Linear.Matrix · linear-1.22 · horizon-platform

value(!*!)

:: (Functor m, Foldable t, Additive t, Additive n, Num a)
=> m (t a)
-> t (n a)
-> m (n a)

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

Example1 expression

>>> 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)

Example1 expression

>>> V2 (IntMap.fromList [(1,2)]) (IntMap.fromList [(2,3)]) !*! IntMap.fromList [(1,V3 0 0 1), (2, V3 0 0 5)]V2 (V3 0 0 2) (V3 0 0 15)

value(!+!) :: (Additive m, Additive n, Num a) => m (n a) -> m (n a) -> m (n a)

#

Entry-wise matrix addition.

Example1 expression

>>> 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)

value(!-!) :: (Additive m, Additive n, Num a) => m (n a) -> m (n a) -> m (n a)

#

Entry-wise matrix subtraction.

Example1 expression

>>> 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)

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

#

Matrix * column vector

Example1 expression

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

value(*!) :: (Num a, Foldable t, Additive f, Additive t) => t a -> t (f a) -> f a

#

Row vector * matrix

Example1 expression

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

value(!!*) :: (Functor m, Functor r, Num a) => m (r a) -> a -> m (r a)

#

Matrix-scalar product

Example1 expression

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

value(*!!) :: (Functor m, Functor r, Num a) => a -> m (r a) -> m (r a)

#

Scalar-matrix product

Example1 expression

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

value(!!/) :: (Functor m, Functor r, Fractional a) => m (r a) -> a -> m (r a)

#

Matrix-scalar division

valuecolumn

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

#

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.

Example1 expression

>>> V2 (V3 1 2 3) (V3 4 5 6) ^._xV3 1 2 3

Example1 expression

>>> V2 (V3 1 2 3) (V3 4 5 6) ^.column _xV2 1 4

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

#

Hermitian conjugate or conjugate transpose

Example1 expression

>>> 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)))

typetype M22 a = V2 (V2 a)

#

A 2x2 matrix with row-major representation

typetype M23 a = V2 (V3 a)

#

A 2x3 matrix with row-major representation

typetype M24 a = V2 (V4 a)

#

A 2x4 matrix with row-major representation

typetype M32 a = V3 (V2 a)

#

A 3x2 matrix with row-major representation

typetype M33 a = V3 (V3 a)

#

A 3x3 matrix with row-major representation

typetype M34 a = V3 (V4 a)

#

A 3x4 matrix with row-major representation

typetype M42 a = V4 (V2 a)

#

A 4x2 matrix with row-major representation

typetype M43 a = V4 (V3 a)

#

A 4x3 matrix with row-major representation

typetype M44 a = V4 (V4 a)

#

A 4x4 matrix with row-major representation

valuem33_to_m44 :: Num a => M33 a -> M44 a

#

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

valuem43_to_m44 :: Num a => M43 a -> M44 a

#

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

valuedet22 :: Num a => M22 a -> a

#

2x2 matrix determinant.

Example1 expression

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

valuedet33 :: Num a => M33 a -> a

#

3x3 matrix determinant.

Example1 expression

>>> 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)

valuedet44 :: Num a => M44 a -> a

#

4x4 matrix determinant.

valueinv22 :: Fractional a => M22 a -> M22 a

#

2x2 matrix inverse.

Example1 expression

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

valueinv33 :: Fractional a => M33 a -> M33 a

#

3x3 matrix inverse.

Example1 expression

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

valueinv44 :: Fractional a => M44 a -> M44 a

#

4x4 matrix inverse.

valueidentity :: (Num a, Traversable t, Applicative t) => t (t a)

#

The identity matrix for any dimension vector.

