Data.Matrix.Static

Data.Matrix.Static wraps matrix's Data.Matrix functions and adds size information on the type level. The name of the functions are mostly the same as in Data.Matrix. Exceptions are, when there is a safer version of a function due to the additional type-level information. In that case, there may be an unsafe variant of the function with the postfix Unsafe.

Matrix type

data Matrix (m :: Nat) (n :: Nat) (a :: Type)

A matrix over the type f with m rows and n columns. This just wraps the Matrix constructor and adds size information to the type

Instances

Functor (Matrix m n)

Defined in Data.Matrix.Static

Methods

fmap :: (a -> b) -> Matrix m n a -> Matrix m n b

(<$) :: a -> Matrix m n b -> Matrix m n a

Applicative (Matrix m n)

Defined in Data.Matrix.Static

Methods

pure :: a -> Matrix m n a

(<*>) :: Matrix m n (a -> b) -> Matrix m n a -> Matrix m n b

liftA2 :: (a -> b -> c) -> Matrix m n a -> Matrix m n b -> Matrix m n c

(*>) :: Matrix m n a -> Matrix m n b -> Matrix m n b

(<*) :: Matrix m n a -> Matrix m n b -> Matrix m n a

Foldable (Matrix m n)

Defined in Data.Matrix.Static

Methods

fold :: Monoid m0 => Matrix m n m0 -> m0

foldMap :: Monoid m0 => (a -> m0) -> Matrix m n a -> m0

foldr :: (a -> b -> b) -> b -> Matrix m n a -> b

foldr' :: (a -> b -> b) -> b -> Matrix m n a -> b

foldl :: (b -> a -> b) -> b -> Matrix m n a -> b

foldl' :: (b -> a -> b) -> b -> Matrix m n a -> b

foldr1 :: (a -> a -> a) -> Matrix m n a -> a

foldl1 :: (a -> a -> a) -> Matrix m n a -> a

toList :: Matrix m n a -> [a]

null :: Matrix m n a -> Bool

length :: Matrix m n a -> Int

elem :: Eq a => a -> Matrix m n a -> Bool

maximum :: Ord a => Matrix m n a -> a

minimum :: Ord a => Matrix m n a -> a

sum :: Num a => Matrix m n a -> a

product :: Num a => Matrix m n a -> a

Traversable (Matrix m n)

Defined in Data.Matrix.Static

Methods

traverse :: Applicative f => (a -> f b) -> Matrix m n a -> f (Matrix m n b)

sequenceA :: Applicative f => Matrix m n (f a) -> f (Matrix m n a)

mapM :: Monad m0 => (a -> m0 b) -> Matrix m n a -> m0 (Matrix m n b)

sequence :: Monad m0 => Matrix m n (m0 a) -> m0 (Matrix m n a)

Eq a => Eq (Matrix m n a)

Defined in Data.Matrix.Static

Methods

(==) :: Matrix m n a -> Matrix m n a -> Bool

(/=) :: Matrix m n a -> Matrix m n a -> Bool

Num f => Num (Matrix m n f)

Defined in Data.Matrix.Static

Methods

(+) :: Matrix m n f -> Matrix m n f -> Matrix m n f

(-) :: Matrix m n f -> Matrix m n f -> Matrix m n f

(*) :: Matrix m n f -> Matrix m n f -> Matrix m n f

negate :: Matrix m n f -> Matrix m n f

abs :: Matrix m n f -> Matrix m n f

signum :: Matrix m n f -> Matrix m n f

fromInteger :: Integer -> Matrix m n f

Ord f => Ord (Matrix m n f)

Defined in Data.Matrix.Static

Methods

compare :: Matrix m n f -> Matrix m n f -> Ordering

(<) :: Matrix m n f -> Matrix m n f -> Bool

(<=) :: Matrix m n f -> Matrix m n f -> Bool

(>) :: Matrix m n f -> Matrix m n f -> Bool

(>=) :: Matrix m n f -> Matrix m n f -> Bool

max :: Matrix m n f -> Matrix m n f -> Matrix m n f

min :: Matrix m n f -> Matrix m n f -> Matrix m n f

Show f => Show (Matrix m n f)

Defined in Data.Matrix.Static

Methods

showsPrec :: Int -> Matrix m n f -> ShowS

show :: Matrix m n f -> String

showList :: [Matrix m n f] -> ShowS

Monoid a => Semigroup (Matrix m n a)

