Copyright	(c) Andrew Lelechenko 2014-2020
License	GPL-3
Maintainer	andrew.lelechenko@gmail.com
Safe Haskell	Safe
Language	Haskell2010

Math.ExpPairs.Matrix3

Description

Matrices of order 3 and efficient multiplication algorithms.

Synopsis

data Matrix3 t = Matrix3 {
- a11 :: !t
- a12 :: !t
- a13 :: !t
- a21 :: !t
- a22 :: !t
- a23 :: !t
- a31 :: !t
- a32 :: !t
- a33 :: !t
}
fromList :: [t] -> Matrix3 t
det :: Num t => Matrix3 t -> t
multCol :: Num t => Matrix3 t -> (t, t, t) -> (t, t, t)
normalize :: Integral t => Matrix3 t -> Matrix3 t
makarovMult :: Num t => Matrix3 t -> Matrix3 t -> Matrix3 t
ladermanMult :: Num t => Matrix3 t -> Matrix3 t -> Matrix3 t

Documentation

data Matrix3 t Source #

Matrix of order 3.

Constructors

Matrix3
Fields a11 :: !t a12 :: !t a13 :: !t a21 :: !t a22 :: !t a23 :: !t a31 :: !t a32 :: !t a33 :: !t

Instances

Functor Matrix3 Source #
Instance details Defined in Math.ExpPairs.Matrix3 Methods fmap :: (a -> b) -> Matrix3 a -> Matrix3 b # (<$) :: a -> Matrix3 b -> Matrix3 a #
Foldable Matrix3 Source #
Instance details Defined in Math.ExpPairs.Matrix3 Methods fold :: Monoid m => Matrix3 m -> m # foldMap :: Monoid m => (a -> m) -> Matrix3 a -> m # foldr :: (a -> b -> b) -> b -> Matrix3 a -> b # foldr' :: (a -> b -> b) -> b -> Matrix3 a -> b # foldl :: (b -> a -> b) -> b -> Matrix3 a -> b # foldl' :: (b -> a -> b) -> b -> Matrix3 a -> b # foldr1 :: (a -> a -> a) -> Matrix3 a -> a # foldl1 :: (a -> a -> a) -> Matrix3 a -> a # toList :: Matrix3 a -> [a] # null :: Matrix3 a -> Bool # length :: Matrix3 a -> Int # elem :: Eq a => a -> Matrix3 a -> Bool # maximum :: Ord a => Matrix3 a -> a # minimum :: Ord a => Matrix3 a -> a # sum :: Num a => Matrix3 a -> a # product :: Num a => Matrix3 a -> a #
Traversable Matrix3 Source #
Instance details Defined in Math.ExpPairs.Matrix3 Methods traverse :: Applicative f => (a -> f b) -> Matrix3 a -> f (Matrix3 b) # sequenceA :: Applicative f => Matrix3 (f a) -> f (Matrix3 a) # mapM :: Monad m => (a -> m b) -> Matrix3 a -> m (Matrix3 b) # sequence :: Monad m => Matrix3 (m a) -> m (Matrix3 a) #
Eq t => Eq (Matrix3 t) Source #
Instance details Defined in Math.ExpPairs.Matrix3 Methods (==) :: Matrix3 t -> Matrix3 t -> Bool # (/=) :: Matrix3 t -> Matrix3 t -> Bool #
Fractional t => Fractional (Matrix3 t) Source #	`fromRational` returns a diagonal matrix
Instance details Defined in Math.ExpPairs.Matrix3 Methods (/) :: Matrix3 t -> Matrix3 t -> Matrix3 t # recip :: Matrix3 t -> Matrix3 t # fromRational :: Rational -> Matrix3 t #
Num t => Num (Matrix3 t) Source #	`fromInteger` returns a diagonal matrix
Instance details Defined in Math.ExpPairs.Matrix3 Methods (+) :: Matrix3 t -> Matrix3 t -> Matrix3 t # (-) :: Matrix3 t -> Matrix3 t -> Matrix3 t # (*) :: Matrix3 t -> Matrix3 t -> Matrix3 t # negate :: Matrix3 t -> Matrix3 t # abs :: Matrix3 t -> Matrix3 t # signum :: Matrix3 t -> Matrix3 t # fromInteger :: Integer -> Matrix3 t #
Show t => Show (Matrix3 t) Source #
Instance details Defined in Math.ExpPairs.Matrix3 Methods showsPrec :: Int -> Matrix3 t -> ShowS # show :: Matrix3 t -> String # showList :: [Matrix3 t] -> ShowS #
Generic (Matrix3 t) Source #
Instance details Defined in Math.ExpPairs.Matrix3 Associated Types type Rep (Matrix3 t) :: Type -> Type # Methods from :: Matrix3 t -> Rep (Matrix3 t) x # to :: Rep (Matrix3 t) x -> Matrix3 t #
NFData t => NFData (Matrix3 t) Source #
Instance details Defined in Math.ExpPairs.Matrix3 Methods rnf :: Matrix3 t -> () #
Pretty t => Pretty (Matrix3 t) Source #
Instance details Defined in Math.ExpPairs.Matrix3 Methods pretty :: Matrix3 t -> Doc ann # prettyList :: [Matrix3 t] -> Doc ann #
type Rep (Matrix3 t) Source #
Instance details Defined in Math.ExpPairs.Matrix3 type Rep (Matrix3 t) = D1 (MetaData "Matrix3" "Math.ExpPairs.Matrix3" "exp-pairs-0.2.1.0-J4IGbuSTVwXCgBqjoU0P5n" False) (C1 (MetaCons "Matrix3" PrefixI True) (((S1 (MetaSel (Just "a11") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t) :: S1 (MetaSel (Just "a12") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t)) :: (S1 (MetaSel (Just "a13") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t) :: S1 (MetaSel (Just "a21") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t))) :: ((S1 (MetaSel (Just "a22") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t) :: S1 (MetaSel (Just "a23") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t)) :: (S1 (MetaSel (Just "a31") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t) :: (S1 (MetaSel (Just "a32") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t) :: S1 (MetaSel (Just "a33") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 t))))))

