Copyright	(c) Alberto Ruiz 2015
License	BSD3
Maintainer	Alberto Ruiz
Stability	provisional
Safe Haskell	None
Language	Haskell98

Numeric.LinearAlgebra.Data

Contents

Elements
Vector
Matrix
Dimensions
Conversion from/to lists
Conversions vector/matrix
Indexing
Construction
Diagonal
Vector extraction
Matrix extraction
Block matrix
Mapping functions
Find elements
Sparse
IO
Element conversion
Misc

Description

This module provides functions for creation and manipulation of vectors and matrices, IO, and other utilities.

Synopsis

Elements

type R = Double Source

type C = Complex Double Source

type I = CInt Source

type Z = Int64 Source

type (./.) x n = Mod n x infixr 5 Source

Vector

1D arrays are storable vectors directly reexported from the vector package.

fromList :: Storable a => [a] -> Vector a

toList :: Storable a => Vector a -> [a] Source

(|>) :: Storable a => Int -> [a] -> Vector a infixl 9 Source

Create a vector from a list of elements and explicit dimension. The input list is truncated if it is too long, so it may safely be used, for instance, with infinite lists.

>>> 5 |> [1..]
fromList [1.0,2.0,3.0,4.0,5.0]

vector :: [R] -> Vector R Source

Create a real vector.

>>> vector [1..5]
fromList [1.0,2.0,3.0,4.0,5.0]

range :: Int -> Vector I Source

>>> range 5
fromList [0,1,2,3,4]

idxs :: [Int] -> Vector I Source

Create a vector of indexes, useful for matrix extraction using '(??)'

Matrix

The main data type of hmatrix is a 2D dense array defined on top of a storable vector. The internal representation is suitable for direct interface with standard numeric libraries.

(><) :: Storable a => Int -> Int -> [a] -> Matrix a Source

Create a matrix from a list of elements

>>> (2><3) [2, 4, 7+2*iC,   -3, 11, 0]
(2><3)
 [       2.0 :+ 0.0,  4.0 :+ 0.0, 7.0 :+ 2.0
 , (-3.0) :+ (-0.0), 11.0 :+ 0.0, 0.0 :+ 0.0 ]

The input list is explicitly truncated, so that it can safely be used with lists that are too long (like infinite lists).

>>> (2><3)[1..]
(2><3)
 [ 1.0, 2.0, 3.0
 , 4.0, 5.0, 6.0 ]

This is the format produced by the instances of Show (Matrix a), which can also be used for input.

matrix Source

Arguments

:: Int	number of columns
-> [R]	elements in row order
-> Matrix R

Create a real matrix.

>>> matrix 5 [1..15]
(3><5)
 [  1.0,  2.0,  3.0,  4.0,  5.0
 ,  6.0,  7.0,  8.0,  9.0, 10.0
 , 11.0, 12.0, 13.0, 14.0, 15.0 ]

tr :: Transposable m mt => m -> mt Source

conjugate transpose

tr' :: Transposable m mt => m -> mt Source

transpose

Dimensions

size :: Container c t => c t -> IndexOf c Source

>>> size $ vector [1..10]
10
>>> size $ (2><5)[1..10::Double]
(2,5)

rows :: Matrix t -> Int Source

cols :: Matrix t -> Int Source

Conversion from/to lists

fromLists :: Element t => [[t]] -> Matrix t Source

Creates a Matrix from a list of lists (considered as rows).

>>> fromLists [[1,2],[3,4],[5,6]]
(3><2)
 [ 1.0, 2.0
 , 3.0, 4.0
 , 5.0, 6.0 ]

toLists :: Element t => Matrix t -> [[t]] Source

the inverse of fromLists

row :: [Double] -> Matrix Double Source

create a single row real matrix from a list

>>> row [2,3,1,8]
(1><4)
 [ 2.0, 3.0, 1.0, 8.0 ]

col :: [Double] -> Matrix Double Source

create a single column real matrix from a list

>>> col [7,-2,4]
(3><1)
 [  7.0
 , -2.0
 ,  4.0 ]

Conversions vector/matrix

flatten :: Element t => Matrix t -> Vector t Source

Creates a vector by concatenation of rows. If the matrix is ColumnMajor, this operation requires a transpose.

