Safe Haskell	None
Language	Haskell2010

Agda.Termination.SparseMatrix

Contents

Basic data types
Generating and creating matrices
Combining and querying matrices
Modifying matrices

Description

Sparse matrices.

We assume the matrices to be very sparse, so we just implement them as sorted association lists.

Most operations are linear in the number of non-zero elements.

An exception is transposition, which needs to sort the association list again; it has the complexity of sorting: n log n where n is the number of non-zero elements.

Another exception is matrix multiplication, of course.

Synopsis

data Matrix i b = Matrix (Size i) [(MIx i, b)]
unM :: Matrix i b -> [(MIx i, b)]
data Size i = Size {
- rows :: i
- cols :: i
}
data MIx i = MIx {
- row :: i
- col :: i
}
fromLists :: (Ord i, Num i, Enum i, HasZero b) => Size i -> [[b]] -> Matrix i b
fromIndexList :: (Ord i, HasZero b) => Size i -> [(MIx i, b)] -> Matrix i b
toLists :: (Integral i, HasZero b) => Matrix i b -> [[b]]
size :: Matrix i b -> Size i
square :: Ix i => Matrix i b -> Bool
isEmpty :: (Num i, Ix i) => Matrix i b -> Bool
isSingleton :: (Eq i, Num i, HasZero b) => Matrix i b -> Maybe b
zipMatrices :: forall a b c i. Ord i => (a -> c) -> (b -> c) -> (a -> b -> c) -> (c -> Bool) -> Matrix i a -> Matrix i b -> Matrix i c
add :: (Ord i, HasZero a) => (a -> a -> a) -> Matrix i a -> Matrix i a -> Matrix i a
intersectWith :: Ord i => (a -> a -> a) -> Matrix i a -> Matrix i a -> Matrix i a
interAssocWith :: Ord i => (a -> a -> a) -> [(i, a)] -> [(i, a)] -> [(i, a)]
mul :: (Ix i, Eq a) => Semiring a -> Matrix i a -> Matrix i a -> Matrix i a
transpose :: Transpose a => a -> a
class Diagonal m e | m -> e where
- diagonal :: m -> [e]
toSparseRows :: Eq i => Matrix i b -> [(i, [(i, b)])]
supSize :: Ord i => Matrix i a -> Matrix i b -> Size i
zipAssocWith :: Ord i => ([(i, a)] -> [(i, c)]) -> ([(i, b)] -> [(i, c)]) -> (a -> Maybe c) -> (b -> Maybe c) -> (a -> b -> Maybe c) -> [(i, a)] -> [(i, b)] -> [(i, c)]
addRow :: (Num i, HasZero b) => b -> Matrix i b -> Matrix i b
addColumn :: (Num i, HasZero b) => b -> Matrix i b -> Matrix i b

Basic data types

data Matrix i b Source #

Type of matrices, parameterised on the type of values.

Sparse matrices are implemented as an ordered association list, mapping coordinates to values.

Constructors

Matrix (Size i) [(MIx i, b)]

Instances

Instances details

Functor (Matrix i) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods fmap :: (a -> b) -> Matrix i a -> Matrix i b (<$) :: a -> Matrix i b -> Matrix i a #
Foldable (Matrix i) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods fold :: Monoid m => Matrix i m -> m foldMap :: Monoid m => (a -> m) -> Matrix i a -> m foldMap' :: Monoid m => (a -> m) -> Matrix i a -> m foldr :: (a -> b -> b) -> b -> Matrix i a -> b foldr' :: (a -> b -> b) -> b -> Matrix i a -> b foldl :: (b -> a -> b) -> b -> Matrix i a -> b foldl' :: (b -> a -> b) -> b -> Matrix i a -> b foldr1 :: (a -> a -> a) -> Matrix i a -> a foldl1 :: (a -> a -> a) -> Matrix i a -> a toList :: Matrix i a -> [a] null :: Matrix i a -> Bool length :: Matrix i a -> Int elem :: Eq a => a -> Matrix i a -> Bool maximum :: Ord a => Matrix i a -> a minimum :: Ord a => Matrix i a -> a sum :: Num a => Matrix i a -> a product :: Num a => Matrix i a -> a
Traversable (Matrix i) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods traverse :: Applicative f => (a -> f b) -> Matrix i a -> f (Matrix i b) sequenceA :: Applicative f => Matrix i (f a) -> f (Matrix i a) mapM :: Monad m => (a -> m b) -> Matrix i a -> m (Matrix i b) sequence :: Monad m => Matrix i (m a) -> m (Matrix i a)
(Integral i, HasZero b, Pretty b) => Pretty (Matrix i b) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods pretty :: Matrix i b -> Doc Source # prettyPrec :: Int -> Matrix i b -> Doc Source # prettyList :: [Matrix i b] -> Doc Source #
(Ord i, HasZero o, NotWorse o) => NotWorse (Matrix i o) Source #	We assume the matrices have the same dimension.
Instance details Defined in Agda.Termination.Order Methods notWorse :: Matrix i o -> Matrix i o -> Bool Source #
(Ord i, PartialOrd a) => PartialOrd (Matrix i a) Source #	Pointwise comparison. Only matrices with the same dimension are comparable.
Instance details Defined in Agda.Termination.SparseMatrix Methods comparable :: Comparable (Matrix i a) Source #
(Integral i, HasZero b, Show i, Show b) => Show (Matrix i b) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods showsPrec :: Int -> Matrix i b -> ShowS show :: Matrix i b -> String showList :: [Matrix i b] -> ShowS
(Eq i, Eq b) => Eq (Matrix i b) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods (==) :: Matrix i b -> Matrix i b -> Bool (/=) :: Matrix i b -> Matrix i b -> Bool
(Ord i, Ord b) => Ord (Matrix i b) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods compare :: Matrix i b -> Matrix i b -> Ordering (<) :: Matrix i b -> Matrix i b -> Bool (<=) :: Matrix i b -> Matrix i b -> Bool (>) :: Matrix i b -> Matrix i b -> Bool (>=) :: Matrix i b -> Matrix i b -> Bool max :: Matrix i b -> Matrix i b -> Matrix i b min :: Matrix i b -> Matrix i b -> Matrix i b
(Integral i, HasZero b) => Diagonal (Matrix i b) b Source #	Diagonal of sparse matrix. `O(n)` where `n` is the number of non-zero elements in the matrix.
Instance details Defined in Agda.Termination.SparseMatrix Methods diagonal :: Matrix i b -> [b] Source #