Defined in Data.Matrix.Static

Methods

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

sconcat :: NonEmpty (Matrix m n a) -> Matrix m n a

stimes :: Integral b => b -> Matrix m n a -> Matrix m n a

Monoid a => Monoid (Matrix m n a)

Defined in Data.Matrix.Static

Methods

mempty :: Matrix m n a

mappend :: Matrix m n a -> Matrix m n a -> Matrix m n a

mconcat :: [Matrix m n a] -> Matrix m n a

NFData a => NFData (Matrix m n a)

Defined in Data.Matrix.Static

Methods

rnf :: Matrix m n a -> ()

prettyMatrix :: forall m n a. Show a => Matrix m n a -> String

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

nrows :: forall m n a. KnownNat m => Matrix m n a -> Int

ncols :: forall m n a. KnownNat n => Matrix m n a -> Int

forceMatrix :: forall m n a. Matrix m n a -> Matrix m n 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

:: (KnownNat m, KnownNat n)

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

Generator function

-> Matrix m n a

O(rows*cols). Generate a matrix from a generator function. | The elements are 1-indexed, i.e. top-left element is (1,1). Example of usage:

matrix (\(i,j) -> 2*i - j) :: Matrix 2 4 Int
( 1  0 -1 -2 )
( 3  2  1  0 )

rowVector :: forall m a. KnownNat m => Vector a -> Maybe (RowVector m a)

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

colVector :: forall n a. KnownNat n => Vector a -> Maybe (ColumnVector n a)

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

Special matrices

zero :: forall m n a. (Num a, KnownNat n, KnownNat m) => Matrix m n a

O(rows*cols). The zero matrix This produces a zero matrix of the size given by the type. Often, the correct dimensions can be inferred by the compiler. If you want a specific size, give a type.

zero :: Matrix 2 2 Int
( 0 0 )
( 0 0 )

identity :: forall n a. (Num a, KnownNat n) => Matrix n n a

O(rows*cols). Identity matrix

identitiy @n =
( 1 0 0 ... 0 0 )
( 0 1 0 ... 0 0 )
(       ...     )
( 0 0 0 ... 1 0 )
( 0 0 0 ... 0 1 )

diagonal

Arguments

:: KnownNat n

=> a

Default element

-> Vector a

Diagonal vector

-> Maybe (Matrix n n a)

Similar to diagonalList, but using Vector, which should be more efficient. The size of the vector is not checked and will lead to an exception if it's not of size n.

diagonalUnsafe

Arguments

:: a

Default element

-> Vector a

Diagonal vector

-> Matrix n n a

Similar to diagonalList, but using Vector, which should be more efficient. The size of the vector is not checked and will lead to an exception if it's not of size n.

permMatrix :: forall n i j a. (Num a, KnownNat n, KnownNat i, KnownNat j, 1 <= i, i <= n, 1 <= j, j <= n) => Matrix n n a

O(rows*cols). Permutation matrix. The parameters are given as type level Nats. To use this, use -XDataKinds and -XTypeApplications. The first type parameter gives the matrix' size, the two following give the rows (or columns) to permute.

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.

permMatrixUnsafe

Arguments

:: (Num a, KnownNat n)

=> Int

Permuted row 1.

-> Int

Permuted row 2.

-> Matrix n n a

Permutation matrix.

O(rows*cols). Permutation matrix. The values of the row and column identifiers are not checked and if they are out of range (not between 1 and n) an exception will be thrown.

permMatrixUnsafe @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 :: forall m n a. (KnownNat m, KnownNat n) => [a] -> Maybe (Matrix m n a)

Create a matrix from a list of elements. The list must have exactly length n*m or this returns Nothing. An example:

fromList [1..9] :: Maybe (Matrix 3 3 Int)
Just ( 1 2 3 )
     ( 4 5 6 )
     ( 7 8 9 )

fromListUnsafe

Arguments

:: (KnownNat m, KnownNat n)

=> [a]

List of elements

-> Matrix m n a

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

fromListUnsafe [1..9] :: Matrix 3 3 Int
( 1 2 3 )
( 4 5 6 )
( 7 8 9 )

fromLists :: forall m n a. (KnownNat m, KnownNat n) => [[a]] -> Maybe (Matrix m n a)

Create a matrix from a list of rows. The list must have exactly m lists of length n. Nothing is returned otherwise Example:

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

fromListsUnsafe :: [[a]] -> Matrix m n a

Create a matrix from a list of rows. The list must have exactly m lists of length n. If this does not hold, the resulting Matrix will have different static dimensions that the runtime dimension and will result in hard to debug errors. Use fromLists whenever you're unsure. Example:

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

toList :: forall m n a. Matrix m n a -> [a]

Get the elements of a matrix stored in a list.