>>> flatten (ident 3)
fromList [1.0,0.0,0.0,0.0,1.0,0.0,0.0,0.0,1.0]

reshape :: Storable t => Int -> Vector t -> Matrix t Source

Creates a matrix from a vector by grouping the elements in rows with the desired number of columns. (GNU-Octave groups by columns. To do it you can define reshapeF r = tr' . reshape r where r is the desired number of rows.)

>>> reshape 4 (fromList [1..12])
(3><4)
 [ 1.0,  2.0,  3.0,  4.0
 , 5.0,  6.0,  7.0,  8.0
 , 9.0, 10.0, 11.0, 12.0 ]

asRow :: Storable a => Vector a -> Matrix a Source

creates a 1-row matrix from a vector

>>> asRow (fromList [1..5])
 (1><5)
  [ 1.0, 2.0, 3.0, 4.0, 5.0 ]

asColumn :: Storable a => Vector a -> Matrix a Source

creates a 1-column matrix from a vector

>>> asColumn (fromList [1..5])
(5><1)
 [ 1.0
 , 2.0
 , 3.0
 , 4.0
 , 5.0 ]

fromRows :: Element t => [Vector t] -> Matrix t Source

Create a matrix from a list of vectors. All vectors must have the same dimension, or dimension 1, which is are automatically expanded.

toRows :: Element t => Matrix t -> [Vector t] Source

extracts the rows of a matrix as a list of vectors

fromColumns :: Element t => [Vector t] -> Matrix t Source

Creates a matrix from a list of vectors, as columns

toColumns :: Element t => Matrix t -> [Vector t] Source

Creates a list of vectors from the columns of a matrix

Indexing

atIndex :: Container c e => c e -> IndexOf c -> e Source

generic indexing function

>>> vector [1,2,3] `atIndex` 1
2.0

>>> matrix 3 [0..8] `atIndex` (2,0)
6.0

class Indexable c t | c -> t, t -> c where Source

Alternative indexing function.

>>> vector [1..10] ! 3
4.0

On a matrix it gets the k-th row as a vector:

>>> matrix 5 [1..15] ! 1
fromList [6.0,7.0,8.0,9.0,10.0]

>>> matrix 5 [1..15] ! 1 ! 3
9.0

Methods

(!) :: c -> Int -> t infixl 9 Source

Instances

Indexable (Vector Double) Double Source
Indexable (Vector Float) Float Source
Indexable (Vector Z) Z Source
Indexable (Vector I) I Source
Indexable (Vector (Complex Double)) (Complex Double) Source
Indexable (Vector (Complex Float)) (Complex Float) Source
Element t => Indexable (Matrix t) (Vector t) Source
(Storable t, Indexable (Vector t) t) => Indexable (Vector (Mod m t)) (Mod m t) Source

Construction

scalar :: Container c e => e -> c e Source

create a structure with a single element

>>> let v = fromList [1..3::Double]
>>> v / scalar (norm2 v)
fromList [0.2672612419124244,0.5345224838248488,0.8017837257372732]

class Konst e d c | d -> c, c -> d where Source

Methods

konst :: e -> d -> c e Source

>>> konst 7 3 :: Vector Float
fromList [7.0,7.0,7.0]

>>> konst i (3::Int,4::Int)
(3><4)
 [ 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0
 , 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0
 , 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0 ]

Instances

Container Vector e => Konst e Int Vector Source
(Num e, Container Vector e) => Konst e (Int, Int) Matrix Source

class Build d f c e | d -> c, c -> d, f -> e, f -> d, f -> c, c e -> f, d e -> f where Source

Methods

build :: d -> f -> c e Source

>>> build 5 (**2) :: Vector Double
fromList [0.0,1.0,4.0,9.0,16.0]

Hilbert matrix of order N:

>>> let hilb n = build (n,n) (\i j -> 1/(i+j+1)) :: Matrix Double
>>> putStr . dispf 2 $ hilb 3
3x3
1.00  0.50  0.33
0.50  0.33  0.25
0.33  0.25  0.20