unM :: Matrix i b -> [(MIx i, b)] Source #

Association of indices to values.

data Size i Source #

Size of a matrix.

Constructors

Size
Fields rows :: i Number of rows, `>= 0`. cols :: i Number of columns, `>= 0`.

Instances

Instances details

Show i => Show (Size i) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods showsPrec :: Int -> Size i -> ShowS show :: Size i -> String showList :: [Size i] -> ShowS
Eq i => Eq (Size i) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods (==) :: Size i -> Size i -> Bool (/=) :: Size i -> Size i -> Bool
Ord i => Ord (Size i) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods compare :: Size i -> Size i -> Ordering (<) :: Size i -> Size i -> Bool (<=) :: Size i -> Size i -> Bool (>) :: Size i -> Size i -> Bool (>=) :: Size i -> Size i -> Bool max :: Size i -> Size i -> Size i min :: Size i -> Size i -> Size i

data MIx i Source #

Type of matrix indices (row, column).

Constructors

MIx
Fields row :: i Row index, `1 <= row <= rows`. col :: i Column index `1 <= col <= cols`.

Instances

Instances details

Ix i => Ix (MIx i) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods range :: (MIx i, MIx i) -> [MIx i] # index :: (MIx i, MIx i) -> MIx i -> Int # unsafeIndex :: (MIx i, MIx i) -> MIx i -> Int inRange :: (MIx i, MIx i) -> MIx i -> Bool # rangeSize :: (MIx i, MIx i) -> Int # unsafeRangeSize :: (MIx i, MIx i) -> Int
Show i => Show (MIx i) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods showsPrec :: Int -> MIx i -> ShowS show :: MIx i -> String showList :: [MIx i] -> ShowS
Eq i => Eq (MIx i) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods (==) :: MIx i -> MIx i -> Bool (/=) :: MIx i -> MIx i -> Bool
Ord i => Ord (MIx i) Source #
Instance details Defined in Agda.Termination.SparseMatrix Methods compare :: MIx i -> MIx i -> Ordering (<) :: MIx i -> MIx i -> Bool (<=) :: MIx i -> MIx i -> Bool (>) :: MIx i -> MIx i -> Bool (>=) :: MIx i -> MIx i -> Bool max :: MIx i -> MIx i -> MIx i min :: MIx i -> MIx i -> MIx i

Generating and creating matrices

fromLists :: (Ord i, Num i, Enum i, HasZero b) => Size i -> [[b]] -> Matrix i b Source #

fromLists sz rs constructs a matrix from a list of lists of values (a list of rows). O(size) where size = rows × cols.

Precondition: length rs == rows sz and all ((cols sz ==) . length) rs.

fromIndexList :: (Ord i, HasZero b) => Size i -> [(MIx i, b)] -> Matrix i b Source #

Constructs a matrix from a list of (index, value)-pairs. O(n) where n is size of the list.

Precondition: indices are unique.

toLists :: (Integral i, HasZero b) => Matrix i b -> [[b]] Source #

Converts a matrix to a list of row lists. O(size) where size = rows × cols.

