Data.Matrix

Matrix datatype and operations.

Every provided example has been tested. Run cabal test for further tests.

Matrix type

data Matrix a

Type of matrices.

Elements can be of any type. Rows and columns are indexed starting by 1. This means that, if m :: Matrix a and i,j :: Int, then m ! (i,j) is the element in the i-th row and j-th column of m.

Instances

Functor Matrix

Applicative Matrix

Foldable Matrix

Traversable Matrix

Eq a => Eq (Matrix a)

Num a => Num (Matrix a)

Show a => Show (Matrix a)

Monoid a => Monoid (Matrix a)

NFData a => NFData (Matrix a)

prettyMatrix :: Show a => Matrix a -> String

Display a matrix as a String using the Show instance of its elements.

nrows :: Matrix a -> Int

Number of rows.

ncols :: Matrix a -> Int

Number of columns.

forceMatrix :: Matrix a -> Matrix a

O(rows*cols). Similar to force. It copies the matrix content dropping any extra memory.

Useful when using submatrix from a big matrix.

Builders

matrix

Arguments

:: Int

Rows

-> Int

Columns

-> ((Int, Int) -> a)

Generator function

-> Matrix a

O(rows*cols). Generate a matrix from a generator function. Example of usage:

                                 (  1  0 -1 -2 )
                                 (  3  2  1  0 )
                                 (  5  4  3  2 )
matrix 4 4 $ \(i,j) -> 2*i - j = (  7  6  5  4 )

rowVector :: Vector a -> Matrix a

O(1). Represent a vector as a one row matrix.

colVector :: Vector a -> Matrix a

O(1). Represent a vector as a one column matrix.

Special matrices

zero

Arguments

:: Num a

=> Int

Rows

-> Int

Columns

-> Matrix a

O(rows*cols). The zero matrix of the given size.

zero n m =
                m
  1 ( 0 0 ... 0 0 )
  2 ( 0 0 ... 0 0 )
    (     ...     )
    ( 0 0 ... 0 0 )
  n ( 0 0 ... 0 0 )

identity :: Num a => Int -> Matrix a

O(rows*cols). Identity matrix of the given order.

identity n =
                n
  1 ( 1 0 ... 0 0 )
  2 ( 0 1 ... 0 0 )
    (     ...     )
    ( 0 0 ... 1 0 )
  n ( 0 0 ... 0 1 )

diagonalList :: Int -> a -> [a] -> Matrix a

Diagonal matrix from a non-empty list given the desired size. Non-diagonal elements will be filled with the given default element. The list must have at least order elements.

diagonalList n 0 [1..] =
                  n
  1 ( 1 0 ... 0   0 )
  2 ( 0 2 ... 0   0 )
    (     ...       )
    ( 0 0 ... n-1 0 )
  n ( 0 0 ... 0   n )

diagonal

Arguments

:: a

Default element

-> Vector a

Diagonal vector

-> Matrix a

Similar to diagonalList, but using Vector, which should be more efficient.

permMatrix

Arguments

:: Num a

=> Int

Size of the matrix.

-> Int

Permuted row 1.

-> Int

Permuted row 2.

-> Matrix a

Permutation matrix.

O(rows*cols). Permutation matrix.

permMatrix n i j =
              i     j       n
  1 ( 1 0 ... 0 ... 0 ... 0 0 )
  2 ( 0 1 ... 0 ... 0 ... 0 0 )
    (     ...   ...   ...     )
  i ( 0 0 ... 0 ... 1 ... 0 0 )
    (     ...   ...   ...     )
  j ( 0 0 ... 1 ... 0 ... 0 0 )
    (     ...   ...   ...     )
    ( 0 0 ... 0 ... 0 ... 1 0 )
  n ( 0 0 ... 0 ... 0 ... 0 1 )

When i == j it reduces to identity n.