Instances

Container Vector e => Build Int (e -> e) Vector e Source
Container Matrix e => Build (Int, Int) (e -> e -> e) Matrix e Source

assoc Source

Arguments

:: Container c e
=> IndexOf c	size
-> e	default value
-> [(IndexOf c, e)]	association list
-> c e	result

Create a structure from an association list

>>> assoc 5 0 [(3,7),(1,4)] :: Vector Double
fromList [0.0,4.0,0.0,7.0,0.0]

>>> assoc (2,3) 0 [((0,2),7),((1,0),2*i-3)] :: Matrix (Complex Double)
(2><3)
 [    0.0 :+ 0.0, 0.0 :+ 0.0, 7.0 :+ 0.0
 , (-3.0) :+ 2.0, 0.0 :+ 0.0, 0.0 :+ 0.0 ]

accum Source

Arguments

:: Container c e
=> c e	initial structure
-> (e -> e -> e)	update function
-> [(IndexOf c, e)]	association list
-> c e	result

Modify a structure using an update function

>>> accum (ident 5) (+) [((1,1),5),((0,3),3)] :: Matrix Double
(5><5)
 [ 1.0, 0.0, 0.0, 3.0, 0.0
 , 0.0, 6.0, 0.0, 0.0, 0.0
 , 0.0, 0.0, 1.0, 0.0, 0.0
 , 0.0, 0.0, 0.0, 1.0, 0.0
 , 0.0, 0.0, 0.0, 0.0, 1.0 ]

computation of histogram:

>>> accum (konst 0 7) (+) (map (flip (,) 1) [4,5,4,1,5,2,5]) :: Vector Double
fromList [0.0,1.0,1.0,0.0,2.0,3.0,0.0]

linspace :: (Fractional e, Container Vector e) => Int -> (e, e) -> Vector e Source

Creates a real vector containing a range of values:

>>> linspace 5 (-3,7::Double)
fromList [-3.0,-0.5,2.0,4.5,7.0]@

>>> linspace 5 (8,2+i) :: Vector (Complex Double)
fromList [8.0 :+ 0.0,6.5 :+ 0.25,5.0 :+ 0.5,3.5 :+ 0.75,2.0 :+ 1.0]

Logarithmic spacing can be defined as follows:

logspace n (a,b) = 10 ** linspace n (a,b)

Diagonal

ident :: (Num a, Element a) => Int -> Matrix a Source

creates the identity matrix of given dimension

diag :: (Num a, Element a) => Vector a -> Matrix a Source

Creates a square matrix with a given diagonal.

diagl :: [Double] -> Matrix Double Source

create a real diagonal matrix from a list

>>> diagl [1,2,3]
(3><3)
 [ 1.0, 0.0, 0.0
 , 0.0, 2.0, 0.0
 , 0.0, 0.0, 3.0 ]

diagRect :: Storable t => t -> Vector t -> Int -> Int -> Matrix t Source

creates a rectangular diagonal matrix:

>>> diagRect 7 (fromList [10,20,30]) 4 5 :: Matrix Double
(4><5)
 [ 10.0,  7.0,  7.0, 7.0, 7.0
 ,  7.0, 20.0,  7.0, 7.0, 7.0
 ,  7.0,  7.0, 30.0, 7.0, 7.0
 ,  7.0,  7.0,  7.0, 7.0, 7.0 ]

takeDiag :: Element t => Matrix t -> Vector t Source

extracts the diagonal from a rectangular matrix

Vector extraction

subVector Source

Arguments

:: Storable t
=> Int	index of the starting element
-> Int	number of elements to extract
-> Vector t	source
-> Vector t	result

takes a number of consecutive elements from a Vector

>>> subVector 2 3 (fromList [1..10])
fromList [3.0,4.0,5.0]

takesV :: Storable t => [Int] -> Vector t -> [Vector t] Source

Extract consecutive subvectors of the given sizes.

