Numeric.Matrix

Efficient matrix operations in 100% pure Haskell.

This package uses miscellaneous implementations, depending on the type of its components. Typically unboxed arrays will perform best, while unboxed arrays give you certain features such as Rational or Complex components.

The following component types are supported by Matrix:

Int

Uses unboxed arrays internally. inv will always return Nothing.

Integer

Uses boxed arrays internally. inv will always return Nothing.

Double and Float

Uses unboxed arrays internally. All matrix operations will work as expected. Matrix Double will probably yield the best peformance.

Rational

Best choice if precision is what you aim for. Uses boxed arrays internally. All matrix operations will work as expected.

Complex

Experimental. Uses boxed arrays internally. The current implementation of inv requires an instance of Ord for the component type, therefor it is currently not possible to calculate the inverse of a complex matrix (on my to do list).

Documentation

data family Matrix e

Matrices are represented by a type which fits best the component type. For example a Matrix Double is represented by unboxed arrays, Matrix Integer by boxed arrays.

Data instances exist for Int, Float, Double, Integer, Ratio, and Complex. Certain types do have certain disadvantages, like for example you can not compute the inverse matrix of a Matrix Int.

Every matrix (regardless of the component type) has instances for Show, Read, Num, Fractional, Eq, Typeable, Binary, and NFData. This means that you can use arithmetic operations like +, *, and /, as well as functions like show, read, or typeOf.

Show (Matrix e)

Note that a Show instance for the component type e must exist.

Read (Matrix e)

You can read a matrix like so:

 read "1 0\n0 1\n" :: Matrix Double

Num (Matrix e)

+, -, *, negate, abs, signum, and fromInteger.

signum will compute the determinant and return the signum of it.

abs applies map abs on the matrix (that is, it applies abs on every component in the matrix and returns a new matrix without negative components).

fromInteger yields a 1-x-1-matrix.

Fractional (Matrix e)

Only available if there exists an instance Fractional e (the component type needs to have a Fractional instance, too). Note that while the Num operations are safe, recip and / will fail (with an error) if the involved matrix is not invertible or not a square matrix.

NFData (Matrix e)

Matrices have instances for NFData so that you can use a matrix in parallel computations using the Control.Monad.Par monad (see the monad-par package for details).

Typeable (Matrix e)

Allows you to use matrices as Dynamic values.

Binary (Matrix e)

Serialize and unserialize matrices using the binary package. See encode and decode.

class (Eq e, Num e) => MatrixElement e where

Methods

matrix :: (Int, Int) -> ((Int, Int) -> e) -> Matrix e

Creates a matrix of the given size using a generator function for the value of each component.

select :: ((Int, Int) -> Bool) -> Matrix e -> [e]

Builds a list from a matrix for the indices for which the given predicate matches.

 trace == select (uncurry (==))

at :: Matrix e -> (Int, Int) -> e

Returns the component at the given position in the matrix. Note that indices start at one, not at zero.

row :: Int -> Matrix e -> [e]

Returns the row at the given index in the matrix. Note that indices start at one, not at zero.

col :: Int -> Matrix e -> [e]

Returns the row at the given index in the matrix. Note that indices start at one, not at zero.

dimensions :: Matrix e -> (Int, Int)

The dimensions of a given matrix.

numRows :: Matrix e -> Int

The number of rows in the given matrix.

numCols :: Matrix e -> Int

The number of columns in the given matrix.

fromList :: [[e]] -> Matrix e

Builds a matrix from a list of lists.

The innermost lists represent the rows. This function will create a m-n-matrix, where m is the number of rows, which is the minimum length of the row lists and n is the number of columns, i.e. the length of the outer list.

 fromList [[1,2,3],[2,1,3],[3,2,1]] :: Matrix Rational

toList :: Matrix e -> [[e]]

unit :: Int -> Matrix e

An identity square matrix of the given size.

>>> unit 4
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

zero :: Int -> Matrix e

A square matrix of the given size consisting of all zeros.

>>> zero 2
0 0
0 0

diag :: [e] -> Matrix e

A square matrix which trace is the given list, all other components set to zero.

