Copyright | (c) Matti A. Eskelinen 2016 |
---|---|

License | MIT |

Maintainer | matti.a.eskelinen@gmail.com |

Stability | experimental |

Portability | POSIX |

Safe Haskell | None |

Language | Haskell2010 |

This module implements the various `Clif`

operations on the underlying type. Currently `Clif`

is implemented on top of Data.Map, with blades as keys and scalar multipliers as values.

# Warning

**This module is not intended to be imported by end users and may change drastically in the future. It is currently exposed (and documented) only to help development.**

As development continues, some of the definitions here may be exported from the other modules. Comments and suggestions are welcomed.

- newtype Clif b a = Clif {}
- fromList :: (Eq a, Basis b a) => [([b], a)] -> Clif b a
- blade :: (Eq a, Basis b a) => [b] -> a -> Clif b a
- (*:) :: (Eq a, Basis b a) => a -> [b] -> Clif b a
- vec :: (Eq a, Basis b a) => b -> a -> Clif b a
- gMul :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a
- gMul' :: (Eq a, Basis b a) => Map [b] a -> Map [b] a -> Map [b] a
- gPlus :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a
- gPlus' :: (Eq a, Basis b a) => Map [b] a -> Map [b] a -> Map [b] a
- rev :: Ord b => Clif b a -> Clif b a
- rev' :: Ord b => Map [b] a -> Map [b] a
- canon :: (Eq a, Basis b a) => Clif b a -> Clif b a
- canon' :: (Eq a, Basis b a) => Map [b] a -> Map [b] a
- grade :: (Eq a, Basis b a) => Int -> Clif b a -> Clif b a
- grade' :: (Eq a, Basis b a) => Int -> Map [b] a -> Map [b] a
- contractWith :: (Eq a, Basis b a) => (Int -> Int -> Int) -> Clif b a -> Clif b a -> Clif b a
- filledGrades :: (Eq a, Basis b a) => Clif b a -> [Int]
- filledGrades' :: (Eq a, Basis b a) => Map [b] a -> [Int]
- grades :: (Eq a, Basis b a) => Clif b a -> [(Int, Clif b a)]
- isScalar :: (Eq a, Basis b a) => Clif b a -> Bool
- isZero :: (Eq a, Basis b a) => Clif b a -> Bool
- maxGrade :: (Eq a, Basis b a) => Clif b a -> Int
- (.:) :: (a -> b) -> (c -> d -> a) -> c -> d -> b

# Type `Clif`

A data type representing a Clif (multivector) composed of direct sum of scaled blades

Functor (Clif b) Source # | |

(Eq b, Eq a, Basis b a) => Eq (Clif b a) Source # | The Eq instance calculates the canonical forms of the compared Clifs before comparison. |

(Eq a, Basis b a, Fractional a) => Fractional (Clif b a) Source # | Inverse elements only exist for Clifs c for which c times c is scalar. For others, recip does not terminate. |

(Eq a, Basis b a) => Num (Clif b a) Source # | Note that abs and signum are only well-defined on the scalar component of each Clif, and zero otherwise. |

(Show b, Show a) => Show (Clif b a) Source # | |

# Constructors

fromList :: (Eq a, Basis b a) => [([b], a)] -> Clif b a Source #

Constructs a `Clif`

from a list of blades and their multipliers in canonical form.

`>>>`

42 *: [] + 1 *: [E 1,E 2]`fromList [([], 42), ([E 1, E 2], 1)]`

# Operations

gMul :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a Source #

The Clifford (geometric) product on `Clif`

s.

gMul' :: (Eq a, Basis b a) => Map [b] a -> Map [b] a -> Map [b] a Source #

The Clifford product on `Map`

s of blades and multipliers. Filter out zero values

gPlus :: (Eq a, Basis b a) => Clif b a -> Clif b a -> Clif b a Source #

Addition of `Clif`

values (direct sum).

gPlus' :: (Eq a, Basis b a) => Map [b] a -> Map [b] a -> Map [b] a Source #

Direct sum of matching keys from two `Map`

s. Filter out zero values.

rev :: Ord b => Clif b a -> Clif b a Source #

Reverse of a `Clif`

, i.e. the reverse of all its component blades.

canon' :: (Eq a, Basis b a) => Map [b] a -> Map [b] a Source #

Returns the canonical representation of a `Clif`

(blades simplified and in canonical order)

grade :: (Eq a, Basis b a) => Int -> Clif b a -> Clif b a Source #

Grade projection on the given grade. For negative values, returns zero.

Note that this always calculates the canonical form of a Clif before projecting it.

grade' :: (Eq a, Basis b a) => Int -> Map [b] a -> Map [b] a Source #

Filter blades (keys) by their length.

contractWith :: (Eq a, Basis b a) => (Int -> Int -> Int) -> Clif b a -> Clif b a -> Clif b a Source #

# Properties

filledGrades :: (Eq a, Basis b a) => Clif b a -> [Int] Source #

List of nonzero grades.

Note that this always calculates the canonical form of a Clif before recovering the filled grades.

filledGrades' :: (Eq a, Basis b a) => Map [b] a -> [Int] Source #

List of nonzero grades

Note that this always calculates the canonical form of a Clif before recovering the filled grades.

isScalar :: (Eq a, Basis b a) => Clif b a -> Bool Source #

True if the `Clif`

contains no nonzero blades of grade greater than zero.

Note that this always calculates the canonical form of a Clif before testing whether it is scalar.

isZero :: (Eq a, Basis b a) => Clif b a -> Bool Source #

True for a zero multivector.

Note that this always calculates the canonical form of a Clif before testing whether it is zero.

maxGrade :: (Eq a, Basis b a) => Clif b a -> Int Source #

The highest nonempty nonzero grade of a Clif.

Note that this always calculates the canonical form of a Clif before recovering the highest grade.