>>> takesV [3,4] (linspace 10 (1,10::Double))
[fromList [1.0,2.0,3.0],fromList [4.0,5.0,6.0,7.0]]

vjoin :: Storable t => [Vector t] -> Vector t Source

concatenate a list of vectors

>>> vjoin [fromList [1..5::Double], konst 1 3]
fromList [1.0,2.0,3.0,4.0,5.0,1.0,1.0,1.0]

Matrix extraction

data Extractor Source

Specification of indexes for the operator ??.

Constructors

All
Range Int Int Int
Pos (Vector I)
PosCyc (Vector I)
Take Int
TakeLast Int
Drop Int
DropLast Int

Instances

Show Extractor Source

(??) :: Element t => Matrix t -> (Extractor, Extractor) -> Matrix t infixl 9 Source

General matrix slicing.

>>> m
(4><5)
 [  0,  1,  2,  3,  4
 ,  5,  6,  7,  8,  9
 , 10, 11, 12, 13, 14
 , 15, 16, 17, 18, 19 ]

>>> m ?? (Take 3, DropLast 2)
(3><3)
 [  0,  1,  2
 ,  5,  6,  7
 , 10, 11, 12 ]

>>> m ?? (Pos (idxs[2,1]), All)
(2><5)
 [ 10, 11, 12, 13, 14
 ,  5,  6,  7,  8,  9 ]

>>> m ?? (PosCyc (idxs[-7,80]), Range 4 (-2) 0)
(2><3)
 [ 9, 7, 5
 , 4, 2, 0 ]

(?) :: Element t => Matrix t -> [Int] -> Matrix t infixl 9 Source

extract rows

>>> (20><4) [1..] ? [2,1,1]
(3><4)
 [ 9.0, 10.0, 11.0, 12.0
 , 5.0,  6.0,  7.0,  8.0
 , 5.0,  6.0,  7.0,  8.0 ]

¿ :: Element t => Matrix t -> [Int] -> Matrix t infixl 9 Source

extract columns

(unicode 0x00bf, inverted question mark, Alt-Gr ?)

>>> (3><4) [1..] ¿ [3,0]
(3><2)
 [  4.0, 1.0
 ,  8.0, 5.0
 , 12.0, 9.0 ]

fliprl :: Element t => Matrix t -> Matrix t Source

Reverse columns

flipud :: Element t => Matrix t -> Matrix t Source

Reverse rows

subMatrix Source

Arguments

:: Element a
=> (Int, Int)	(r0,c0) starting position
-> (Int, Int)	(rt,ct) dimensions of submatrix
-> Matrix a	input matrix
-> Matrix a	result

reference to a rectangular slice of a matrix (no data copy)

takeRows :: Element t => Int -> Matrix t -> Matrix t Source

dropRows :: Element t => Int -> Matrix t -> Matrix t Source

takeColumns :: Element t => Int -> Matrix t -> Matrix t Source

dropColumns :: Element t => Int -> Matrix t -> Matrix t Source

remap :: Element t => Matrix I -> Matrix I -> Matrix t -> Matrix t Source

Extract elements from positions given in matrices of rows and columns.

>>> r
(3><3)
 [ 1, 1, 1
 , 1, 2, 2
 , 1, 2, 3 ]
>>> c
(3><3)
 [ 0, 1, 5
 , 2, 2, 1
 , 4, 4, 1 ]
>>> m
(4><6)
 [  0,  1,  2,  3,  4,  5
 ,  6,  7,  8,  9, 10, 11
 , 12, 13, 14, 15, 16, 17
 , 18, 19, 20, 21, 22, 23 ]

>>> remap r c m
(3><3)
 [  6,  7, 11
 ,  8, 14, 13
 , 10, 16, 19 ]

The indexes are autoconformable.

>>> c'
(3><1)
 [ 1
 , 2
 , 4 ]
>>> remap r c' m
(3><3)
 [  7,  7,  7
 ,  8, 14, 14
 , 10, 16, 22 ]

Block matrix

fromBlocks :: Element t => [[Matrix t]] -> Matrix t Source

Create a matrix from blocks given as a list of lists of matrices.

