linear-1.16.2: Linear Algebra

Copyright (C) 2012-2013 Edward Kmett, BSD-style (see the file LICENSE) Edward Kmett experimental non-portable Trustworthy Haskell98

Linear.Matrix

Description

Simple matrix operation for low-dimensional primitives.

Synopsis

# Documentation

(!*!) :: (Functor m, Foldable t, Additive t, Additive n, Num a) => m (t a) -> t (n a) -> m (n a) infixl 7 Source

Matrix product. This can compute any combination of sparse and dense multiplication.

````>>> ````V2 (V3 1 2 3) (V3 4 5 6) !*! V3 (V2 1 2) (V2 3 4) (V2 4 5)
```V2 (V2 19 25) (V2 43 58)
```
````>>> ````V2 (fromList [(1,2)]) (fromList [(2,3)]) !*! fromList [(1,V3 0 0 1), (2, V3 0 0 5)]
```V2 (V3 0 0 2) (V3 0 0 15)
```

(!+!) :: (Additive m, Additive n, Num a) => m (n a) -> m (n a) -> m (n a) infixl 6 Source

Entry-wise matrix addition.

````>>> ````V2 (V3 1 2 3) (V3 4 5 6) !+! V2 (V3 7 8 9) (V3 1 2 3)
```V2 (V3 8 10 12) (V3 5 7 9)
```

(!-!) :: (Additive m, Additive n, Num a) => m (n a) -> m (n a) -> m (n a) infixl 6 Source

Entry-wise matrix subtraction.

````>>> ````V2 (V3 1 2 3) (V3 4 5 6) !-! V2 (V3 7 8 9) (V3 1 2 3)
```V2 (V3 (-6) (-6) (-6)) (V3 3 3 3)
```

(!*) :: (Functor m, Foldable r, Additive r, Num a) => m (r a) -> r a -> m a infixl 7 Source

Matrix * column vector

````>>> ````V2 (V3 1 2 3) (V3 4 5 6) !* V3 7 8 9
```V2 50 122
```

(*!) :: (Num a, Foldable t, Additive f, Additive t) => t a -> t (f a) -> f a infixl 7 Source

Row vector * matrix

````>>> ````V2 1 2 *! V2 (V3 3 4 5) (V3 6 7 8)
```V3 15 18 21
```

(!!*) :: (Functor m, Functor r, Num a) => m (r a) -> a -> m (r a) infixl 7 Source

Matrix-scalar product

````>>> ````V2 (V2 1 2) (V2 3 4) !!* 5
```V2 (V2 5 10) (V2 15 20)
```

(*!!) :: (Functor m, Functor r, Num a) => a -> m (r a) -> m (r a) infixl 7 Source

Scalar-matrix product

````>>> ````5 *!! V2 (V2 1 2) (V2 3 4)
```V2 (V2 5 10) (V2 15 20)
```

(!!/) :: (Functor m, Functor r, Fractional a) => m (r a) -> a -> m (r a) infixl 7 Source

Matrix-scalar division

column :: Representable f => LensLike (Context a b) s t a b -> Lens (f s) (f t) (f a) (f b) Source

This is a generalization of `inside` to work over any corepresentable `Functor`.

````column` :: `Representable` f => `Lens` s t a b -> `Lens` (f s) (f t) (f a) (f b)
```

In practice it is used to access a column of a matrix.

````>>> ````V2 (V3 1 2 3) (V3 4 5 6) ^._x
```V3 1 2 3
```
````>>> ````V2 (V3 1 2 3) (V3 4 5 6) ^.column _x
```V2 1 4
```

