Numeric.Container

Basic numeric operations on Vector and Matrix, including conversion routines.

The Container class is used to define optimized generic functions which work on Vector and Matrix with real or complex elements.

Some of these functions are also available in the instances of the standard numeric Haskell classes provided by Numeric.LinearAlgebra.

Basic functions

module Data.Packed

constant :: Element a => a -> Int -> Vector a

creates a vector with a given number of equal components:

> constant 2 7
7 |> [2.0,2.0,2.0,2.0,2.0,2.0,2.0]

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

Creates a real vector containing a range of values:

> linspace 5 (-3,7)
5 |> [-3.0,-0.5,2.0,4.5,7.0]

Logarithmic spacing can be defined as follows:

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

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

Creates a square matrix with a given diagonal.

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

creates the identity matrix of given dimension

ctrans :: (Container Vector e, Element e) => Matrix e -> Matrix e

conjugate transpose

Generic operations

class (Complexable c, Fractional e, Element e) => Container c e where

Basic element-by-element functions for numeric containers

Methods

scalar :: e -> c e

create a structure with a single element

conj :: c e -> c e

complex conjugate

scale :: e -> c e -> c e

scaleRecip :: e -> c e -> c e

scale the element by element reciprocal of the object:

scaleRecip 2 (fromList [5,i]) == 2 |> [0.4 :+ 0.0,0.0 :+ (-2.0)]

addConstant :: e -> c e -> c e

add :: c e -> c e -> c e

sub :: c e -> c e -> c e

mul :: c e -> c e -> c e

element by element multiplication

divide :: c e -> c e -> c e

element by element division

equal :: c e -> c e -> Bool

arctan2 :: c e -> c e -> c e

cmap :: Element b => (e -> b) -> c e -> c b

cannot implement instance Functor because of Element class constraint

konst :: e -> IndexOf c -> c e

constant structure of given size

build :: IndexOf c -> ArgOf c e -> c e

create a structure using a function

Hilbert matrix of order N:

hilb n = build (n,n) (\i j -> 1/(i+j+1))

atIndex :: c e -> IndexOf c -> e

indexing function

minIndex :: c e -> IndexOf c

index of min element

maxIndex :: c e -> IndexOf c

index of max element

minElement :: c e -> e

value of min element

maxElement :: c e -> e

value of max element

sumElements :: c e -> e

the sum of elements (faster than using fold)

prodElements :: c e -> e

the product of elements (faster than using fold)

step :: RealElement e => c e -> c e

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

Arguments

:: RealElement e

=> c e

a

-> c e

b

-> c e

l

-> c e

e

-> c e

g

-> c e

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 ]

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

Find index of elements which satisfy a predicate

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

assoc

Arguments

:: IndexOf c

size

-> e

default value

-> [(IndexOf c, e)]

association list

-> c e

result

Create a structure from an association list

> assoc 5 0 [(2,7),(1,3)] :: Vector Double
 5 |> [0.0,3.0,7.0,0.0,0.0]

accum

Arguments

:: 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 ]

Instances

Container Vector Double

Container Vector Float

Container Vector a => Container Matrix a

Container Vector (Complex Double)

Container Vector (Complex Float)

Matrix product

class Element e => Product e where

Matrix product and related functions

Methods

multiply :: Matrix e -> Matrix e -> Matrix e

matrix product

dot :: Vector e -> Vector e -> e

dot (inner) product

absSum :: Vector e -> RealOf e

sum of absolute value of elements (differs in complex case from norm1)

norm1 :: Vector e -> RealOf e

sum of absolute value of elements

norm2 :: Vector e -> RealOf e

euclidean norm

normInf :: Vector e -> RealOf e

element of maximum magnitude

Instances

Product Double

Product Float

Product (Complex Double)

Product (Complex Float)

optimiseMult :: Product t => [Matrix t] -> Matrix t

Provide optimal association order for a chain of matrix multiplications and apply the multiplications.