Single row-column components are automatically expanded to match the corresponding common row and column:

disp = putStr . dispf 2

>>> disp $ fromBlocks [[ident 5, 7, row[10,20]], [3, diagl[1,2,3], 0]]
8x10
1  0  0  0  0  7  7  7  10  20
0  1  0  0  0  7  7  7  10  20
0  0  1  0  0  7  7  7  10  20
0  0  0  1  0  7  7  7  10  20
0  0  0  0  1  7  7  7  10  20
3  3  3  3  3  1  0  0   0   0
3  3  3  3  3  0  2  0   0   0
3  3  3  3  3  0  0  3   0   0

(|||) :: Element t => Matrix t -> Matrix t -> Matrix t infixl 3 Source

horizontal concatenation

>>> ident 3 ||| konst 7 (3,4)
(3><7)
 [ 1.0, 0.0, 0.0, 7.0, 7.0, 7.0, 7.0
 , 0.0, 1.0, 0.0, 7.0, 7.0, 7.0, 7.0
 , 0.0, 0.0, 1.0, 7.0, 7.0, 7.0, 7.0 ]

(===) :: Element t => Matrix t -> Matrix t -> Matrix t infixl 2 Source

vertical concatenation

diagBlock :: (Element t, Num t) => [Matrix t] -> Matrix t Source

create a block diagonal matrix

>>> disp 2 $ diagBlock [konst 1 (2,2), konst 2 (3,5), col [5,7]]
7x8
1  1  0  0  0  0  0  0
1  1  0  0  0  0  0  0
0  0  2  2  2  2  2  0
0  0  2  2  2  2  2  0
0  0  2  2  2  2  2  0
0  0  0  0  0  0  0  5
0  0  0  0  0  0  0  7

>>> diagBlock [(0><4)[], konst 2 (2,3)]  :: Matrix Double
(2><7)
 [ 0.0, 0.0, 0.0, 0.0, 2.0, 2.0, 2.0
 , 0.0, 0.0, 0.0, 0.0, 2.0, 2.0, 2.0 ]

repmat :: Element t => Matrix t -> Int -> Int -> Matrix t Source

creates matrix by repetition of a matrix a given number of rows and columns

>>> repmat (ident 2) 2 3
(4><6)
 [ 1.0, 0.0, 1.0, 0.0, 1.0, 0.0
 , 0.0, 1.0, 0.0, 1.0, 0.0, 1.0
 , 1.0, 0.0, 1.0, 0.0, 1.0, 0.0
 , 0.0, 1.0, 0.0, 1.0, 0.0, 1.0 ]

toBlocks :: Element t => [Int] -> [Int] -> Matrix t -> [[Matrix t]] Source

Partition a matrix into blocks with the given numbers of rows and columns. The remaining rows and columns are discarded.

toBlocksEvery :: Element t => Int -> Int -> Matrix t -> [[Matrix t]] Source

Fully partition a matrix into blocks of the same size. If the dimensions are not a multiple of the given size the last blocks will be smaller.

Mapping functions

conj :: Container c e => c e -> c e Source

complex conjugate

cmap :: (Element b, Container c e) => (e -> b) -> c e -> c b Source

like fmap (cannot implement instance Functor because of Element class constraint)

cmod :: (Integral e, Container c e) => e -> c e -> c e Source

mod for integer arrays

>>> cmod 3 (range 5)
fromList [0,1,2,0,1]

step :: (Ord e, Container c e) => c e -> c e Source

A more efficient implementation of cmap (\x -> if x>0 then 1 else 0)

>>> step $ linspace 5 (-1,1::Double)
5 |> [0.0,0.0,0.0,1.0,1.0]

cond Source

Arguments

:: (Ord e, Container c e, Container c x)
=> c e	a
-> c e	b
-> c x	l
-> c x	e
-> c x	g
-> c x	result

Element by element version of case compare a b of {LT -> l; EQ -> e; GT -> g}.