Example2 expressions

>>> identity :: M44 IntV4 (V4 1 0 0 0) (V4 0 1 0 0) (V4 0 0 1 0) (V4 0 0 0 1)>>> identity :: V3 (V3 Int)V3 (V3 1 0 0) (V3 0 1 0) (V3 0 0 1)

classclass Functor m => Trace (m :: Type -> Type) where

#

Methods

trace :: Num a => m (m a) -> a
Compute the trace of a matrix
Example1 expression
```
>>> trace (V2 (V2 a b) (V2 c d))a + d
```
diagonal :: m (m a) -> m a
Compute the diagonal of a matrix
Example1 expression
```
>>> diagonal (V2 (V2 a b) (V2 c d))V2 a d
```

Instances14Trace, …

Trace ComplexDefined in linear-1.22 · Linear.Trace
Trace IntMapDefined in linear-1.22 · Linear.Trace
Trace PluckerDefined in linear-1.22 · Linear.Trace
Trace QuaternionDefined in linear-1.22 · Linear.Trace
Trace V0Defined in linear-1.22 · Linear.Trace
Trace V1Defined in linear-1.22 · Linear.Trace
Trace V2Defined in linear-1.22 · Linear.Trace
Trace V3Defined in linear-1.22 · Linear.Trace
Trace V4Defined in linear-1.22 · Linear.Trace
Ord k => Trace (Map k)Defined in linear-1.22 · Linear.Trace
(Eq k, Hashable k) => Trace (HashMap k)Defined in linear-1.22 · Linear.Trace
Dim n => Trace (V n)Defined in linear-1.22 · Linear.Trace
(Trace f, Trace g) => Trace (Product f g)Defined in linear-1.22 · Linear.Trace
(Distributive g, Trace g, Trace f) => Trace (Compose g f)Defined in linear-1.22 · Linear.Trace

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

#

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

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

#

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)

valuefromQuaternion :: Num a => Quaternion a -> M33 a

#

Build a rotation matrix from a unit Quaternion.

valuemkTransformation :: Num a => Quaternion a -> V3 a -> M44 a

#

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

valuemkTransformationMat :: Num a => M33 a -> V3 a -> M44 a

#

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

value_m22 :: (Representable t, R2 t, R2 v) => Lens' (t (v a)) (M22 a)

#

Extract a 2x2 matrix from a matrix of higher dimensions by dropping excess rows and columns.

value_m23 :: (Representable t, R2 t, R3 v) => Lens' (t (v a)) (M23 a)

#

Extract a 2x3 matrix from a matrix of higher dimensions by dropping excess rows and columns.

value_m24 :: (Representable t, R2 t, R4 v) => Lens' (t (v a)) (M24 a)

#

Extract a 2x4 matrix from a matrix of higher dimensions by dropping excess rows and columns.

value_m32 :: (Representable t, R3 t, R2 v) => Lens' (t (v a)) (M32 a)

#

Extract a 3x2 matrix from a matrix of higher dimensions by dropping excess rows and columns.

value_m33 :: (Representable t, R3 t, R3 v) => Lens' (t (v a)) (M33 a)

#

Extract a 3x3 matrix from a matrix of higher dimensions by dropping excess rows and columns.

value_m34 :: (Representable t, R3 t, R4 v) => Lens' (t (v a)) (M34 a)

#

Extract a 3x4 matrix from a matrix of higher dimensions by dropping excess rows and columns.

value_m42 :: (Representable t, R4 t, R2 v) => Lens' (t (v a)) (M42 a)

#

Extract a 4x2 matrix from a matrix of higher dimensions by dropping excess rows and columns.

value_m43 :: (Representable t, R4 t, R3 v) => Lens' (t (v a)) (M43 a)

#

Extract a 4x3 matrix from a matrix of higher dimensions by dropping excess rows and columns.

value_m44 :: (Representable t, R4 t, R4 v) => Lens' (t (v a)) (M44 a)

#

Extract a 4x4 matrix from a matrix of higher dimensions by dropping excess rows and columns.

valuelu

:: (Num a, Fractional a, Foldable m, Traversable m, Applicative m, Additive m, Ixed (m a), Ixed (m (m a)), i ~ Index (m a), i ~ Index (m (m a)), Eq i, Integral i, a ~ IxValue (m a), m a ~ IxValue (m (m a)), Num (m a))
=> m (m a)
-> (m (m a), m (m a))