Combining and querying matrices

size :: Matrix i b -> Size i Source #

Dimensions of the matrix.

square :: Ix i => Matrix i b -> Bool Source #

True iff the matrix is square.

isEmpty :: (Num i, Ix i) => Matrix i b -> Bool Source #

Returns True iff the matrix is empty.

isSingleton :: (Eq i, Num i, HasZero b) => Matrix i b -> Maybe b Source #

Returns 'Just b' iff it is a 1x1 matrix with just one entry b. O(1).

zipMatrices Source #

Arguments

:: forall a b c i. Ord i
=> (a -> c)	Element only present in left matrix.
-> (b -> c)	Element only present in right matrix.
-> (a -> b -> c)	Element present in both matrices.
-> (c -> Bool)	Result counts as zero?
-> Matrix i a
-> Matrix i b
-> Matrix i c

General pointwise combination function for sparse matrices. O(n1 + n2).

add :: (Ord i, HasZero a) => (a -> a -> a) -> Matrix i a -> Matrix i a -> Matrix i a Source #

add (+) m1 m2 adds m1 and m2, using (+) to add values. O(n1 + n2).

Returns a matrix of size supSize m1 m2.

intersectWith :: Ord i => (a -> a -> a) -> Matrix i a -> Matrix i a -> Matrix i a Source #

intersectWith f m1 m2 build the pointwise conjunction m1 and m2. Uses f to combine non-zero values. O(n1 + n2).

Returns a matrix of size infSize m1 m2.

interAssocWith :: Ord i => (a -> a -> a) -> [(i, a)] -> [(i, a)] -> [(i, a)] Source #

Association list intersection. O(n1 + n2).

interAssocWith f l l' = { (i, f a b) | (i,a) ∈ l and (i,b) ∈ l' }

Used to combine sparse matrices, it might introduce zero elements if f can return zero for non-zero arguments.

mul :: (Ix i, Eq a) => Semiring a -> Matrix i a -> Matrix i a -> Matrix i a Source #

mul semiring m1 m2 multiplies matrices m1 and m2. Uses the operations of the semiring semiring to perform the multiplication.

O(n1 + n2 log n2 + Σ(i <= r1) Σ(j <= c2) d(i,j)) where r1 is the number of non-empty rows in m1 and c2 is the number of non-empty columns in m2 and d(i,j) is the bigger one of the following two quantifies: the length of sparse row i in m1 and the length of sparse column j in m2.

Given dimensions m1 : r1 × c1 and m2 : r2 × c2, a matrix of size r1 × c2 is returned. It is not necessary that c1 == r2, the matrices are implicitly patched with zeros to match up for multiplication. For sparse matrices, this patching is a no-op.

transpose :: Transpose a => a -> a Source #

class Diagonal m e | m -> e where Source #

diagonal m extracts the diagonal of m.

For non-square matrices, the length of the diagonal is the minimum of the dimensions of the matrix.

Methods

diagonal :: m -> [e] Source #

Instances

Instances details

HasZero a => Diagonal (CallMatrix' a) a Source #
Instance details Defined in Agda.Termination.CallMatrix Methods diagonal :: CallMatrix' a -> [a] Source #
Diagonal (CallMatrixAug cinfo) Order Source #
Instance details Defined in Agda.Termination.CallMatrix Methods diagonal :: CallMatrixAug cinfo -> [Order] Source #
(Integral i, HasZero b) => Diagonal (Matrix i b) b Source #	Diagonal of sparse matrix. `O(n)` where `n` is the number of non-zero elements in the matrix.
Instance details Defined in Agda.Termination.SparseMatrix Methods diagonal :: Matrix i b -> [b] Source #

toSparseRows :: Eq i => Matrix i b -> [(i, [(i, b)])] Source #

Converts a sparse matrix to a sparse list of rows. O(n) where n is the number of non-zero entries of the matrix.

Only non-empty rows are generated.

supSize :: Ord i => Matrix i a -> Matrix i b -> Size i Source #

Compute the matrix size of the union of two matrices.

zipAssocWith Source #

Arguments

:: Ord i
=> ([(i, a)] -> [(i, c)])	Only left map remaining.
-> ([(i, b)] -> [(i, c)])	Only right map remaining.
-> (a -> Maybe c)	Element only present in left map.
-> (b -> Maybe c)	Element only present in right map.
-> (a -> b -> Maybe c)	Element present in both maps.
-> [(i, a)]
-> [(i, b)]
-> [(i, c)]

General pointwise combination function for association lists. O(n1 + n2) where ni is the number of non-zero element in matrix i.

In zipAssocWith fs gs f g h l l',

fs is possibly more efficient version of mapMaybe ( (i, a) -> (i,) $ f a), and same for gs and g.

Modifying matrices

addRow :: (Num i, HasZero b) => b -> Matrix i b -> Matrix i b Source #

addRow x m adds a new row to m, after the rows already existing in the matrix. All elements in the new row get set to x.

addColumn :: (Num i, HasZero b) => b -> Matrix i b -> Matrix i b Source #

addColumn x m adds a new column to m, after the columns already existing in the matrix. All elements in the new column get set to x.