Would this work for something like linear algebra? Could it support multiplying two 3x3 matrices where each cell can have a different dimension, and only work if they are compatible?
Here is an example:
mat1 = Table[
RandomChoice[{
Quantity[RandomReal[], "Meters"],
Quantity[RandomReal[], "Seconds"],
Quantity[RandomReal[], "Meters/Seconds"],
Quantity[RandomReal[], "Meters/Seconds^2"]
}] , 3, 3];
mat1 // MatrixForm
mat1 . Transpose[mat1]
use Physics::Measure :ALL;
sub infix:<·>(@x1, @x2) {
die "Incompatible dimensions."
unless @x1 == @x2[0] && @x1[0] == @x2;
[for ^@x1 -> $m {
[for ^@x1 -> $n {
[+] @x1[$m;*] >>*<< @x2[*;$n]
}]
}]
}
my @m = [[1m,2m],[3m,4m]];
say @m · [Z] @m; #[[5m^2 11m^2] [11m^2 25m^2]]
Haskell has arrays and matrices of homogeneous types, so it wouldn't work by default.
If you needed this kind of functionality, you would have to create your own Matrix type which would look very similar to a 9-tuple, and then define matrix multiplication. It would then be possible to encode the dimensional constraints in the type signature, but it's already quite involved for 2x2 matrices:
data Mat2 a b c d
= MkMatrix a b c d
matmul :: ???? -- good luck writing this type signature
matmul (MkMatrix a11 a12 a21 a22) (MkMatrix b11 b12 b21 b22)
= MkMatrix ((a11 * b11) + (a12 * b21))
((a11 * b12) + (a12 * b22))
((a21 * b11) + (a22 * b21))
((a21 * b12) + (a22 * b22))
I wonder if there is any plausible approach that would work closer to how units are treated in physics calculations - not as types, but as special numerical values, more or less (values that you can multiply/divide by any other unit, but can only add/subtract with themselves).
What's so cool about `dimensional` is that the types work the way you describe!
Normally, in Haskell, the addition and multiplication functions are typed like so:
(+) :: Double -> Double -> Double
(*) :: Double -> Double -> Double
(+) :: Quantity d Double -> Quantity d Double -> Quantity d Double
(*) :: Quantity d1 Double -> Quantity d2 Double -> Quantity (d1 * d2) Double
This means that `dimensional` isn't only about 'statically checking' dimensions and units, but also inferring them.
You can see this interactively. What is the type of 1 meter divided by 1 second:
ghci> :t (1 *~ meter) / (1 *~ second)
Quantity (Dim Pos1 Zero Neg1 Zero Zero Zero Zero) Double