage.haskell.org/cgi-bin/hackage-scripts/package/vector-space

hmatrix, by Alberto Ruiz :
  http://hackage.haskell.org/cgi-bin/hackage-scripts/package/hmatrix

blas bindings, by Patrick Perry :
  http://hackage.haskell.org/cgi-bin/hackage-scripts/package/blas

templatized geometry library, by Oliver Kreylos (C++) :
  http://graphics.cs.ucdavis.edu/~okreylos/ResDev/Geometry/index.html