Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	Safe-Inferred
Language	Haskell2010

OAlg.Entity.Matrix.Entries

Contents

Entries
Row
Col
Col Row
Duality

Description

entries of matrices and viewing them as a column of rows respectively as a row of columns.

Synopsis

newtype Entries i j x = Entries (PSequence (i, j) x)
etsxs :: Entries i j x -> [(x, i, j)]
etsEmpty :: Entries i j x
etsAdd :: (Additive x, Ord i, Ord j) => Entries i j x -> Entries i j x -> Entries i j x
etsMlt :: (Distributive x, Ord k) => Col i (Row k x) -> Row j (Col k x) -> Col i (Row j x)
etsJoin :: (i ~ N, j ~ N) => ProductSymbol i -> ProductSymbol j -> Entries i j (Entries i j x) -> Entries i j x
etscr :: Eq i => Entries i j x -> Col i (Row j x)
etsrc :: (Ord i, Ord j) => Entries i j x -> Row j (Col i x)
crets :: Col i (Row j x) -> Entries i j x
rcets :: (Ord i, Ord j) => Row j (Col i x) -> Entries i j x
etsElimZeros :: Additive x => Entries i j x -> Entries i j x
newtype Row j x = Row (PSequence j x)
rowxs :: Row j x -> [(x, j)]
rowEmpty :: Row j x
rowIsEmpty :: Row j x -> Bool
rowHead :: Row j x -> (x, j)
rowTail :: Row j x -> Row j x
rowFilter :: (x -> Bool) -> Row j x -> Row j x
rowMapShift :: Number j => j -> ((x, j) -> y) -> Row j x -> Row j y
rowAppend :: Row j x -> Row j x -> Row j x
rowInterlace :: Ord j => (x -> y -> z) -> (x -> z) -> (y -> z) -> Row j x -> Row j y -> Row j z
rowElimZeros :: Additive a => Row i a -> Row i a
rowSwap :: Ord j => j -> j -> Row j x -> Row j x
rowAdd :: (Additive a, Ord j) => Row j a -> Row j a -> Row j a
rowMltl :: Distributive a => a -> Row j a -> Row j a
rowShear :: Ord j => (Maybe x -> s -> Maybe x) -> (Maybe x -> Maybe x -> Maybe x) -> j -> j -> s -> s -> s -> s -> Row j x -> Row j x
rowScale :: Ord j => (x -> s -> Maybe x) -> j -> s -> Row j x -> Row j x
newtype Col i x = Col (PSequence i x)
colxs :: Col i x -> [(x, i)]
colEmpty :: Col i x
colIsEmpty :: Col i x -> Bool
colHead :: Col i x -> (x, i)
colTail :: Col i x -> Col i x
colFilter :: (x -> Bool) -> Col i x -> Col i x
colMapShift :: Number i => i -> ((x, i) -> y) -> Col i x -> Col i y
colAppend :: Col i x -> Col i x -> Col i x
colInterlace :: Ord i => (x -> y -> z) -> (x -> z) -> (y -> z) -> Col i x -> Col i y -> Col i z
colElimZeros :: Additive a => Col i a -> Col i a
colSwap :: Ord i => i -> i -> Col i x -> Col i x
colAdd :: (Additive a, Ord i) => Col i a -> Col i a -> Col i a
colMltr :: Distributive a => Col i a -> a -> Col i a
colShear :: Ord i => (s -> Maybe x -> Maybe x) -> (Maybe x -> Maybe x -> Maybe x) -> i -> i -> s -> s -> s -> s -> Col i x -> Col i x
colScale :: Ord i => (s -> x -> Maybe x) -> i -> s -> Col i x -> Col i x
crHeadColAt :: Eq j => j -> Col i (Row j a) -> Col i a
crHeadRowAt :: Eq i => i -> Col i (Row j a) -> Row j a
coEntries :: (Ord i, Ord j) => Entries i j x -> Dual (Entries i j x)
coEntriesInv :: (Ord i, Ord j) => Dual (Entries i j x) -> Entries i j x

Entries

newtype Entries i j x Source #

two dimensional partial sequence.

