Copyright | (c) Wanja Chresta 2018 |
---|---|

License | BSD-3 |

Maintainer | wanja dot hs at chrummibei dot ch |

Stability | experimental |

Portability | POSIX |

Safe Haskell | None |

Language | Haskell2010 |

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`

.

## Synopsis

- data Matrix (m :: Nat) (n :: Nat) (a :: Type)
- prettyMatrix :: forall m n a. Show a => Matrix m n a -> String
- 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
- matrix :: forall m n a. (KnownNat m, KnownNat n) => ((Int, Int) -> a) -> Matrix m n a
- type RowVector = Matrix 1
- rowVector :: forall m a. KnownNat m => Vector a -> Maybe (RowVector m a)
- type ColumnVector m = Matrix m 1
- colVector :: forall n a. KnownNat n => Vector a -> Maybe (ColumnVector n a)
- zero :: forall m n a. (Num a, KnownNat n, KnownNat m) => Matrix m n a
- identity :: forall n a. (Num a, KnownNat n) => Matrix n n a
- diagonal :: forall n a. KnownNat n => a -> Vector a -> Maybe (Matrix n n a)
- diagonalUnsafe :: forall n a. a -> Vector a -> Matrix n n a
- 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
- permMatrixUnsafe :: forall n a. (Num a, KnownNat n) => Int -> Int -> Matrix n n a
- fromList :: forall m n a. (KnownNat m, KnownNat n) => [a] -> Maybe (Matrix m n a)
- fromListUnsafe :: forall m n a. (KnownNat m, KnownNat n) => [a] -> Matrix m n a
- fromLists :: forall m n a. (KnownNat m, KnownNat n) => [[a]] -> Maybe (Matrix m n a)
- fromListsUnsafe :: [[a]] -> Matrix m n a
- toList :: forall m n a. Matrix m n a -> [a]
- toLists :: forall m n a. Matrix m n a -> [[a]]
- getElem :: forall i j m n a. (KnownNat i, KnownNat j, 1 <= i, i <= m, 1 <= j, j <= n) => Matrix m n a -> a
- (!) :: Matrix m n a -> (Int, Int) -> a
- unsafeGet :: Int -> Int -> Matrix m n a -> a
- (!.) :: Matrix m n a -> (Int, Int) -> a
- safeGet :: forall m n a. (KnownNat n, KnownNat m) => Int -> Int -> Matrix m n a -> Maybe a
- safeSet :: forall m n a. a -> (Int, Int) -> Matrix m n a -> Maybe (Matrix m n a)
- getRow :: Int -> Matrix m n a -> Vector a
- getCol :: Int -> Matrix m n a -> Vector a
- safeGetRow :: Int -> Matrix m n a -> Maybe (Vector a)
- safeGetCol :: Int -> Matrix m n a -> Maybe (Vector a)
- getDiag :: Matrix m n a -> Vector a
- getMatrixAsVector :: Matrix m n a -> Vector a
- (.*) :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a
- (^*) :: forall m n a. Num a => a -> Matrix m n a -> Matrix m n a
- setElem :: forall i j m n a. (KnownNat i, KnownNat j, 1 <= i, i <= m, 1 <= j, j <= n) => a -> Matrix m n a -> Matrix m n a
- unsafeSet :: a -> (Int, Int) -> Matrix m n a -> Matrix m n a
- transpose :: forall m n a. Matrix m n a -> Matrix n m a
- setSize :: forall newM newN m n a. (KnownNat newM, KnownNat newN, 1 <= newM, 1 <= newN) => a -> Matrix m n a -> Matrix newM newN a
- extendTo :: forall newM newN m n a. (KnownNat newM, KnownNat newN, n <= newN, m <= newM) => a -> Matrix m n a -> Matrix newM newN a
- inverse :: forall n a. (Fractional a, Eq a) => Matrix n n a -> Either String (Matrix n n a)
- rref :: (Fractional a, Eq a) => Matrix m n a -> Either String (Matrix m n a)
- mapRow :: forall i m n a. (KnownNat i, KnownNat m, 1 <= i, i <= m) => (Int -> a -> a) -> Matrix m n a -> Matrix m n a
- mapRowUnsafe :: forall m n a. (Int -> a -> a) -> Int -> Matrix m n a -> Matrix m n a
- mapCol :: forall j m n a. (KnownNat j, KnownNat m, 1 <= j, j <= n) => (Int -> a -> a) -> Matrix m n a -> Matrix m n a
- mapColUnsafe :: forall m n a. (Int -> a -> a) -> Int -> Matrix m n a -> Matrix m n a
- mapPos :: ((Int, Int) -> a -> b) -> Matrix m n a -> Matrix m n b
- 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
- submatrixUnsafe :: forall rows cols m n a. (KnownNat rows, KnownNat cols, 1 <= rows, rows <= m, 1 <= cols, cols <= n) => Int -> Int -> Matrix m n a -> Matrix rows cols a
- minorMatrix :: forall delRow delCol m n a. (KnownNat delRow, KnownNat delCol, 1 <= delRow, 1 <= delCol, delRow <= m, delCol <= n, 2 <= n, 2 <= m) => Matrix m n a -> Matrix (m - 1) (n - 1) a
- minorMatrixUnsafe :: (2 <= n, 2 <= m) => Int -> Int -> Matrix m n a -> Matrix (m - 1) (n - 1) a
- splitBlocks :: forall mt nl mb nr a. (KnownNat mt, KnownNat nl, 1 <= mt, 1 <= mb, 1 <= nl, 1 <= nr) => Matrix (mt + mb) (nl + nr) a -> (Matrix mt nl a, Matrix mt nr a, Matrix mb nl a, Matrix mb nr a)
- (<|>) :: forall m n k a. Matrix m n a -> Matrix m k a -> Matrix m (k + n) a
- (<->) :: forall m k n a. Matrix m n a -> Matrix k n a -> Matrix (m + k) n a
- 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
- elementwise :: forall m n a b c. (a -> b -> c) -> Matrix m n a -> Matrix m n b -> Matrix m n c
- multStd :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a
- multStd2 :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a
- multStrassen :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a
- multStrassenMixed :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a
- scaleMatrix :: Num a => a -> Matrix m n a -> Matrix m n a
- scaleRow :: forall i m n a. (KnownNat i, Num a) => a -> Matrix m n a -> Matrix m n a
- scaleRowUnsafe :: Num a => a -> Int -> Matrix m n a -> Matrix m n a
- combineRows :: forall i k m n a. (KnownNat i, KnownNat k, Num a) => a -> Matrix m n a -> Matrix m n a
- combineRowsUnsafe :: Num a => Int -> a -> Int -> Matrix m n a -> Matrix m n a
- switchRows :: forall i k m n a. (KnownNat i, KnownNat k, 1 <= i, i <= m, 1 <= k, k <= m) => Matrix m n a -> Matrix m n a
- switchRowsUnsafe :: Int -> Int -> Matrix m n a -> Matrix m n a
- switchCols :: forall i k m n a. (KnownNat i, KnownNat k, 1 <= i, i <= n, 1 <= k, k <= n) => Matrix m n a -> Matrix m n a
- switchColsUnsafe :: Int -> Int -> Matrix m n a -> Matrix m n a
- luDecomp :: (Ord a, Fractional a) => Matrix m n a -> Maybe (Matrix m n a, Matrix m n a, Matrix m n a, a)
- luDecompUnsafe :: (Ord a, Fractional a) => Matrix m n a -> (Matrix m n a, Matrix m n a, Matrix m n a, a)
- 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)
- 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)
- cholDecomp :: Floating a => Matrix n n a -> Matrix n n a
- trace :: Num a => Matrix m n a -> a
- diagProd :: Num a => Matrix m n a -> a
- detLaplace :: Num a => Matrix n n a -> a
- detLU :: (Ord a, Fractional a) => Matrix n n a -> a
- flatten :: forall m' n' m n a. Matrix m' n' (Matrix m n a) -> Matrix (m' * m) (n' * n) a
- applyUnary :: forall m n m' n' a b. (Matrix a -> Matrix b) -> Matrix m n a -> Matrix m' n' b
- 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
- unpackStatic :: forall m n a. Matrix m n a -> Matrix a

