Determinants

[jeffreyrosenbluth]

I think this is what was intended by the trello card "implement determinants".

Not the most efficient algorithm, but most of our transformations are of low dimension and the cost of converting to a more efficient data structure seems to out weigh the benefits, fro example the following implementation is about 4X slower on 4 x 4 matrices when taking into account the time needed to convert to Data.Vectors. Additionally, something like an LU decomposition seems like overkill.

minorV ::  Int -> Int -> Vector (Vector a) -> Vector (Vector a)
minorV i j v = removeV j $ V.map (removeV i) v
  where removeV n = V.ifilter (\k _ -> n /= k)

detV :: Num a => Vector (Vector a) -> a
detV m
  | V.length m == 1 = V.head . V.head $ m
  | otherwise = sum [(-1)^i * (c1 ! i) * detV (minorV i 0 m) | i <- [0 .. (n-1)]]
      where
        c1 = V.head m
        n = V.length m

It is also possible that matrixRep or something like it can be used instead of the definitions of onBasis in the individual vector spaces, e.g. R2, R3.

[bergey]

Looks good to me. I'm surprised by the vector result; thanks for profiling that. Do you want to export determinant from Diagrams.Core? I think it's mostly useful inside Backends, and it's easier to find if exported there.

I'd like to give other folks more time to review, but I'm OK merging this tomorrow if they're busy.

onBasis looks the way it does because I don't like exporting lists with a promise that they'll always be a known length. I like having the type checker verify that. But I don't feel that strongly about it.

[byorgey]

The code looks good to me; has anyone tested it to make sure it produces correct results?

[jeffreyrosenbluth]

I didn't export determinant from Diagrams.Core because I was following the lead of onBasis but I am happy to do so.
I tested the code on Transformations of type T2 and T3 and det on some 4 x 4s. All giving correct results.

[bergey]

I didn't export onBasis because I wanted to steer Backend writers to the better-typed versions in TwoD.Transform and ThreeD.Transform.

[jeffreyrosenbluth]

Got it, I'll export determinate then.