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

Scott E. Dillard sedillard@gmail.com

The darcs repository is at http://graphics.cs.ucdavis.edu/~sdillard/Vec

Some examples can be found at http://graphics.cs.ucdavis.edu/~sdillard/Vec/examples

*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*

- 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.

To the point 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.

constructs a packed vector of unboxed
Floats. Functions `Vec3F`

x y z

and `pack`

convert between packed and unpacked
types. When vector operations are bracketed by `unpack`

`pack`

and `unpack`

, GHC can
unfold them into very efficient code. The `Storable`

and `UArray`

instances for
vectors also store them efficiently and 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 (C++), by Oliver Kreylos : http://graphics.cs.ucdavis.edu/~okreylos/ResDev/Geometry/index.html