-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/
-- | Type-safe matrix operations
--
-- Please see the README on GitHub at
-- https://github.com/wchresta/matrix-static#readme
@package matrix-static
@version 0.2
-- | 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.
module Data.Matrix.Static
-- | 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
data Matrix (m :: Nat) (n :: Nat) (a :: Type)
-- | Display a matrix as a String using the Show instance of
-- its elements.
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
-- | O(rows*cols). Similar to force. It copies the matrix
-- content dropping any extra memory.
--
-- Useful when using submatrix from a big matrix.
forceMatrix :: forall m n a. Matrix m n a -> 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 )
--
matrix :: forall m n a. (KnownNat m, KnownNat n) => ((Int, Int) -> a) -> Matrix m n a
-- | O(1). Represent a vector as a one row matrix.
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(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 )
--
zero :: forall m n a. (Num a, KnownNat n, KnownNat m) => Matrix m 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 )
--
identity :: forall n a. (Num a, KnownNat n) => 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.
diagonal :: forall n a. KnownNat n => a -> Vector a -> 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 :: forall n a. a -> Vector a -> 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.
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 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.
permMatrixUnsafe :: forall n a. (Num a, KnownNat n) => Int -> Int -> Matrix n 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 )
--
fromList :: forall m n a. (KnownNat m, KnownNat n) => [a] -> Maybe (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 )
--
fromListUnsafe :: forall m n a. (KnownNat m, KnownNat n) => [a] -> 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 )
--
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. 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 )
--
fromListsUnsafe :: [[a]] -> Matrix m n a
-- | Get the elements of a matrix stored in a list.
--
--
-- ( 1 2 3 )
-- ( 4 5 6 )
-- toList ( 7 8 9 ) = [1..9]
--
toList :: 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] ]
--
toLists :: forall m n a. Matrix m n a -> [[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
--
getElem :: forall i j m n a. (KnownNat i, KnownNat j, 1 <= i, i <= m, 1 <= j, j <= n) => Matrix m n a -> a
-- | Short alias for unsafeGet. Careful: This has no bounds checking
-- This deviates from Data.Matrix, where (!) does check bounds
-- on runtime.
(!) :: Matrix m n a -> (Int, Int) -> a
-- | O(1). Unsafe variant of getElem. This will do no bounds
-- checking
unsafeGet :: Int -> Int -> Matrix m n a -> a
-- | Alias for '(!)'. This exists to keep the interface similar to
-- Data.Matrix but serves no other purpose. Use '(!)' (or even
-- better getElem) instead.
(!.) :: Matrix m n a -> (Int, Int) -> a
-- | Variant of unsafeGet that returns Maybe instead of an error.
safeGet :: forall m n a. (KnownNat n, KnownNat m) => Int -> Int -> Matrix m n a -> Maybe a
-- | Variant of setElem that returns Maybe instead of an error.
safeSet :: forall m n a. a -> (Int, Int) -> Matrix m n a -> Maybe (Matrix m n 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
getRow :: 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
getCol :: Int -> Matrix m n a -> Vector a
-- | Varian of getRow that returns a maybe instead of an error Only
-- available when used with matrix >= 0.3.6!
safeGetRow :: 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!
safeGetCol :: Int -> Matrix m n a -> Maybe (Vector a)
-- | O(min rows cols). Diagonal of a not necessarily square
-- matrix.
getDiag :: 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.
getMatrixAsVector :: Matrix m n a -> Vector 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 k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a
-- | Type safe scalar multiplication
(^*) :: forall m n a. Num a => a -> Matrix m n a -> Matrix m n a
-- | 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)
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
-- | Unsafe variant of setElem, without bounds checking.
unsafeSet :: a -> (Int, Int) -> Matrix m n a -> Matrix m n 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 )
--
transpose :: forall m n a. Matrix m n a -> Matrix n m a
-- | Set the size of a matrix to given parameters. Use a default element
-- for undefined entries if the matrix has been extended.
setSize :: forall newM newN m n a. (KnownNat newM, KnownNat newN, 1 <= newM, 1 <= newN) => a -> 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 )
--
extendTo :: forall newM newN m n a. (KnownNat newM, KnownNat newN, n <= newN, m <= newM) => a -> Matrix m n a -> Matrix newM newN a
-- | O(rows^4). The inverse of a square matrix Uses naive Gaussian
-- elimination formula.
inverse :: forall n a. (Fractional a, Eq a) => Matrix n n a -> Either String (Matrix n 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).
rref :: (Fractional a, Eq a) => Matrix m n a -> Either String (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 )
--
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
-- | 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 )
--
mapRowUnsafe :: forall m n a. (Int -> a -> a) -> Int -> 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 )
--
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
-- | 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 )
--
mapColUnsafe :: forall m n a. (Int -> a -> a) -> Int -> Matrix m n a -> Matrix m n a
-- | 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!
mapPos :: ((Int, Int) -> a -> b) -> Matrix m n a -> Matrix m n b
-- | O(1). Extract a submatrix from the given position. The type
-- parameters expected are the starting and ending indices of row and
-- column elements.
