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# Matrix multiplication

Futhark has no builtin operator or function for multiplying
matrices. Indeed, it does not have matrices as a distinct type at
all. Instead, matrices are represented simply as two-dimensional
arrays of some type that supports multiplication and addition. We
can write a function for multiplying integer matrices via the usual
`map`

/`reduce`

constructs:

```
def matmul_i32 [n][m][p] (A: [n][m]i32) (B: [m][p]i32) : [n][p]i32 =
map (\A_row ->
map (\B_col ->0 (map2 (*) A_row B_col))
reduce (+)
(transpose B)) A
```

We can also write a polymorphic higher-order function that encapsulates the general pattern:

```
def matmul [n][m][p] 'a
(add: a -> a -> a) (mul: a -> a -> a) (zero: a)
(A: [n][m]a) (B: [m][p]a) : [n][p]a =
map (\A_row ->
map (\B_col ->
reduce add zero (map2 mul A_row B_col))
(transpose B)) A
```

We can then partially apply `matmul`

to obtain a matrix
multiplication function for a specific type.

`def matmul_f32 = matmul (+) (*) 0f32`