The algorithm is the well-known O(n^3) dynamic programming algorithm that builds a pyramid of optimal associations.

m1, m2, m3, m4 :: Matrix Double
m1 = (10><15) [1..]
m2 = (15><20) [1..]
m3 = (20><5) [1..]
m4 = (5><10) [1..]

>>> optimiseMult [m1,m2,m3,m4]

will perform ((m1 multiply (m2 multiply m3)) multiply m4)

The naive left-to-right multiplication would take 4500 scalar multiplications whereas the optimised version performs 2750 scalar multiplications. The complexity in this case is 32 (= 4^3/2) * (2 comparisons, 3 scalar multiplications, 3 scalar additions, 5 lookups, 2 updates) + a constant (= three table allocations)

mXm :: Product t => Matrix t -> Matrix t -> Matrix t

mXv :: Product t => Matrix t -> Vector t -> Vector t

vXm :: Product t => Vector t -> Matrix t -> Vector t

(<.>) :: Product t => Vector t -> Vector t -> t infixl 7

Dot product: u <.> v = dot u v

class Mul a b c | a b -> c where

Methods

(<>) :: Product t => a t -> b t -> c t infixl 7

Matrix-matrix, matrix-vector, and vector-matrix products.

Instances

Mul Vector Matrix Vector

Mul Matrix Vector Vector

Mul Matrix Matrix Matrix

class LSDiv b c | b -> c, c -> b where

Methods

(<\>) :: Field t => Matrix t -> b t -> c t infixl 7

least squares solution of a linear system, similar to the \ operator of Matlab/Octave (based on linearSolveSVD)

Instances

LSDiv Vector Vector

LSDiv Matrix Matrix

outer :: Product t => Vector t -> Vector t -> Matrix t

Outer product of two vectors.

> fromList [1,2,3] `outer` fromList [5,2,3]
(3><3)
 [  5.0, 2.0, 3.0
 , 10.0, 4.0, 6.0
 , 15.0, 6.0, 9.0 ]

kronecker :: Product t => Matrix t -> Matrix t -> Matrix t

Kronecker product of two matrices.

m1=(2><3)
 [ 1.0,  2.0, 0.0
 , 0.0, -1.0, 3.0 ]
m2=(4><3)
 [  1.0,  2.0,  3.0
 ,  4.0,  5.0,  6.0
 ,  7.0,  8.0,  9.0
 , 10.0, 11.0, 12.0 ]

> kronecker m1 m2
(8><9)
 [  1.0,  2.0,  3.0,   2.0,   4.0,   6.0,  0.0,  0.0,  0.0
 ,  4.0,  5.0,  6.0,   8.0,  10.0,  12.0,  0.0,  0.0,  0.0
 ,  7.0,  8.0,  9.0,  14.0,  16.0,  18.0,  0.0,  0.0,  0.0
 , 10.0, 11.0, 12.0,  20.0,  22.0,  24.0,  0.0,  0.0,  0.0
 ,  0.0,  0.0,  0.0,  -1.0,  -2.0,  -3.0,  3.0,  6.0,  9.0
 ,  0.0,  0.0,  0.0,  -4.0,  -5.0,  -6.0, 12.0, 15.0, 18.0
 ,  0.0,  0.0,  0.0,  -7.0,  -8.0,  -9.0, 21.0, 24.0, 27.0
 ,  0.0,  0.0,  0.0, -10.0, -11.0, -12.0, 30.0, 33.0, 36.0 ]

Random numbers

data RandDist

Constructors

Uniform

uniform distribution in [0,1)

Gaussian

normal distribution with mean zero and standard deviation one

Instances

Enum RandDist

randomVector

Arguments

:: Int

seed

-> RandDist

distribution

-> Int

vector size

-> Vector Double

Obtains a vector of pseudorandom elements from the the mt19937 generator in GSL, with a given seed. Use randomIO to get a random seed.

gaussianSample

Arguments

:: Seed

-> Int

number of rows

-> Vector Double

mean vector

-> Matrix Double

covariance matrix

-> Matrix Double

result

Obtains a matrix whose rows are pseudorandom samples from a multivariate Gaussian distribution.

uniformSample

Arguments

:: Seed

-> Int

number of rows

-> [(Double, Double)]

ranges for each column

-> Matrix Double

result

Obtains a matrix whose rows are pseudorandom samples from a multivariate uniform distribution.

meanCov :: Matrix Double -> (Vector Double, Matrix Double)

Compute mean vector and covariance matrix of the rows of a matrix.

