Safe Haskell	None
Language	Haskell2010

Eigen.Solver.LA

Synopsis

data Decomposition
- = PartialPivLU
- | FullPivLU
- | HouseholderQR
- | ColPivHouseholderQR
- | FullPivHouseholderQR
- | LLT
- | LDLT
- | JacobiSVD
solve :: (KnownNat n, KnownNat m, KnownNat n1, KnownNat m1, Elem a) => Decomposition -> Matrix n m a -> Matrix n1 m1 a -> Matrix m 1 a
relativeError :: (KnownNat n, KnownNat m, KnownNat n1, KnownNat m1, KnownNat n2, KnownNat m2, Elem a) => Matrix n m a -> Matrix n1 m1 a -> Matrix n2 m2 a -> a
rank :: (KnownNat n, KnownNat m, Elem a) => Decomposition -> Matrix n m a -> Int
kernel :: forall a n m. (Elem a, KnownNat n, KnownNat m) => Decomposition -> Matrix n m a -> Matrix n m a
image :: forall a n m. (Elem a, KnownNat n, KnownNat m) => Decomposition -> Matrix n m a -> Matrix n m a
linearRegression :: forall r. KnownNat r => Row r -> [[Double]] -> Maybe ([Double], Double)

Documentation

data Decomposition Source #

Decomposition           Requirements on the matrix          Speed   Accuracy  Rank  Kernel  Image

PartialPivLU            Invertible                          ++      +         -     -       -
FullPivLU               None                                -       +++       +     +       +
HouseholderQR           None                                ++      +         -     -       -
ColPivHouseholderQR     None                                +       ++        +     -       -
FullPivHouseholderQR    None                                -       +++       +     -       -
LLT                     Positive definite                   +++     +         -     -       -
LDLT                    Positive or negative semidefinite   +++     ++        -     -       -
JacobiSVD               None                                -       +++       +     -       -

The best way to do least squares solving for square matrices is with a SVD decomposition (JacobiSVD)

Constructors

PartialPivLU	LU decomposition of a matrix with partial pivoting.
FullPivLU	LU decomposition of a matrix with complete pivoting.
HouseholderQR	Householder QR decomposition of a matrix.
ColPivHouseholderQR	Householder rank-revealing QR decomposition of a matrix with column-pivoting.
FullPivHouseholderQR	Householder rank-revealing QR decomposition of a matrix with full pivoting.
LLT	Standard Cholesky decomposition (LL^T) of a matrix.
LDLT	Robust Cholesky decomposition of a matrix with pivoting.
JacobiSVD	Two-sided Jacobi SVD decomposition of a rectangular matrix.

Instances

Enum Decomposition Source #
Instance details Defined in Eigen.Solver.LA Methods succ :: Decomposition -> Decomposition # pred :: Decomposition -> Decomposition # toEnum :: Int -> Decomposition # fromEnum :: Decomposition -> Int # enumFrom :: Decomposition -> [Decomposition] # enumFromThen :: Decomposition -> Decomposition -> [Decomposition] # enumFromTo :: Decomposition -> Decomposition -> [Decomposition] # enumFromThenTo :: Decomposition -> Decomposition -> Decomposition -> [Decomposition] #
Eq Decomposition Source #
Instance details Defined in Eigen.Solver.LA Methods (==) :: Decomposition -> Decomposition -> Bool # (/=) :: Decomposition -> Decomposition -> Bool #
Read Decomposition Source #
Instance details Defined in Eigen.Solver.LA Methods readsPrec :: Int -> ReadS Decomposition # readList :: ReadS [Decomposition] # readPrec :: ReadPrec Decomposition # readListPrec :: ReadPrec [Decomposition] #
Show Decomposition Source #
Instance details Defined in Eigen.Solver.LA Methods showsPrec :: Int -> Decomposition -> ShowS # show :: Decomposition -> String # showList :: [Decomposition] -> ShowS #

solve :: (KnownNat n, KnownNat m, KnownNat n1, KnownNat m1, Elem a) => Decomposition -> Matrix n m a -> Matrix n1 m1 a -> Matrix m 1 a Source #

x = solve d a b: finds a solution x of ax = b equation using decomposition d

relativeError :: (KnownNat n, KnownNat m, KnownNat n1, KnownNat m1, KnownNat n2, KnownNat m2, Elem a) => Matrix n m a -> Matrix n1 m1 a -> Matrix n2 m2 a -> a Source #

e = relativeError x a b: computes norm (ax - b) / norm b where norm is L2 norm

rank :: (KnownNat n, KnownNat m, Elem a) => Decomposition -> Matrix n m a -> Int Source #

The rank of the matrix.

kernel :: forall a n m. (Elem a, KnownNat n, KnownNat m) => Decomposition -> Matrix n m a -> Matrix n m a Source #

Return the matrix whose columns form a basis of the null-space of A.

image :: forall a n m. (Elem a, KnownNat n, KnownNat m) => Decomposition -> Matrix n m a -> Matrix n m a Source #

Return a matrix whose columns form a basis of the column-space of A.

linearRegression :: forall r. KnownNat r => Row r -> [[Double]] -> Maybe ([Double], Double) Source #

(coeffs, error) = linearRegression points: computes multiple linear regression y = a1 x1 + a2 x2 + ... + an xn + b using ColPivHouseholderQR decomposition

point format is [y, x1..xn]
coeffs format is [b, a1..an]
error is calculated using relativeError

import Data.Eigen.LA
main = print $ linearRegression (Row @5)
  [
    [-4.32, 3.02, 6.89],
    [-3.79, 2.01, 5.39],
    [-4.01, 2.41, 6.01],
    [-3.86, 2.09, 5.55],
    [-4.10, 2.58, 6.32]
  ]

produces the following output

Just ([-2.3466569233817127,-0.2534897541434826,-0.1749653335680988],1.8905965120153139e-3)