#

Compute the (L, U) decomposition of a square matrix using Crout's algorithm. The Index of the vectors must be Integral.

valueluFinite

:: (Num a, Fractional a, Functor m, Finite m, n ~ Size m, KnownNat n, Num (m a))
=> m (m a)
-> (m (m a), m (m a))

#

Compute the (L, U) decomposition of a square matrix using Crout's algorithm, using the vector's Finite instance to provide an index.

valueforwardSub

:: (Num a, Fractional a, Foldable m, Additive m, Ixed (m a), Ixed (m (m a)), i ~ Index (m a), i ~ Index (m (m a)), Eq i, Ord i, Integral i, a ~ IxValue (m a), m a ~ IxValue (m (m a)))
=> m (m a)
-> m a
-> m a

#

Solve a linear system with a lower-triangular matrix of coefficients with forwards substitution.

valueforwardSubFinite

:: (Num a, Fractional a, Foldable m, n ~ Size m, KnownNat n, Additive m, Finite m)
=> m (m a)
-> m a
-> m a

#

Solve a linear system with a lower-triangular matrix of coefficients with forwards substitution, using the vector's Finite instance to provide an index.

valuebackwardSub

:: (Num a, Fractional a, Foldable m, Additive m, Ixed (m a), Ixed (m (m a)), i ~ Index (m a), i ~ Index (m (m a)), Eq i, Ord i, Integral i, a ~ IxValue (m a), m a ~ IxValue (m (m a)))
=> m (m a)
-> m a
-> m a

#

Solve a linear system with an upper-triangular matrix of coefficients with backwards substitution.

valuebackwardSubFinite

:: (Num a, Fractional a, Foldable m, n ~ Size m, KnownNat n, Additive m, Finite m)
=> m (m a)
-> m a
-> m a

#

Solve a linear system with an upper-triangular matrix of coefficients with backwards substitution, using the vector's Finite instance to provide an index.

valueluSolve

:: (Num a, Fractional a, Foldable m, Traversable m, Applicative m, Additive m, Ixed (m a), Ixed (m (m a)), i ~ Index (m a), i ~ Index (m (m a)), Eq i, Integral i, a ~ IxValue (m a), m a ~ IxValue (m (m a)), Num (m a))
=> m (m a)
-> m a
-> m a

#

Solve a linear system with LU decomposition.

valueluSolveFinite

:: (Num a, Fractional a, Functor m, Finite m, n ~ Size m, KnownNat n, Num (m a))
=> m (m a)
-> m a
-> m a

#

Solve a linear system with LU decomposition, using the vector's Finite instance to provide an index.

valueluInv

:: (Num a, Fractional a, Foldable m, Traversable m, Applicative m, Additive m, Distributive m, Ixed (m a), Ixed (m (m a)), i ~ Index (m a), i ~ Index (m (m a)), Eq i, Integral i, a ~ IxValue (m a), m a ~ IxValue (m (m a)), Num (m a))
=> m (m a)
-> m (m a)

#

Invert a matrix with LU decomposition.

valueluInvFinite

:: (Num a, Fractional a, Functor m, Finite m, n ~ Size m, KnownNat n, Num (m a))
=> m (m a)
-> m (m a)

#

Invert a matrix with LU decomposition, using the vector's Finite instance to provide an index.

valueluDet

:: (Num a, Fractional a, Foldable m, Traversable m, Applicative m, Additive m, Trace m, Ixed (m a), Ixed (m (m a)), i ~ Index (m a), i ~ Index (m (m a)), Eq i, Integral i, a ~ IxValue (m a), m a ~ IxValue (m (m a)), Num (m a))
=> m (m a)
-> a

#

Compute the determinant of a matrix using LU decomposition.

valueluDetFinite

:: (Num a, Fractional a, Functor m, Finite m, n ~ Size m, KnownNat n, Num (m a))
=> m (m a)
-> a

#

Compute the determinant of a matrix using LU decomposition, using the vector's Finite instance to provide an index.