Constructors

Entries (PSequence (i, j) x)

Instances

Instances details

Functor (Entries i j) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods fmap :: (a -> b) -> Entries i j a -> Entries i j b # (<$) :: a -> Entries i j b -> Entries i j a #
(Show x, Show i, Show j) => Show (Entries i j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods showsPrec :: Int -> Entries i j x -> ShowS # show :: Entries i j x -> String # showList :: [Entries i j x] -> ShowS #
(Eq x, Eq i, Eq j) => Eq (Entries i j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods (==) :: Entries i j x -> Entries i j x -> Bool # (/=) :: Entries i j x -> Entries i j x -> Bool #
(Ord x, Ord i, Ord j) => Ord (Entries i j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods compare :: Entries i j x -> Entries i j x -> Ordering # (<) :: Entries i j x -> Entries i j x -> Bool # (<=) :: Entries i j x -> Entries i j x -> Bool # (>) :: Entries i j x -> Entries i j x -> Bool # (>=) :: Entries i j x -> Entries i j x -> Bool # max :: Entries i j x -> Entries i j x -> Entries i j x # min :: Entries i j x -> Entries i j x -> Entries i j x #
(Transposable x, Ord n) => Transposable (Entries n n x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods transpose :: Entries n n x -> Entries n n x Source #
LengthN (Entries i j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods lengthN :: Entries i j x -> N Source #
(Entity x, Entity i, Entity j, Ord i, Ord j) => Validable (Entries i j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods valid :: Entries i j x -> Statement Source #
(Entity x, Entity i, Entity j, Ord i, Ord j) => Entity (Entries i j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries
type Dual (Entries i j x :: TYPE LiftedRep) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries type Dual (Entries i j x :: TYPE LiftedRep) = Entries j i (Op x)

etsxs :: Entries i j x -> [(x, i, j)] Source #

underlying list of indexed entries.

etsEmpty :: Entries i j x Source #

the empty sequence of entries.

etsAdd :: (Additive x, Ord i, Ord j) => Entries i j x -> Entries i j x -> Entries i j x Source #

adding two entries.

Property Let zs = etsAdd xs ys, then holds:

Pre: For all (i,j) in (i,j) where there exists an (x,i,j) in xs and a (y,i,j) in ys holds: root x == root y.
Post

zs is valid.
For all (i,j) in (i,j) holds:
1. If exists a (x,i,j) in xs but not exists a (y,i,j) in ys then there exists a (z,i,j) in zs with z == x.
2. If exists a (y,i,j) in ys but not exists a (x,i,j) in xs then there exists a (z,i,j) in zs with z == y.
3. If exists a (x,i,j) in xs and (y,i,j) in ys then there exists a (z,i,j) in zs with z == x + y.

etsMlt :: (Distributive x, Ord k) => Col i (Row k x) -> Row j (Col k x) -> Col i (Row j x) Source #

multiplication.

etsJoin :: (i ~ N, j ~ N) => ProductSymbol i -> ProductSymbol j -> Entries i j (Entries i j x) -> Entries i j x Source #

joining entries of entries.

Property Let xs' = etsJoin r c xs

Pre

For all (xij,i,j) in xs holds:

i < lengthN r and j < lengthN c
For all (_,i',j') in xij holds: i' < ri and j' < cj where ..ri.. = r, ..cj.. = c.

Post

xs' is valid.

etscr :: Eq i => Entries i j x -> Col i (Row j x) Source #

the underlying column of rows.

etsrc :: (Ord i, Ord j) => Entries i j x -> Row j (Col i x) Source #

the underlying row of columns.

crets :: Col i (Row j x) -> Entries i j x Source #

the entries given by a column of rows.

rcets :: (Ord i, Ord j) => Row j (Col i x) -> Entries i j x Source #

the entries given by a row of columns.

etsElimZeros :: Additive x => Entries i j x -> Entries i j x Source #

elimination of zeros.

Row

newtype Row j x Source #

viewing a partial sequence as a row.

Constructors

Row (PSequence j x)

Instances

Instances details

Functor (Row j) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods fmap :: (a -> b) -> Row j a -> Row j b # (<$) :: a -> Row j b -> Row j a #
Ord j => Sequence (Row j) j x Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods graph :: p j -> Row j x -> Graph j x Source # list :: p j -> Row j x -> [(x, j)] Source # (??) :: Row j x -> j -> Maybe x Source #
(Entity x, Entity j, Ord j) => PermutableSequence (Row j) j x Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods permuteBy :: p j -> (w -> w -> Ordering) -> (x -> w) -> Row j x -> (Row j x, Permutation j) Source #
Ord j => Opr (Permutation j) (Row j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods (<*) :: Row j x -> Permutation j -> Row j x Source #
(Entity x, Entity j, Ord j) => TotalOpr (Permutation j) (Row j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries
(Show x, Show j) => Show (Row j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods showsPrec :: Int -> Row j x -> ShowS # show :: Row j x -> String # showList :: [Row j x] -> ShowS #
(Eq x, Eq j) => Eq (Row j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods (==) :: Row j x -> Row j x -> Bool # (/=) :: Row j x -> Row j x -> Bool #
LengthN (Row j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods lengthN :: Row j x -> N Source #
(Entity x, Entity j, Ord j) => Validable (Row j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods valid :: Row j x -> Statement Source #
(Entity x, Entity j, Ord j) => Entity (Row j x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries
type Dual (Row j (Col i x) :: TYPE LiftedRep) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries type Dual (Row j (Col i x) :: TYPE LiftedRep) = Col j (Row i (Op x))