List conversions

fromList

Arguments

:: Int

Rows

-> Int

Columns

-> [a]

List of elements

-> Matrix a

Create a matrix from a non-empty list given the desired size. The list must have at least rows*cols elements. An example:

                      ( 1 2 3 )
                      ( 4 5 6 )
fromList 3 3 [1..] =  ( 7 8 9 )

fromLists :: [[a]] -> Matrix a

Create a matrix from a non-empty list of non-empty lists. Each list must have at least as many elements as the first list. Examples:

fromLists [ [1,2,3]      ( 1 2 3 )
          , [4,5,6]      ( 4 5 6 )
          , [7,8,9] ] =  ( 7 8 9 )

fromLists [ [1,2,3  ]     ( 1 2 3 )
          , [4,5,6,7]     ( 4 5 6 )
          , [8,9,0  ] ] = ( 8 9 0 )

toList :: Matrix a -> [a]

Get the elements of a matrix stored in a list.

       ( 1 2 3 )
       ( 4 5 6 )
toList ( 7 8 9 ) = [1,2,3,4,5,6,7,8,9]

toLists :: Matrix a -> [[a]]

Get the elements of a matrix stored in a list of lists, where each list contains the elements of a single row.

        ( 1 2 3 )   [ [1,2,3]
        ( 4 5 6 )   , [4,5,6]
toLists ( 7 8 9 ) = , [7,8,9] ]

Accessing

getElem

Arguments

:: Int

Row

-> Int

Column

-> Matrix a

Matrix

-> a

O(1). Get an element of a matrix. Indices range from (1,1) to (n,m). It returns an error if the requested element is outside of range.

(!) :: Matrix a -> (Int, Int) -> a

Short alias for getElem.

unsafeGet

Arguments

:: Int

Row

-> Int

Column

-> Matrix a

Matrix

-> a

O(1). Unsafe variant of getElem, without bounds checking.

safeGet :: Int -> Int -> Matrix a -> Maybe a

Variant of getElem that returns Maybe instead of an error.

safeSet :: a -> (Int, Int) -> Matrix a -> Maybe (Matrix a)

Variant of setElem that returns Maybe instead of an error.

getRow :: Int -> Matrix a -> Vector a

O(1). Get a row of a matrix as a vector.

getCol :: Int -> Matrix a -> Vector a

O(rows). Get a column of a matrix as a vector.

getDiag :: Matrix a -> Vector a

O(min rows cols). Diagonal of a not necessarily square matrix.

getMatrixAsVector :: Matrix a -> Vector a

O(rows*cols). Transform a Matrix to a Vector of size rows*cols. This is equivalent to get all the rows of the matrix using getRow and then append them, but far more efficient.

Manipulating matrices

setElem

Arguments

:: a

New value.

-> (Int, Int)

Position to replace.

-> Matrix a

Original matrix.

-> Matrix a

Matrix with the given position replaced with the given value.

Replace the value of a cell in a matrix.

unsafeSet

Arguments

:: a

New value.

-> (Int, Int)

Position to replace.

-> Matrix a

Original matrix.

-> Matrix a

Matrix with the given position replaced with the given value.

Unsafe variant of setElem, without bounds checking.

transpose :: Matrix a -> Matrix a

O(rows*cols). The transpose of a matrix. Example:

          ( 1 2 3 )   ( 1 4 7 )
          ( 4 5 6 )   ( 2 5 8 )
transpose ( 7 8 9 ) = ( 3 6 9 )

setSize

Arguments

:: a

Default element.

-> Int

Number of rows.

-> Int

Number of columns.

-> Matrix a

-> Matrix a

Set the size of a matrix to given parameters. Use a default element for undefined entries if the matrix has been extended.

extendTo

Arguments

:: a

Element to add when extending.

-> Int

Minimal number of rows.

-> Int

Minimal number of columns.