Arguments with any dimension = 1 are automatically expanded:

>>> cond ((1><4)[1..]) ((3><1)[1..]) 0 100 ((3><4)[1..]) :: Matrix Double
(3><4)
[ 100.0,   2.0,   3.0,  4.0
,   0.0, 100.0,   7.0,  8.0
,   0.0,   0.0, 100.0, 12.0 ]

>>> let chop x = cond (abs x) 1E-6 0 0 x

Find elements

find :: Container c e => (e -> Bool) -> c e -> [IndexOf c] Source

Find index of elements which satisfy a predicate

>>> find (>0) (ident 3 :: Matrix Double)
[(0,0),(1,1),(2,2)]

maxIndex :: Container c e => c e -> IndexOf c Source

index of maximum element

minIndex :: Container c e => c e -> IndexOf c Source

index of minimum element

maxElement :: Container c e => c e -> e Source

value of maximum element

minElement :: Container c e => c e -> e Source

value of minimum element

sortVector :: (Ord t, Element t) => Vector t -> Vector t Source

sortIndex :: (Ord t, Element t) => Vector t -> Vector I Source

>>> m <- randn 4 10
>>> disp 2 m
4x10
-0.31   0.41   0.43  -0.19  -0.17  -0.23  -0.17  -1.04  -0.07  -1.24
 0.26   0.19   0.14   0.83  -1.54  -0.09   0.37  -0.63   0.71  -0.50
-0.11  -0.10  -1.29  -1.40  -1.04  -0.89  -0.68   0.35  -1.46   1.86
 1.04  -0.29   0.19  -0.75  -2.20  -0.01   1.06   0.11  -2.09  -1.58

>>> disp 2 $ m ?? (All, Pos $ sortIndex (m!1))
4x10
-0.17  -1.04  -1.24  -0.23   0.43   0.41  -0.31  -0.17  -0.07  -0.19
-1.54  -0.63  -0.50  -0.09   0.14   0.19   0.26   0.37   0.71   0.83
-1.04   0.35   1.86  -0.89  -1.29  -0.10  -0.11  -0.68  -1.46  -1.40
-2.20   0.11  -1.58  -0.01   0.19  -0.29   1.04   1.06  -2.09  -0.75

Sparse

type AssocMatrix = [((Int, Int), Double)] Source

toDense :: AssocMatrix -> Matrix Double Source

mkSparse :: AssocMatrix -> GMatrix Source

mkDiagR :: Int -> Int -> Vector Double -> GMatrix Source

mkDense :: Matrix Double -> GMatrix Source

IO

disp :: Int -> Matrix Double -> IO () Source

print a real matrix with given number of digits after the decimal point

>>> disp 5 $ ident 2 / 3
2x2
0.33333  0.00000
0.00000  0.33333

loadMatrix :: FilePath -> IO (Matrix Double) Source

load a matrix from an ASCII file formatted as a 2D table.

loadMatrix' :: FilePath -> IO (Maybe (Matrix Double)) Source

saveMatrix Source

Arguments

:: FilePath
-> String	"printf" format (e.g. "%.2f", "%g", etc.)
-> Matrix Double
-> IO ()

save a matrix as a 2D ASCII table

latexFormat Source

Arguments

:: String	type of braces: "matrix", "bmatrix", "pmatrix", etc.
-> String	Formatted matrix, with elements separated by spaces and newlines
-> String

Tool to display matrices with latex syntax.

>>> latexFormat "bmatrix" (dispf 2 $ ident 2)
"\\begin{bmatrix}\n1  &  0\n\\\\\n0  &  1\n\\end{bmatrix}"

dispf :: Int -> Matrix Double -> String Source

Show a matrix with a given number of decimal places.

>>> dispf 2 (1/3 + ident 3)
"3x3\n1.33  0.33  0.33\n0.33  1.33  0.33\n0.33  0.33  1.33\n"

>>> putStr . dispf 2 $ (3><4)[1,1.5..]
3x4
1.00  1.50  2.00  2.50
3.00  3.50  4.00  4.50
5.00  5.50  6.00  6.50

>>> putStr . unlines . tail . lines . dispf 2 . asRow $ linspace 10 (0,1)
0.00  0.11  0.22  0.33  0.44  0.56  0.67  0.78  0.89  1.00

disps :: Int -> Matrix Double -> String Source

Show a matrix with "autoscaling" and a given number of decimal places.