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

toLists :: forall m n a. Matrix m n 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

:: (KnownNat i, KnownNat j, 1 <= i, i <= m, 1 <= j, j <= n)

=> Matrix m n a

Matrix

-> a

O(1). Get an element of a matrix. Indices range from (1,1) to (m,n). The parameters are given as type level Nats. To use this, use -XDataKinds and -XTypeApplications.

The type parameters are: row, column

Example:

              ( 1 2 )
getElem @2 @1 ( 3 4 ) = 3

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

Short alias for unsafeGet. Careful: This has no bounds checking This deviates from Data.Matrix, where (!) does check bounds on runtime.

unsafeGet

Arguments

:: Int

Row

-> Int

Column

-> Matrix m n a

Matrix

-> a

O(1). Unsafe variant of getElem. This will do no bounds checking

(!.) :: Matrix m n a -> (Int, Int) -> a

Alias for '(!)'. This exists to keep the interface similar to Data.Matrix but serves no other purpose. Use '(!)' (or even better getElem) instead.

safeGet :: forall m n a. (KnownNat n, KnownNat m) => Int -> Int -> Matrix m n a -> Maybe a

Variant of unsafeGet that returns Maybe instead of an error.

safeSet :: forall m n a. a -> (Int, Int) -> Matrix m n a -> Maybe (Matrix m n a)

Variant of setElem that returns Maybe instead of an error.

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

O(1). Get a row of a matrix as a vector. The range of the input is not checked and must be between 1 and m

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

O(1). Get a column of a matrix as a vector. The range of the input is not checked and must be between 1 and n

safeGetRow :: Int -> Matrix m n a -> Maybe (Vector a)

Varian of getRow that returns a maybe instead of an error Only available when used with matrix >= 0.3.6!

safeGetCol :: Int -> Matrix m n a -> Maybe (Vector a)

Variant of getCol that returns a maybe instead of an error Only available when used with matrix >= 0.3.6!

getDiag :: Matrix m n a -> Vector a

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

getMatrixAsVector :: Matrix m n 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

(.*) :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a

Type safe matrix multiplication This is called (*) in matrix. Since the dimensions of the input matrices differ, they are not the same type and we cannot use Num's (*)

(^*) :: forall m n a. Num a => a -> Matrix m n a -> Matrix m n a

Type safe scalar multiplication

setElem

Arguments

:: (KnownNat i, KnownNat j, 1 <= i, i <= m, 1 <= j, j <= n)

=> a

New value.

-> Matrix m n a

Original matrix.

-> Matrix m n a

Matrix with the given position replaced with the given value.

Replace the value of a cell in a matrix. The position to be replaced is given by TypeLevel Nats. To use this, use -XDataKinds and -XTypeApplications.

Example: setElem 1 2 0 (1 2 3) = (1 0 3)

unsafeSet

Arguments

:: a

New value.

-> (Int, Int)

Position to replace.

-> Matrix m n a

Original matrix.

-> Matrix m n a

Matrix with the given position replaced with the given value.

Unsafe variant of setElem, without bounds checking.

transpose :: forall m n a. Matrix m n a -> Matrix n m 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

:: (KnownNat newM, KnownNat newN, 1 <= newM, 1 <= newN)

=> a

Default element.

-> Matrix m n a

-> Matrix newM newN 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

:: (KnownNat newM, KnownNat newN, n <= newN, m <= newM)

=> a

Element to add when extending.

-> Matrix m n a

-> Matrix newM newN a

Extend a matrix to the expected size adding a default element. If the matrix already has the required size, nothing happens. Example:

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

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

O(rows^4). The inverse of a square matrix Uses naive Gaussian elimination formula.