-> Matrix a

-> Matrix a

Extend a matrix to a given size adding a default element. If the matrix already has the required size, nothing happens. The matrix is never reduced in size. Example:

                           ( 1 2 3 0 0 )
               ( 1 2 3 )   ( 4 5 6 0 0 )
               ( 4 5 6 )   ( 7 8 9 0 0 )
extendTo 0 4 5 ( 7 8 9 ) = ( 0 0 0 0 0 )

The definition of extendTo is based on setSize:

extendTo e n m a = setSize e (max n $ nrows a) (max m $ ncols a) a

inverse :: (Fractional a, Eq a) => Matrix a -> Either String (Matrix a)

O(rows*rows*rows) = O(cols*cols*cols). The inverse of a square matrix. Uses naive Gaussian elimination formula.

rref :: (Fractional a, Eq a) => Matrix a -> Either String (Matrix a)

O(rows*rows*cols). Converts a matrix to reduced row echelon form, thus solving a linear system of equations. This requires that (cols > rows) if cols < rows, then there are fewer variables than equations and the problem cannot be solved consistently. If rows = cols, then it is basically a homogenous system of equations, so it will be reduced to identity or an error depending on whether the marix is invertible (this case is allowed for robustness).

mapRow

Arguments

:: (Int -> a -> a)

Function takes the current column as additional argument.

-> Int

Row to map.

-> Matrix a

-> Matrix a

O(rows*cols). Map a function over a row. Example:

                         ( 1 2 3 )   ( 1 2 3 )
                         ( 4 5 6 )   ( 5 6 7 )
mapRow (\_ x -> x + 1) 2 ( 7 8 9 ) = ( 7 8 9 )

mapCol

Arguments

:: (Int -> a -> a)

Function takes the current row as additional argument.

-> Int

Column to map.

-> Matrix a

-> Matrix a

O(rows*cols). Map a function over a column. Example:

                         ( 1 2 3 )   ( 1 3 3 )
                         ( 4 5 6 )   ( 4 6 6 )
mapCol (\_ x -> x + 1) 2 ( 7 8 9 ) = ( 7 9 9 )

Submatrices

Splitting blocks

submatrix

Arguments

:: Int

Starting row

-> Int

Ending row

-> Int

Starting column

-> Int

Ending column

-> Matrix a

-> Matrix a

O(1). Extract a submatrix given row and column limits. Example:

                  ( 1 2 3 )
                  ( 4 5 6 )   ( 2 3 )
submatrix 1 2 2 3 ( 7 8 9 ) = ( 5 6 )

minorMatrix

Arguments

:: Int

Row r to remove.

-> Int

Column c to remove.

-> Matrix a

Original matrix.

-> Matrix a

Matrix with row r and column c removed.

O(rows*cols). Remove a row and a column from a matrix. Example:

                ( 1 2 3 )
                ( 4 5 6 )   ( 1 3 )
minorMatrix 2 2 ( 7 8 9 ) = ( 7 9 )

splitBlocks

Arguments

:: Int

Row of the splitting element.

-> Int

Column of the splitting element.

-> Matrix a

Matrix to split.

-> (Matrix a, Matrix a, Matrix a, Matrix a)

(TL,TR,BL,BR)

O(1). Make a block-partition of a matrix using a given element as reference. The element will stay in the bottom-right corner of the top-left corner matrix.

                (             )   (      |      )
                (             )   ( ...  | ...  )
                (    x        )   (    x |      )
splitBlocks i j (             ) = (-------------) , where x = a_{i,j}
                (             )   (      |      )
                (             )   ( ...  | ...  )
                (             )   (      |      )

Note that some blocks can end up empty. We use the following notation for these blocks:

( TL | TR )
(---------)
( BL | BR )

Where T = Top, B = Bottom, L = Left, R = Right.

Joining blocks

(<|>) :: Matrix a -> Matrix a -> Matrix a

Horizontally join two matrices. Visually:

( A ) <|> ( B ) = ( A | B )

Where both matrices A and B have the same number of rows. This condition is not checked.