>>> putStr . disps 2 $ 120 * (3><4) [1..]
3x4  E3
 0.12  0.24  0.36  0.48
 0.60  0.72  0.84  0.96
 1.08  1.20  1.32  1.44

dispcf :: Int -> Matrix (Complex Double) -> String Source

Pretty print a complex matrix with at most n decimal digits.

format :: Element t => String -> (t -> String) -> Matrix t -> String Source

Creates a string from a matrix given a separator and a function to show each entry. Using this function the user can easily define any desired display function:

import Text.Printf(printf)

disp = putStr . format "  " (printf "%.2f")

dispDots :: Int -> Matrix Double -> IO () Source

dispBlanks :: Int -> Matrix Double -> IO () Source

dispShort :: Int -> Int -> Int -> Matrix Double -> IO () Source

Element conversion

class Convert t where Source

Methods

real :: Complexable c => c (RealOf t) -> c t Source

complex :: Complexable c => c t -> c (ComplexOf t) Source

single :: Complexable c => c t -> c (SingleOf t) Source

double :: Complexable c => c t -> c (DoubleOf t) Source

toComplex :: (Complexable c, RealElement t) => (c t, c t) -> c (Complex t) Source

fromComplex :: (Complexable c, RealElement t) => c (Complex t) -> (c t, c t) Source

Instances

Convert Double Source
Convert Float Source
Convert (Complex Double) Source
Convert (Complex Float) Source

roundVector :: Vector Double -> Vector Double Source

fromInt :: Container c e => c I -> c e Source

>>> fromInt ((2><2) [0..3]) :: Matrix (Complex Double)
(2><2)
[ 0.0 :+ 0.0, 1.0 :+ 0.0
, 2.0 :+ 0.0, 3.0 :+ 0.0 ]

toInt :: Container c e => c e -> c I Source

fromZ :: Container c e => c Z -> c e Source

toZ :: Container c e => c e -> c Z Source

Misc

arctan2 :: (Fractional e, Container c e) => c e -> c e -> c e Source

separable :: Element t => (Vector t -> Vector t) -> Matrix t -> Matrix t Source

matrix computation implemented as separated vector operations by rows and columns.

fromArray2D :: Storable e => Array (Int, Int) e -> Matrix e Source

module Data.Complex

data Mod n t Source

Wrapper with a phantom integer for statically checked modular arithmetic.

Instances

KnownNat m => Container Vector (Mod m Z) Source
KnownNat m => Container Vector (Mod m I) Source
KnownNat m => Num (Vector (Mod m Z)) Source
KnownNat m => Num (Vector (Mod m I)) Source
KnownNat m => Testable (Matrix (Mod m I)) Source
KnownNat m => Normed (Vector (Mod m Z)) Source
KnownNat m => Normed (Vector (Mod m I)) Source
(Storable t, Indexable (Vector t) t) => Indexable (Vector (Mod m t)) (Mod m t) Source
(Integral t, Enum t, KnownNat m) => Enum (Mod m t) Source
(Eq t, KnownNat m) => Eq (Mod m t) Source
(Show (Mod m t), Num (Mod m t), Eq t, KnownNat m) => Fractional (Mod m t) Source	this instance is only valid for prime m
(Integral t, KnownNat m, Num (Mod m t)) => Integral (Mod m t) Source
(Integral t, KnownNat n) => Num (Mod n t) Source
(Ord t, KnownNat m) => Ord (Mod m t) Source
(Real t, KnownNat m, Integral (Mod m t)) => Real (Mod m t) Source
Show t => Show (Mod n t) Source
Storable t => Storable (Mod n t) Source
NFData t => NFData (Mod n t) Source
KnownNat m => Element (Mod m Z) Source
KnownNat m => Element (Mod m I) Source
KnownNat m => Product (Mod m Z) Source
KnownNat m => Product (Mod m I) Source
KnownNat m => Numeric (Mod m Z) Source
KnownNat m => Numeric (Mod m I) Source
type RealOf (Mod n Z) = Z Source
type RealOf (Mod n I) = I Source