adjoint :: (Functor m, Distributive n, Conjugate a) => m (n a) -> n (m a) Source

Hermitian conjugate or conjugate transpose

````>>> ````adjoint (V2 (V2 (1 :+ 2) (3 :+ 4)) (V2 (5 :+ 6) (7 :+ 8)))
```V2 (V2 (1.0 :+ (-2.0)) (5.0 :+ (-6.0))) (V2 (3.0 :+ (-4.0)) (7.0 :+ (-8.0)))
```

type M22 a = V2 (V2 a) Source

A 2x2 matrix with row-major representation

type M23 a = V2 (V3 a) Source

A 2x3 matrix with row-major representation

type M24 a = V2 (V4 a) Source

A 2x3 matrix with row-major representation

type M32 a = V3 (V2 a) Source

A 3x2 matrix with row-major representation

type M33 a = V3 (V3 a) Source

A 3x3 matrix with row-major representation

type M34 a = V3 (V4 a) Source

A 3x4 matrix with row-major representation

type M42 a = V4 (V2 a) Source

A 4x2 matrix with row-major representation

type M43 a = V4 (V3 a) Source

A 4x3 matrix with row-major representation

type M44 a = V4 (V4 a) Source

A 4x4 matrix with row-major representation

m33_to_m44 :: Num a => M33 a -> M44 a Source

Convert a 3x3 matrix to a 4x4 matrix extending it with 0's in the new row and column.

m43_to_m44 :: Num a => M43 a -> M44 a Source

Convert from a 4x3 matrix to a 4x4 matrix, extending it with the `[ 0 0 0 1 ]` column vector

det22 :: Num a => M22 a -> a Source

2x2 matrix determinant.

````>>> ````det22 (V2 (V2 a b) (V2 c d))
```a * d - b * c
```

det33 :: Num a => M33 a -> a Source

3x3 matrix determinant.

````>>> ````det33 (V3 (V3 a b c) (V3 d e f) (V3 g h i))
```a * (e * i - f * h) - d * (b * i - c * h) + g * (b * f - c * e)
```

det44 :: Num a => M44 a -> a Source

4x4 matrix determinant.

inv22 :: (Epsilon a, Floating a) => M22 a -> Maybe (M22 a) Source

2x2 matrix inverse.

````>>> ````inv22 \$ V2 (V2 1 2) (V2 3 4)
```Just (V2 (V2 (-2.0) 1.0) (V2 1.5 (-0.5)))
```

inv33 :: (Epsilon a, Floating a) => M33 a -> Maybe (M33 a) Source

3x3 matrix inverse.

````>>> ````inv33 \$ V3 (V3 1 2 4) (V3 4 2 2) (V3 1 1 1)
```Just (V3 (V3 0.0 0.5 (-1.0)) (V3 (-0.5) (-0.75) 3.5) (V3 0.5 0.25 (-1.5)))
```

inv44 :: (Epsilon a, Fractional a) => M44 a -> Maybe (M44 a) Source

4x4 matrix inverse.

eye2 :: Num a => M22 a Source

2x2 identity matrix.

````>>> ````eye2
```V2 (V2 1 0) (V2 0 1)
```

eye3 :: Num a => M33 a Source

3x3 identity matrix.

````>>> ````eye3
```V3 (V3 1 0 0) (V3 0 1 0) (V3 0 0 1)
```

eye4 :: Num a => M44 a Source

4x4 identity matrix.

````>>> ````eye4
```V4 (V4 1 0 0 0) (V4 0 1 0 0) (V4 0 0 1 0) (V4 0 0 0 1)
```

class Functor m => Trace m where Source

Minimal complete definition

Nothing

Methods

trace :: Num a => m (m a) -> a Source

Compute the trace of a matrix

````>>> ````trace (V2 (V2 a b) (V2 c d))
```a + d
```

diagonal :: m (m a) -> m a Source

Compute the diagonal of a matrix

````>>> ````diagonal (V2 (V2 a b) (V2 c d))
```V2 a d
```

Instances

 Trace Complex Trace IntMap Trace V0 Trace V1 Trace V2 Trace V3 Trace V4 Trace Plucker Trace Quaternion Ord k => Trace (Map k) (Eq k, Hashable k) => Trace (HashMap k) (Distributive g, Trace g, Trace f) => Trace (Compose g f) (Trace f, Trace g) => Trace (Product f g) Dim * n => Trace (V * n)

translation :: (Representable t, R3 t, R4 v) => Lens' (t (v a)) (V3 a) Source

Extract the translation vector (first three entries of the last column) from a 3x4 or 4x4 matrix.

transpose :: (Distributive g, Functor f) => f (g a) -> g (f a) Source

`transpose` is just an alias for `distribute`

`transpose (V3 (V2 1 2) (V2 3 4) (V2 5 6))`

V2 (V3 1 3 5) (V3 2 4 6)

fromQuaternion :: Num a => Quaternion a -> M33 a Source

Build a rotation matrix from a unit `Quaternion`.

mkTransformation :: Num a => Quaternion a -> V3 a -> M44 a Source

Build a transformation matrix from a rotation expressed as a `Quaternion` and a translation vector.

mkTransformationMat :: Num a => M33 a -> V3 a -> M44 a Source

Build a transformation matrix from a rotation matrix and a translation vector.