(<->) :: Matrix a -> Matrix a -> Matrix a

Vertically join two matrices. Visually:

                  ( A )
( A ) <-> ( B ) = ( - )
                  ( B )

Where both matrices A and B have the same number of columns. This condition is not checked.

joinBlocks :: (Matrix a, Matrix a, Matrix a, Matrix a) -> Matrix a

Join blocks of the form detailed in splitBlocks. Precisely:

joinBlocks (tl,tr,bl,br) =
  (tl <|> tr)
      <->
  (bl <|> br)

Matrix operations

elementwise :: (a -> b -> c) -> Matrix a -> Matrix b -> Matrix c

Perform an operation element-wise. The second matrix must have at least as many rows and columns as the first matrix. If it's bigger, the leftover items will be ignored. If it's smaller, it will cause a run-time error. You may want to use elementwiseUnsafe if you are definitely sure that a run-time error won't arise.

elementwiseUnsafe :: (a -> b -> c) -> Matrix a -> Matrix b -> Matrix c

Unsafe version of elementwise, but faster.

Matrix multiplication

About matrix multiplication

Four methods are provided for matrix multiplication.

• multStd: Matrix multiplication following directly the definition. This is the best choice when you know for sure that your matrices are small.
• multStd2: Matrix multiplication following directly the definition. However, using a different definition from multStd. According to our benchmarks with this version, multStd2 is around 3 times faster than multStd.
• multStrassen: Matrix multiplication following the Strassen's algorithm. Complexity grows slower but also some work is added partitioning the matrix. Also, it only works on square matrices of order 2^n, so if this condition is not met, it is zero-padded until this is accomplished. Therefore, its use is not recommended.
• multStrassenMixed: This function mixes the previous methods. It provides a better performance in general. Method (*) of the Num class uses this function because it gives the best average performance. However, if you know for sure that your matrices are small (size less than 500x500), you should use multStd or multStd2 instead, since multStrassenMixed is going to switch to those functions anyway.

We keep researching how to get better performance for matrix multiplication. If you want to be on the safe side, use (*).

Functions

multStd :: Num a => Matrix a -> Matrix a -> Matrix a

Standard matrix multiplication by definition.

multStd2 :: Num a => Matrix a -> Matrix a -> Matrix a

Standard matrix multiplication by definition.

multStrassen :: Num a => Matrix a -> Matrix a -> Matrix a

Strassen's matrix multiplication.

multStrassenMixed :: Num a => Matrix a -> Matrix a -> Matrix a

Mixed Strassen's matrix multiplication.

Linear transformations

scaleMatrix :: Num a => a -> Matrix a -> Matrix a

Scale a matrix by a given factor. Example:

              ( 1 2 3 )   (  2  4  6 )
              ( 4 5 6 )   (  8 10 12 )
scaleMatrix 2 ( 7 8 9 ) = ( 14 16 18 )

scaleRow :: Num a => a -> Int -> Matrix a -> Matrix a

Scale a row by a given factor. Example:

             ( 1 2 3 )   (  1  2  3 )
             ( 4 5 6 )   (  8 10 12 )
scaleRow 2 2 ( 7 8 9 ) = (  7  8  9 )

combineRows :: Num a => Int -> a -> Int -> Matrix a -> Matrix a

Add to one row a scalar multiple of another row. Example:

                  ( 1 2 3 )   (  1  2  3 )
                  ( 4 5 6 )   (  6  9 12 )
combineRows 2 2 1 ( 7 8 9 ) = (  7  8  9 )

switchRows

Arguments

:: Int

Row 1.

-> Int

Row 2.

-> Matrix a

Original matrix.

-> Matrix a

Matrix with rows 1 and 2 switched.

Switch two rows of a matrix. Example:

               ( 1 2 3 )   ( 4 5 6 )
               ( 4 5 6 )   ( 1 2 3 )
switchRows 1 2 ( 7 8 9 ) = ( 7 8 9 )

switchCols

Arguments

:: Int

Col 1.

-> Int

Col 2.

-> Matrix a

Original matrix.

-> Matrix a

Matrix with cols 1 and 2 switched.

Switch two coumns of a matrix. Example:

               ( 1 2 3 )   ( 2 1 3 )
               ( 4 5 6 )   ( 5 4 6 )
switchCols 1 2 ( 7 8 9 ) = ( 8 7 9 )

Decompositions

luDecomp :: (Ord a, Fractional a) => Matrix a -> Maybe (Matrix a, Matrix a, Matrix a, a)

Matrix LU decomposition with partial pivoting. The result for a matrix M is given in the format (U,L,P,d) where:

• U is an upper triangular matrix.
• L is an unit lower triangular matrix.
• P is a permutation matrix.
• d is the determinant of P.
• PM = LU.