data Vector a :: * -> *

Instances

Complexable Vector
LSDiv Vector
Storable a => Vector Vector a
Container Vector t => Linear t Vector
Container Vector Double
Container Vector Float
Container Vector Z
Container Vector I
Container Vector e => Konst e Int Vector
Container Vector (Complex Double)
Container Vector (Complex Float)
KnownNat n => Sized ℝ (R n) Vector
KnownNat m => Container Vector (Mod m Z)
KnownNat m => Container Vector (Mod m I)
Container Vector e => Build Int (e -> e) Vector e
Storable a => IsList (Vector a)
(Storable a, Eq a) => Eq (Vector a)
(Data a, Storable a) => Data (Vector a)
KnownNat m => Num (Vector (Mod m Z))
KnownNat m => Num (Vector (Mod m I))
(Storable a, Ord a) => Ord (Vector a)
(Read a, Storable a) => Read (Vector a)
(Show a, Storable a) => Show (Vector a)
Storable a => Monoid (Vector a)
NFData (Vector a)
Storable t => TransArray (Vector t)
Container Vector t => Additive (Vector t)
Normed (Vector Float)
Normed (Vector (Complex Float))
Normed (Vector C)
Normed (Vector R)
Normed (Vector Z)
Normed (Vector I)
KnownNat m => Normed (Vector (Mod m Z))
KnownNat m => Normed (Vector (Mod m I))
Indexable (Vector Double) Double
Indexable (Vector Float) Float
Indexable (Vector Z) Z
Indexable (Vector I) I
Indexable (Vector (Complex Double)) (Complex Double)
Indexable (Vector (Complex Float)) (Complex Float)
Element t => Indexable (Matrix t) (Vector t)
(Storable t, Indexable (Vector t) t) => Indexable (Vector (Mod m t)) (Mod m t)
type Mutable Vector = MVector
type IndexOf Vector = Int
type Item (Vector a) = a
type Trans (Vector t) b = CInt -> Ptr t -> b
type TransRaw (Vector t) b = CInt -> Ptr t -> b

data Matrix t Source

Matrix representation suitable for BLAS/LAPACK computations.

Instances

Complexable Matrix Source
LSDiv Matrix Source
Container Matrix t => Linear t Matrix Source
(Num a, Element a, Container Vector a) => Container Matrix a Source
(Num e, Container Vector e) => Konst e (Int, Int) Matrix Source
(KnownNat m, KnownNat n) => Sized ℝ (L m n) Matrix Source
(Storable t, NFData t) => NFData (Matrix t) Source
Storable t => TransArray (Matrix t) Source
KnownNat m => Testable (Matrix (Mod m I)) Source
Container Matrix t => Additive (Matrix t) Source
Normed (Matrix C) Source
Normed (Matrix R) Source
(CTrans t, Container Vector t) => Transposable (Matrix t) (Matrix t) Source
Element t => Indexable (Matrix t) (Vector t) Source
Container Matrix e => Build (Int, Int) (e -> e -> e) Matrix e Source
type IndexOf Matrix = (Int, Int) Source
type Trans (Matrix t) b = CInt -> CInt -> CInt -> CInt -> Ptr t -> b Source
type TransRaw (Matrix t) b = CInt -> CInt -> Ptr t -> b Source

data GMatrix Source

General matrix with specialized internal representations for dense, sparse, diagonal, banded, and constant elements.

>>> let m = mkSparse [((0,999),1.0),((1,1999),2.0)]
>>> m
SparseR {gmCSR = CSR {csrVals = fromList [1.0,2.0],
                      csrCols = fromList [1000,2000],
                      csrRows = fromList [1,2,3],
                      csrNRows = 2,
                      csrNCols = 2000},
                      nRows = 2,
                      nCols = 2000}

>>> let m = mkDense (mat 2 [1..4])
>>> m
Dense {gmDense = (2><2)
 [ 1.0, 2.0
 , 3.0, 4.0 ], nRows = 2, nCols = 2}

Instances

Show GMatrix Source
Transposable GMatrix GMatrix Source

nRows :: GMatrix -> Int Source

nCols :: GMatrix -> Int Source