>>> diag [1,4,7,9]
1 0 0 0
0 4 0 0
0 0 7 0
0 0 0 9

empty :: Matrix e

Check whether the matrix is the empty matrix.

 dimensions empty == (0, 0)

minus :: Matrix e -> Matrix e -> Matrix e

Subtract a matrix from another.

plus :: Matrix e -> Matrix e -> Matrix e

Add two matrices.

You may also use the Num instance for matrices, i.e. matrix1 + matrix2 will work, too.

times :: Matrix e -> Matrix e -> Matrix e

Multiply two matrices O(n^3).

You may also use the Num instance for matrices, i.e. matrix1 * matrix2 will work, too.

inv :: Matrix e -> Maybe (Matrix e)

Compute the inverse of a matrix. Returns Nothing if the matrix is not invertible.

det :: Matrix e -> e

Applies Bareiss multistep integer-preserving algorithm for finding the determinant of a matrix. Returns 0 if the matrix is not a square matrix.

transpose :: Matrix e -> Matrix e

Flip rows and columns.

 1 8 9                1 2 3
 2 1 8  --transpose-> 8 1 2
 3 2 1                9 8 1 

rank :: Matrix e -> e

Compute the rank of a matrix.

trace :: Matrix e -> [e]

minor :: MatrixElement e => Matrix e -> (Int, Int) -> e

cofactors :: MatrixElement e => Matrix e -> Matrix e

adjugate :: MatrixElement e => Matrix e -> Matrix e

minorMatrix :: MatrixElement e => Matrix e -> (Int, Int) -> Matrix e

map :: MatrixElement f => (e -> f) -> Matrix e -> Matrix f

Apply a function on every component in the matrix.

all :: (e -> Bool) -> Matrix e -> Bool

Apply a predicate on every component in the matrix and returns True iff all components satisfy it.

any :: (e -> Bool) -> Matrix e -> Bool

Apply a predicate on every component in the matrix and return True if one or more components satisfy it.

mapWithIndex :: MatrixElement f => ((Int, Int) -> e -> f) -> Matrix e -> Matrix f

allWithIndex :: ((Int, Int) -> e -> Bool) -> Matrix e -> Bool

anyWithIndex :: ((Int, Int) -> e -> Bool) -> Matrix e -> Bool

Instances

MatrixElement Double

MatrixElement Float

MatrixElement Int

MatrixElement Int32

MatrixElement Int64

MatrixElement Integer

(Show a, Integral a) => MatrixElement (Ratio a)

(Show a, RealFloat a) => MatrixElement (Complex a)

Matrix property and utility functions.

(<|>) :: MatrixElement e => Matrix e -> Matrix e -> Matrix e

Joins two matrices horizontally.

 1 2 3     1 0 0      1 2 3 1 0 0
 3 4 5 <|> 2 1 0  ->  3 4 5 2 1 0
 5 6 7     3 2 1      5 6 7 3 2 1

(<->) :: MatrixElement e => Matrix e -> Matrix e -> Matrix e

Joins two matrices vertically.

 1 2 3     1 0 0      1 2 3
 3 4 5 <-> 2 1 0  ->  3 4 5
 5 6 7     3 2 1      5 6 7
                      1 0 0
                      2 1 0
                      3 2 1

scale :: MatrixElement e => Matrix e -> e -> Matrix e

Scales a matrix by the given factor.

 scale s == map (*s)

Matrix properties

isUnit :: MatrixElement e => Matrix e -> Bool

Check whether the matrix is an identity matrix.

 1 0 0
 0 1 0
 0 0 1 (True)

isZero :: MatrixElement e => Matrix e -> Bool

Check whether the matrix consists of all zeros.

 isZero == all (== 0)

isDiagonal :: MatrixElement e => Matrix e -> Bool

Checks whether the matrix is a diagonal matrix.

 4 0 0 0
 0 7 0 0
 0 0 3 0
 0 0 0 9 (True)

isEmpty :: MatrixElement e => Matrix e -> Bool

Checks whether the matrix is empty.

 isEmpty m = numCols == 0 || numRows == 0

isSquare :: MatrixElement e => Matrix e -> Bool

Checks whether the matrix is a square matrix.

 isSquare == uncurry (==) . dimensions