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

O(rows*rows*cols*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

:: (KnownNat i, KnownNat m, 1 <= i, i <= m)

=> (Int -> a -> a)

Function takes the current column as additional argument.

-> Matrix m n a

-> Matrix m n a

O(rows*cols). Map a function over a row. The row to map is given by a TypeLevel Nat. To use this, use -XDataKinds and -XTypeApplications. Example:

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

mapRowUnsafe

Arguments

:: (Int -> a -> a)

Function takes the current column as additional argument.

-> Int

Row to map.

-> Matrix m n a

-> Matrix m n a

O(rows*cols). Map a function over a row. The bounds of the row parameter is not checked and might throw an error. Example:

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

mapCol

Arguments

:: (KnownNat j, KnownNat m, 1 <= j, j <= n)

=> (Int -> a -> a)

Function takes the current column as additional argument.

-> Matrix m n a

-> Matrix m n a

O(rows*cols). Map a function over a column. The row to map is given by a TypeLevel Nat. To use this, use -XDataKinds and -XTypeApplications. Example:

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

mapColUnsafe

Arguments

:: (Int -> a -> a)

Function takes the current column as additional argument.

-> Int

Row to map.

-> Matrix m n a

-> Matrix m n a

O(rows*cols). Map a function over a column. The bounds of the row parameter is not checked and might throw an error. Example:

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

mapPos

Arguments

:: ((Int, Int) -> a -> b)

Function takes the current Position as additional argument.

-> Matrix m n a

-> Matrix m n b

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

                           ( 1 2 3 )   ( 0 -1 -2 )
                           ( 4 5 6 )   ( 1  0 -1 )
mapPos (\(r,c) _ -> r - c) ( 7 8 9 ) = ( 2  1  0 )

Only available when used with matrix >= 0.3.6!

Submatrices

Splitting blocks

submatrix :: forall iFrom jFrom iTo jTo m n a. (KnownNat iFrom, KnownNat iTo, KnownNat jFrom, KnownNat jTo, 1 <= iFrom, 1 <= ((iTo - iFrom) + 1), ((iTo - iFrom) + 1) <= m, 1 <= jFrom, 1 <= ((jTo - jFrom) + 1), ((jTo - jFrom) + 1) <= n) => Matrix m n a -> Matrix ((iTo - iFrom) + 1) ((jTo - jFrom) + 1) a

O(1). Extract a submatrix from the given position. The type parameters expected are the starting and ending indices of row and column elements.

submatrixUnsafe

Arguments

:: (KnownNat rows, KnownNat cols, 1 <= rows, rows <= m, 1 <= cols, cols <= n)

=> Int

Starting row

-> Int

Starting column

-> Matrix m n a

-> Matrix rows cols a

O(1). Extract a submatrix from the given position. The type parameters are the dimension of the returned matrix, the run-time indices are the indiced of the top-left element of the new matrix. Example:

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

minorMatrix

Arguments

:: (KnownNat delRow, KnownNat delCol, 1 <= delRow, 1 <= delCol, delRow <= m, delCol <= n, 2 <= n, 2 <= m)

=> Matrix m n a

Original matrix.

-> Matrix (m - 1) (n - 1) 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 )

minorMatrixUnsafe

Arguments

:: (2 <= n, 2 <= m)

=> Int

Row r to remove.

-> Int

Column c to remove.

-> Matrix m n a

Original matrix.

-> Matrix (m - 1) (n - 1) 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 )
minorMatrixUnsafe 2 2 ( 7 8 9 ) = ( 7 9 )

splitBlocks

Arguments

:: (KnownNat i, KnownNat j, 1 <= i, (i + 1) <= m, 1 <= j, (j + 1) <= n)

=> Matrix m n a

Matrix to split.

-> (Matrix i j a, Matrix i (n - j) a, Matrix (n - i) j a, Matrix (m - i) (n - j) 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. This means, the ranges of the pivot elements positions are \[ i <- [1..m-1], j <- [1..n-1] \]

                  (             )   ( TR   | TL   )
                  (             )   ( ...  | ...  )
                  (    x        )   (    x |      )
splitBlocks @i @j (             ) = (-------------) , where x = a_{i,j}
                  (             )   ( BL   | BR   )
                  (             )   ( ...  | ...  )
                  (             )   (      |      )

Note that contrary to the matrix version of this function, blocks will never be empty. Also, because of TypeLits not providing proper dependent types, there is no way to have a type safe variant of this functon where the pivot element is given at run-time.