Element conversion

class Convert t where

Methods

real :: Container c t => c (RealOf t) -> c t

complex :: Container c t => c t -> c (ComplexOf t)

single :: Container c t => c t -> c (SingleOf t)

double :: Container c t => c t -> c (DoubleOf t)

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

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

Instances

Convert Double

Convert Float

Convert (Complex Double)

Convert (Complex Float)

class Complexable c

Structures that may contain complex numbers

Minimal complete definition

toComplex', fromComplex', comp', single', double'

Instances

Complexable Vector

Complexable Matrix

class (Element t, Element (Complex t), RealFloat t) => RealElement t

Supported real types

Instances

RealElement Double

RealElement Float

type family RealOf x

Instances

type RealOf Double = Double

type RealOf Float = Float

type RealOf (Complex Double) = Double

type RealOf (Complex Float) = Float

type family ComplexOf x

Instances

type ComplexOf Double = Complex Double

type ComplexOf Float = Complex Float

type ComplexOf (Complex Double) = Complex Double

type ComplexOf (Complex Float) = Complex Float

type family SingleOf x

Instances

type SingleOf Double = Float

type SingleOf Float = Float

type SingleOf (Complex a) = Complex (SingleOf a)

type family DoubleOf x

Instances

type DoubleOf Double = Double

type DoubleOf Float = Double

type DoubleOf (Complex a) = Complex (DoubleOf a)

type family IndexOf c

Instances

type IndexOf Vector = Int

type IndexOf Matrix = (Int, Int)

module Data.Complex

Input / Output

dispf :: Int -> Matrix Double -> String

Show a matrix with a given number of decimal places.

@disp = putStr . dispf 3

> disp (1/3 + ident 4) 4x4 1.333 0.333 0.333 0.333 0.333 1.333 0.333 0.333 0.333 0.333 1.333 0.333 0.333 0.333 0.333 1.333 @

disps :: Int -> Matrix Double -> String

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

@disp = putStr . disps 2

> disp $ 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

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

vecdisp :: Element t => (Matrix t -> String) -> Vector t -> String

Show a vector using a function for showing matrices.

@disp = putStr . vecdisp (dispf 2)

> disp (linspace 10 (0,1)) 10 |> 0.00 0.11 0.22 0.33 0.44 0.56 0.67 0.78 0.89 1.00 @

latexFormat

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.

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

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")

loadMatrix :: FilePath -> IO (Matrix Double)

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

saveMatrix

Arguments

:: FilePath

-> String

format (%f, %g, %e)

-> Matrix Double

-> IO ()

Saves a matrix as 2D ASCII table.

fromFile :: FilePath -> (Int, Int) -> IO (Matrix Double)

Loads a matrix from an ASCII file (the number of rows and columns must be known in advance).

fileDimensions :: FilePath -> IO (Int, Int)

obtains the number of rows and columns in an ASCII data file (provisionally using unix's wc).

readMatrix :: String -> Matrix Double

reads a matrix from a string containing a table of numbers.

fscanfVector :: FilePath -> Int -> IO (Vector Double)

Loads a vector from an ASCII file (the number of elements must be known in advance).

fprintfVector :: FilePath -> String -> Vector Double -> IO ()

Saves the elements of a vector, with a given format (%f, %e, %g), to an ASCII file.

freadVector :: FilePath -> Int -> IO (Vector Double)

Loads a vector from a binary file (the number of elements must be known in advance).

fwriteVector :: FilePath -> Vector Double -> IO ()

Saves the elements of a vector to a binary file.

Experimental

build' :: Build f => BoundsOf f -> f -> ContainerOf f

konst' :: (Konst s, Element e) => e -> s -> ContainerOf' s e