Goal.Geometry.Map.Multilinear

The Map module provides tools for developing function space Manifolds. A map is a Manifold where the Points of the Manifold represent parametric functions between Manifolds. The defining feature of Maps is that they have a particular Domain and Codomain, which themselves are Manifolds.

Tensors

data Tensor m n

Manifold of Tensors given by the tensor product of the underlying pair of Manifolds.

Constructors

Tensor m n

Instances

(Manifold m, Manifold n) => Apply c d (Tensor m n)

(Eq m, Eq n) => Eq (Tensor m n)

(Read m, Read n) => Read (Tensor m n)

(Show m, Show n) => Show (Tensor m n)

(Manifold m, Manifold n) => Manifold (Tensor n m)

(Manifold m, Manifold n) => Map (Tensor m n)

type Domain (Tensor m n) = n

type Codomain (Tensor m n) = m

Construction

(>.<) :: (Manifold m, Manifold n) => (d :#: m) -> (c :#: n) -> Function (Dual c) d :#: Tensor m n

>.< denotes the outer product between two points. It provides a way of constructing matrices of the Tensor product space.

Matrix Operations

(<#>) :: (Manifold m, Manifold n, Manifold o) => (Function d e :#: Tensor m n) -> (Function c d :#: Tensor n o) -> Function c e :#: Tensor m o

Tensor product composition.

matrixRank :: (Manifold m, Manifold n) => (c :#: Tensor m n) -> Int

matrixInverse :: (Manifold n, Manifold m) => (Function c d :#: Tensor m n) -> Function d c :#: Tensor n m

The inverse of a given Tensor point.

matrixTranspose :: (Manifold m, Manifold n) => (Function c d :#: Tensor m n) -> Function (Dual d) (Dual c) :#: Tensor n m

The transpose of a given Tensor point.

matrixSquareRoot :: Manifold m => (c :#: Tensor m m) -> c :#: Tensor m m

The square root of a matrix.

matrixApply :: (Manifold m, Manifold n) => (Function c d :#: Tensor n m) -> (c :#: m) -> d :#: n

Matrix vector multiplication.

matrixMap :: (Manifold m, Manifold n) => (Function c d :#: Tensor m n) -> [c :#: n] -> [d :#: m]

Mapped matrix vector multiplication, where we first turn the input vectors into a matrix itself (this can greatly improve computation time).

matrixDiagonalConcatenate :: (Manifold m, Manifold n, Manifold o, Manifold p) => (Function c d :#: Tensor m n) -> (Function e f :#: Tensor o p) -> Function (c, e) (d, f) :#: Tensor (m, o) (n, p)

Creates a block diagonal matrix.

Cartesian

coordinateTransform :: Manifold m => [c :#: m] -> Function Cartesian c :#: Tensor m Euclidean

Returns the coordinate transformation from Euclidean space into the space defined by the given basis vectors. This is a glorified fromColumns function.

linearProjection :: Manifold m => [Cartesian :#: m] -> Function Cartesian Cartesian :#: Tensor m m

Returns the linear projection operator for the given subset of basis vectors.

HMatrix Conversion

toHMatrix :: Manifold n => (c :#: Tensor m n) -> Matrix Double

Converts a point on a Tensor product manifold to a matrix for snappy calculation.

fromHMatrix :: (Manifold m, Manifold n) => Tensor m n -> Matrix Double -> c :#: Tensor m n

Affine Functions

data Affine m n

Manifolds of Affine functions.

Constructors

Affine m n

Instances

(Manifold m, Manifold n) => Apply c d (Affine m n)

(Eq m, Eq n) => Eq (Affine m n)

(Read m, Read n) => Read (Affine m n)

(Show m, Show n) => Show (Affine m n)

(Manifold m, Manifold n) => Manifold (Affine m n)

(Manifold m, Manifold n) => Map (Affine m n)

type Domain (Affine m n) = n

type Codomain (Affine m n) = m

splitAffine :: (Manifold m, Manifold n) => (Function c d :#: Affine m n) -> (d :#: m, Function c d :#: Tensor m n)

Splits an Point on an Affine space into a matrix and a constant.

joinAffine :: (Manifold m, Manifold n) => (d :#: m) -> (Function c d :#: Tensor m n) -> Function c d :#: Affine m n

Combines a matrix and a constant into Point on an Affine space.