fromList :: [t] -> Matrix3 t Source #

Convert a list of 9 elements into Matrix3. Reverse conversion can be done by toList.

det :: Num t => Matrix3 t -> t Source #

Compute the determinant of a matrix.

multCol :: Num t => Matrix3 t -> (t, t, t) -> (t, t, t) Source #

Multiplicate a matrix by a column vector.

normalize :: Integral t => Matrix3 t -> Matrix3 t Source #

Divide all elements of the matrix by their greatest common divisor. This is useful for matrices of projective transformations to reduce the magnitude of computations.

makarovMult :: Num t => Matrix3 t -> Matrix3 t -> Matrix3 t Source #

Multiplicate matrices under assumption that multiplication of elements is commutative. Requires 22 multiplications and 66 additions. It becomes faster than vanilla multiplication '(*)', which requires 27 multiplications and 18 additions, when matrix's elements are large (> 700 digits) integers.

An algorithm follows O. M. Makarov. An algorithm for multiplication of $ 3 \times 3 $ matrices. Zh. Vychisl. Mat. i Mat. Fiz., 26(2):293–294, 320, 1986. Our contribution is reducing the number of additions from 105 to 66 by well-thought choice of intermediate variables.

ladermanMult :: Num t => Matrix3 t -> Matrix3 t -> Matrix3 t Source #

Multiplicate matrices. Requires 23 multiplications and 62 additions. It becomes faster than vanilla multiplication '(*)', which requires 27 multiplications and 18 additions, when matrix's elements are large (> 700 digits) integers.

An algorithm follows J. Laderman. A noncommutative algorithm for multiplying $ 3 \times 3 $ matrices using 23 multiplications. Bull. Amer. Math. Soc., 82:126–128, 1976. Our contribution is reducing the number of additions from 98 to 62 by well-thought choice of intermediate variables.