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 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 )
--
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
-- | 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 )
--
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
-- | 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 )
--
minorMatrixUnsafe :: (2 <= n, 2 <= m) => Int -> Int -> Matrix m n a -> Matrix (m - 1) (n - 1) a
-- | 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 <math>
--
--
-- ( ) ( 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.
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)
-- | Horizontally join two matrices. Visually:
--
--
-- ( A ) <|> ( B ) = ( A | B )
--
(<|>) :: forall m n k a. Matrix m n a -> Matrix m k a -> Matrix m (k + n) a
-- | Horizontally join two matrices. Visually:
--
--
-- ( A )
-- ( A ) <-> ( B ) = ( - )
-- ( B )
--
(<->) :: forall m k n a. Matrix m n a -> Matrix k n a -> Matrix (m + k) n a
-- | Join blocks of the form detailed in splitBlocks. Precisely:
--
--
-- joinBlocks (tl,tr,bl,br) =
-- (tl <|> tr)
-- <->
-- (bl <|> br)
--
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
-- | Perform an operation element-wise. This uses matrix's
-- elementwiseUnsafe since we can guarantee proper dimensions at
-- compile time.
elementwise :: forall m n a b c. (a -> b -> c) -> (Matrix m n a -> Matrix m n b -> Matrix m n c)
-- | Standard matrix multiplication by definition.
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
-- | Strassen's matrix multiplication.
multStrassen :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a
-- | Mixed Strassen's matrix multiplication.
multStrassenMixed :: forall m k n a. Num a => Matrix m k a -> Matrix k 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 )
--
scaleMatrix :: 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 )
--
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 )
-- scaleRowUnsafe 3 2 ( 7 8 9 ) = ( 7 8 9 )
--
scaleRowUnsafe :: Num a => 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 )
-- combineRows @2 @1 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 )
-- combineRowsUnsafe 2 2 1 ( 7 8 9 ) = ( 7 8 9 )
--
combineRowsUnsafe :: Num a => Int -> a -> Int -> Matrix m n a -> Matrix m n a
-- | 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 )
--
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
-- | 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 )
--
switchRowsUnsafe :: Int -> Int -> Matrix m n a -> Matrix m n a
-- | 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 )
--
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
-- | 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 )
--
switchColsUnsafe :: Int -> Int -> Matrix m n a -> Matrix m n 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.
luDecomp :: (Ord a, Fractional a) => Matrix m n a -> Maybe (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.
luDecompUnsafe :: (Ord a, Fractional a) => Matrix m n a -> (Matrix m n a, Matrix m n a, Matrix m n 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.
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)
-- | Unsafe version of luDecomp'. It fails when the input matrix is
-- singular.
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)
-- | Simple Cholesky decomposition of a symmetric, positive definite
-- matrix. The result for a matrix M is a lower triangular matrix
-- L such that:
--
--
--
-- 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 )
--
cholDecomp :: (Floating a) => Matrix n n a -> Matrix n n a
-- | Sum of the elements in the diagonal. See also getDiag. Example:
--
--
-- ( 1 2 3 )
-- ( 4 5 6 )
-- trace ( 7 8 9 ) = 15
--
trace :: 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
--
diagProd :: Num a => Matrix m 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.
detLaplace :: Num a => Matrix n n a -> a
-- | Matrix determinant using LU decomposition. It works even when the
-- input matrix is singular.
detLU :: (Ord a, Fractional a) => Matrix n n a -> a
-- | Flatten a matrix of matrices.
flatten :: forall m' n' m n a. Matrix m' n' (Matrix m n a) -> Matrix (m' * m) (n' * n) a
-- | Apply a map function to the unsafe inner matrix type.
applyUnary :: forall m n m' n' a b. (Matrix a -> Matrix b) -> Matrix m n a -> Matrix m' n' b
-- | Transform a binary unstatic function to a binary static function.
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
-- | Forget static information about a matrix. This converts this converts
-- the Matrix type to Data.Matrix.Matrix
unpackStatic :: forall m n a. Matrix m n a -> Matrix a
instance Control.DeepSeq.NFData a => Control.DeepSeq.NFData (Data.Matrix.Static.Matrix m n a)
instance GHC.Base.Monoid a => GHC.Base.Monoid (Data.Matrix.Static.Matrix m n a)
instance Data.Traversable.Traversable (Data.Matrix.Static.Matrix m n)
instance Data.Foldable.Foldable (Data.Matrix.Static.Matrix m n)
instance GHC.Base.Applicative (Data.Matrix.Static.Matrix m n)
instance GHC.Base.Functor (Data.Matrix.Static.Matrix m n)
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Matrix.Static.Matrix m n a)
instance GHC.Base.Monoid a => GHC.Base.Semigroup (Data.Matrix.Static.Matrix m n a)
instance GHC.Show.Show f => GHC.Show.Show (Data.Matrix.Static.Matrix m n f)
instance GHC.Classes.Ord f => GHC.Classes.Ord (Data.Matrix.Static.Matrix m n f)
instance GHC.Num.Num f => GHC.Num.Num (Data.Matrix.Static.Matrix m n f)