# Matrix type

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

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) Source # | |

Applicative (Matrix m n) Source # | |

Defined in Data.Matrix.Static | |

Foldable (Matrix m n) Source # | |

Defined in Data.Matrix.Static 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 # | |

Traversable (Matrix m n) Source # | |

Defined in Data.Matrix.Static | |

Eq a => Eq (Matrix m n a) Source # | |

Num f => Num (Matrix m n f) Source # | |

Defined in Data.Matrix.Static (+) :: 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) Source # | |

Defined in Data.Matrix.Static | |

Show f => Show (Matrix m n f) Source # | |

Monoid a => Semigroup (Matrix m n a) Source # | |

Monoid a => Monoid (Matrix m n a) Source # | |

NFData a => NFData (Matrix m n a) Source # | |

Defined in Data.Matrix.Static |

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

# Builders

*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) Source #

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

type ColumnVector m = Matrix m 1 Source #

A column vector (a matrix with one column).

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

*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 Source #

*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 Source #

*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 )

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.

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 Source #

*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`

.

*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) Source #

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 )

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) Source #

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 Source #

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] Source #

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]] Source #

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

*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 Source #

Short alias for `unsafeGet`

. Careful: This has no bounds checking
This deviates from `Data.Matrix`

, where (!) does check bounds on runtime.

*O(1)*. Unsafe variant of `getElem`

. This will do no bounds checking

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

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 Source #

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) Source #

Variant of `setElem`

that returns Maybe instead of an error.

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

*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 Source #

*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) Source #

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) Source #

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 Source #

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

getMatrixAsVector :: Matrix m n a -> Vector a Source #

# Manipulating matrices

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

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 Source #

Type safe scalar multiplication

:: (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)

:: 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 Source #

*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 )

:: (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.

:: (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) Source #

*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) Source #

*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).

:: (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 )

:: (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 )

:: (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 )

:: (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 )

:: ((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 Source #

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

:: (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 )

:: (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 |

*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 )

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

=> Int | Row |

-> Int | Column |

-> Matrix m n a | Original matrix. |

-> Matrix (m - 1) (n - 1) a | Matrix with row |

*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 )

:: (KnownNat mt, KnownNat nl, 1 <= mt, 1 <= mb, 1 <= nl, 1 <= nr) | |

=> Matrix (mt + mb) (nl + nr) a | Matrix to split. |

-> (Matrix mt nl a, Matrix mt nr a, Matrix mb nl a, Matrix mb nr 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 Source #

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 Source #

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 Source #

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 Source #

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 Source #

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 Source #

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 Source #

Strassen's matrix multiplication.

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

Mixed Strassen's matrix multiplication.

# Linear transformations

scaleMatrix :: Num a => a -> Matrix m n a -> Matrix m n a Source #

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 Source #

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 Source #

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 Source #

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 Source #

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 )

:: (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 )

:: 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 )

:: (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 )

:: 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) Source #

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) Source #

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) Source #

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) Source #

Unsafe version of `luDecomp'`

. It fails when the input matrix is singular.

cholDecomp :: Floating a => Matrix n n a -> Matrix n n a Source #

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 Source #

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 Source #

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 Source #

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 Source #

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 Source #

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 Source #

Apply a map function to the unsafe inner matrix type.