Agda-2.3.0.1: A dependently typed functional programming language and proof assistant

Agda.Termination.SparseMatrix

Description

Sparse matrices.

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

Synopsis

# Basic data types

data Matrix i b Source

Type of matrices, parameterised on the type of values.

Instances

 (Eq i, Eq b) => Eq (Matrix i b) (Ord i, Ord b) => Ord (Matrix i b) (Ord i, Integral i, Enum i, Show i, Show b, HasZero b) => Show (Matrix i b) (Arbitrary i, Num i, Integral i, Arbitrary b, HasZero b) => Arbitrary (Matrix i b) (Show i, Ord i, Integral i, Enum i, CoArbitrary b, HasZero b) => CoArbitrary (Matrix i b) (Show i, Integral i, HasZero b, Pretty b) => Pretty (Matrix i b)

matrixInvariant :: (Num i, Ix i) => Matrix i b -> BoolSource

data Size i Source

Size of a matrix.

Constructors

 Size Fieldsrows :: i cols :: i

Instances

 Eq i => Eq (Size i) Ord i => Ord (Size i) Show i => Show (Size i) (Arbitrary i, Integral i) => Arbitrary (Size i) CoArbitrary i => CoArbitrary (Size i)

sizeInvariant :: (Ord i, Num i) => Size i -> BoolSource

data MIx i Source

Type of matrix indices (row, column).

Constructors

 MIx Fieldsrow :: i col :: i

Instances

 Eq i => Eq (MIx i) Ord i => Ord (MIx i) Show i => Show (MIx i) Ix i => Ix (MIx i) (Arbitrary i, Integral i) => Arbitrary (MIx i) CoArbitrary i => CoArbitrary (MIx i)

mIxInvariant :: (Ord i, Num i) => MIx i -> BoolSource

No nonpositive indices are allowed.

# Generating and creating matrices

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

`fromLists sz rs` constructs a matrix from a list of lists of values (a list of rows).

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

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

Constructs a matrix from a list of (index, value)-pairs.

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

Converts a matrix to a list of row lists.

matrix :: (Arbitrary i, Integral i, Arbitrary b, HasZero b) => Size i -> Gen (Matrix i b)Source

Generates a matrix of the given size.

Arguments

 :: (Arbitrary i, Integral i, Arbitrary b, HasZero b) => Size i -> (i -> Gen [b]) The generator is parameterised on the size of the row. -> Gen (Matrix i b)

Generates a matrix of the given size, using the given generator to generate the rows.

# Combining and querying matrices

size :: Matrix i b -> Size iSource

square :: Ix i => Matrix i b -> BoolSource

`True` iff the matrix is square.

isEmpty :: (Num i, Ix i) => Matrix i b -> BoolSource

Returns `True` iff the matrix is empty.

isSingleton :: (Num i, Ix i) => Matrix i b -> Maybe bSource

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

add :: Ord i => (a -> a -> a) -> Matrix i a -> Matrix i a -> Matrix i aSource

`add (+) m1 m2` adds `m1` and `m2`. Uses `(+)` to add values.

Precondition: `size m1 == size m2`.

intersectWith :: Ord i => (a -> a -> a) -> Matrix i a -> Matrix i a -> Matrix i aSource

`intersectWith f m1 m2` build the pointwise conjunction `m1` and `m2`. Uses `f` to combine non-zero values.

Precondition: `size m1 == size m2`.

mul :: (Enum i, Num i, Ix i, Eq a) => Semiring a -> Matrix i a -> Matrix i a -> Matrix i aSource

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

Precondition: `cols (size m1) == rows (size m2)`.

transpose :: Ord i => Matrix i b -> Matrix i bSource

diagonal :: (Show i, Enum i, Num i, Ix i, HasZero b) => Matrix i b -> Array i bSource

`diagonal m` extracts the diagonal of `m`.

Precondition: `square m`.

# Modifying matrices

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

`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 bSource

`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`.