rowxs :: Row j x -> [(x, j)] Source #

underlying list of indexed entries.

rowEmpty :: Row j x Source #

the empty row.

rowIsEmpty :: Row j x -> Bool Source #

check for being empty.

rowHead :: Row j x -> (x, j) Source #

head.

rowTail :: Row j x -> Row j x Source #

tail.

rowFilter :: (x -> Bool) -> Row j x -> Row j x Source #

filtering a row by the given predicate.

rowMapShift :: Number j => j -> ((x, j) -> y) -> Row j x -> Row j y Source #

mapping and shifting of a row.

rowAppend :: Row j x -> Row j x -> Row j x Source #

appending a row.

Property Let zs = rowAppend xs ys where ..(x,l) = xs and (y,f).. = ys then holds:

If: l < f
Then: zs is valid.

rowInterlace :: Ord j => (x -> y -> z) -> (x -> z) -> (y -> z) -> Row j x -> Row j y -> Row j z Source #

interlacing two rows.

rowElimZeros :: Additive a => Row i a -> Row i a Source #

elimination of zeros.

rowSwap :: Ord j => j -> j -> Row j x -> Row j x Source #

swapping two entries of a row.

Pre k < l.

rowAdd :: (Additive a, Ord j) => Row j a -> Row j a -> Row j a Source #

adding two rows.

rowMltl :: Distributive a => a -> Row j a -> Row j a Source #

multiplies each element of the row by the given factor from the left.

rowShear :: Ord j => (Maybe x -> s -> Maybe x) -> (Maybe x -> Maybe x -> Maybe x) -> j -> j -> s -> s -> s -> s -> Row j x -> Row j x Source #

shears two entries of a row.

Property Let r' = rowShear (<*) (+) k l s t u v r, then holds:

Pre: k < l.

Note rowShear is like multiplying the given row from the right with the matrix given by k l s t u v.

rowScale :: Ord j => (x -> s -> Maybe x) -> j -> s -> Row j x -> Row j x Source #

scales the entry at the given position by the given factor.

Col

newtype Col i x Source #

viewing a partial sequence as a column.

Constructors

Col (PSequence i x)

Instances

Instances details

Functor (Col i) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods fmap :: (a -> b) -> Col i a -> Col i b # (<$) :: a -> Col i b -> Col i a #
Ord i => Sequence (Col i) i x Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods graph :: p i -> Col i x -> Graph i x Source # list :: p i -> Col i x -> [(x, i)] Source # (??) :: Col i x -> i -> Maybe x Source #
(Entity x, Entity i, Ord i) => PermutableSequence (Col i) i x Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods permuteBy :: p i -> (w -> w -> Ordering) -> (x -> w) -> Col i x -> (Col i x, Permutation i) Source #
Ord i => Opr (Permutation i) (Col i x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods (<*) :: Col i x -> Permutation i -> Col i x Source #
(Entity x, Entity i, Ord i) => TotalOpr (Permutation i) (Col i x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries
(Show x, Show i) => Show (Col i x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods showsPrec :: Int -> Col i x -> ShowS # show :: Col i x -> String # showList :: [Col i x] -> ShowS #
(Eq x, Eq i) => Eq (Col i x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods (==) :: Col i x -> Col i x -> Bool # (/=) :: Col i x -> Col i x -> Bool #
LengthN (Col i x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods lengthN :: Col i x -> N Source #
(Entity x, Entity i, Ord i) => Validable (Col i x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries Methods valid :: Col i x -> Statement Source #
(Entity x, Entity i, Ord i) => Entity (Col i x) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries
type Dual (Row j (Col i x) :: TYPE LiftedRep) Source #
Instance details Defined in OAlg.Entity.Matrix.Entries type Dual (Row j (Col i x) :: TYPE LiftedRep) = Col j (Row i (Op x))