These properties are only guaranteed when the input matrix is invertible. An additional property matches thanks to the strategy followed for pivoting:

• L_(i,j) <= 1, for all i,j.

This follows from the maximal property of the selected pivots, which also leads to a better numerical stability of the algorithm.

Example:

         ( 1 2 0 )     ( 2 0  2 )   (   1 0 0 )   ( 0 0 1 )
         ( 0 2 1 )     ( 0 2 -1 )   ( 1/2 1 0 )   ( 1 0 0 )
luDecomp ( 2 0 2 ) = ( ( 0 0  2 ) , (   0 1 1 ) , ( 0 1 0 ) , 1 )

Nothing is returned if no LU decomposition exists.

luDecompUnsafe :: (Ord a, Fractional a) => Matrix a -> (Matrix a, Matrix a, Matrix a, a)

Unsafe version of luDecomp. It fails when the input matrix is singular.

luDecomp' :: (Ord a, Fractional a) => Matrix a -> Maybe (Matrix a, Matrix a, Matrix a, Matrix a, a, a)

Matrix LU decomposition with complete pivoting. The result for a matrix M is given in the format (U,L,P,Q,d,e) where:

• U is an upper triangular matrix.
• L is an unit lower triangular matrix.
• P,Q are permutation matrices.
• d,e are the determinants of P and Q respectively.
• PMQ = LU.

These properties are only guaranteed when the input matrix is invertible. An additional property matches thanks to the strategy followed for pivoting:

• L_(i,j) <= 1, for all i,j.

This follows from the maximal property of the selected pivots, which also leads to a better numerical stability of the algorithm.

Example:

          ( 1 0 )     ( 2 1 )   (   1    0 0 )   ( 0 0 1 )
          ( 0 2 )     ( 0 2 )   (   0    1 0 )   ( 0 1 0 )   ( 1 0 )
luDecomp' ( 2 1 ) = ( ( 0 0 ) , ( 1/2 -1/4 1 ) , ( 1 0 0 ) , ( 0 1 ) , -1 , 1 )

Nothing is returned if no LU decomposition exists.

luDecompUnsafe' :: (Ord a, Fractional a) => Matrix a -> (Matrix a, Matrix a, Matrix a, Matrix a, a, a)

Unsafe version of luDecomp'. It fails when the input matrix is singular.

cholDecomp :: Floating a => Matrix a -> Matrix a

Simple Cholesky decomposition of a symmetric, positive definite matrix. The result for a matrix M is a lower triangular matrix L such that:

• M = LL^T.

Example:

           (  2 -1  0 )   (  1.41  0     0    )
           ( -1  2 -1 )   ( -0.70  1.22  0    )
cholDecomp (  0 -1  2 ) = (  0.00 -0.81  1.15 )

Properties

trace :: Num a => Matrix a -> a

Sum of the elements in the diagonal. See also getDiag. Example:

      ( 1 2 3 )
      ( 4 5 6 )
trace ( 7 8 9 ) = 15

diagProd :: Num a => Matrix a -> a

Product of the elements in the diagonal. See also getDiag. Example:

         ( 1 2 3 )
         ( 4 5 6 )
diagProd ( 7 8 9 ) = 45

Determinants

detLaplace :: Num a => Matrix a -> a

Matrix determinant using Laplace expansion. If the elements of the Matrix are instance of Ord and Fractional consider to use detLU in order to obtain better performance. Function detLaplace is extremely slow.

detLU :: (Ord a, Fractional a) => Matrix a -> a

Matrix determinant using LU decomposition. It works even when the input matrix is singular.

flatten :: Matrix (Matrix a) -> Matrix a

Flatten a matrix of matrices. All sub matrices must have same dimensions This criteria is not checked.