Joining blocks

(<|>) :: forall m n k a. Matrix m n a -> Matrix m k a -> Matrix m (k + n) a

Horizontally join two matrices. Visually:

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

(<->) :: forall m k n a. Matrix m n a -> Matrix k n a -> Matrix (m + k) n a

Horizontally join two matrices. Visually:

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

joinBlocks :: forall mt mb nl nr a. (1 <= mt, 1 <= mb, 1 <= nl, 1 <= nr) => (Matrix mt nl a, Matrix mt nr a, Matrix mb nl a, Matrix mb nr a) -> Matrix (mt + mb) (nl + nr) a

Join blocks of the form detailed in splitBlocks. Precisely:

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

Matrix operations

elementwise :: forall m n a b c. (a -> b -> c) -> Matrix m n a -> Matrix m n b -> Matrix m n c

Perform an operation element-wise. This uses matrix's elementwiseUnsafe since we can guarantee proper dimensions at compile time.

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 a) 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 :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a

Standard matrix multiplication by definition.

multStd2 :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a

Standard matrix multiplication by definition.

multStrassen :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a

Strassen's matrix multiplication.

multStrassenMixed :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a

Mixed Strassen's matrix multiplication.

Linear transformations

scaleMatrix :: Num a => a -> Matrix m n a -> Matrix m n 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 :: forall i m n a. (KnownNat i, Num a) => a -> Matrix m n a -> Matrix m n a

Scale a row by a given factor. The input row is not checked for validity. Example:

              ( 1 2 3 )   (  1  2  3 )
              ( 4 5 6 )   ( 12 15 18 )
scaleRow @2 3 ( 7 8 9 ) = (  7  8  9 )

scaleRowUnsafe :: Num a => a -> Int -> Matrix m n a -> Matrix m n a

Scale a row by a given factor. The input row is not checked for validity. Example:

                   ( 1 2 3 )   (  1  2  3 )
                   ( 4 5 6 )   ( 12 15 18 )
scaleRowUnsafe 3 2 ( 7 8 9 ) = (  7  8  9 )

combineRows :: forall i k m n a. (KnownNat i, KnownNat k, Num a) => a -> Matrix m n a -> Matrix m n 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 @1 2 ( 7 8 9 ) = (  7  8  9 )

combineRowsUnsafe :: Num a => Int -> a -> Int -> Matrix m n a -> Matrix m n a

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

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

switchRows

Arguments

:: (KnownNat i, KnownNat k, 1 <= i, i <= m, 1 <= k, k <= m)

=> Matrix m n a

Original matrix.

-> Matrix m n 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 )

switchRowsUnsafe

Arguments

:: Int

Row 1.

-> Int

Row 2.

-> Matrix m n a

Original matrix.

-> Matrix m n a

Matrix with rows 1 and 2 switched.

Switch two rows of a matrix. The validity of the input row numbers is not checked Example:

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

switchCols

Arguments

:: (KnownNat i, KnownNat k, 1 <= i, i <= n, 1 <= k, k <= n)

=> Matrix m n a

Original matrix.

-> Matrix m n 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 )

switchColsUnsafe

Arguments

:: Int

Col 1.

-> Int

Col 2.

-> Matrix m n a

Original matrix.

-> Matrix m n a

Matrix with cols 1 and 2 switched.

Switch two coumns of a matrix. The validity of the input column numbers is not checked. Example:

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

Decompositions

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

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

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

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

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

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

flatten :: forall m' n' m n a. Matrix m' n' (Matrix m n a) -> Matrix (m' * m) (n' * n) a

Flatten a matrix of matrices.

Helper functions

applyUnary :: forall m n m' n' a b. (Matrix a -> Matrix b) -> Matrix m n a -> Matrix m' n' b

Apply a map function to the unsafe inner matrix type.

applyBinary :: forall m n m' n' m'' n'' a b. (Matrix a -> Matrix a -> Matrix b) -> Matrix m n a -> Matrix m' n' a -> Matrix m'' n'' b

Transform a binary unstatic function to a binary static function.

unpackStatic :: forall m n a. Matrix m n a -> Matrix a

Forget static information about a matrix. This converts this converts the Matrix type to Data.Matrix.Matrix