Numeric.LinearAlgebra.Sparse

Sparsify : remove almost-0 elements (|x| < eps)

sparsifySV :: SpVector Double -> SpVector Double

Sparsify an SpVector

Matrix condition number

conditionNumberSM :: SpMatrix Double -> Double

uses the R matrix from the QR factorization

Householder transformation

hhMat :: Num a => a -> SpVector a -> SpMatrix a

hhRefl :: SpVector Double -> SpMatrix Double

a vector x uniquely defines an orthogonal plane; the Householder operator reflects any point v with respect to this plane: v' = (I - 2 x >< x) v

Givens rotation matrix

hypot :: Floating a => a -> a -> a

sign :: (Ord a, Num a) => a -> a

givensCoef :: (Ord a, Floating a) => a -> a -> (a, a, a)

Givens coefficients (using stable algorithm shown in Anderson, Edward (4 December 2000). "Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem". LAPACK Working Note)

givens :: SpMatrix Double -> IxRow -> IxCol -> SpMatrix Double

Givens method, row version: choose other row index i' s.t. i' is : * below the diagonal * corresponding element is nonzero

QR.C1 ) To zero out entry A(i, j) we must find row k such that A(k, j) is non-zero but A has zeros in row k for all columns less than j.

firstNonZeroColumn :: IntMap a -> IxRow -> Bool

Is the kth the first nonzero column in the row?

candidateRows :: IntMap (IntMap a) -> IxRow -> IxCol -> Maybe [Key]

Returns a set of rows {k} that satisfy QR.C1

QR decomposition

qr :: SpMatrix Double -> (SpMatrix Double, SpMatrix Double)

Applies Givens rotation iteratively to zero out sub-diagonal elements

gmats :: SpMatrix Double -> [SpMatrix Double]

Givens matrices in order [G1, G2, .. , G_N ]

Eigenvalue algorithms

All eigenvalues (QR algorithm)

eigsQR :: Int -> SpMatrix Double -> SpVector Double

`eigsQR n mm` performs n iterations of the QR algorithm on matrix mm

One eigenvalue and eigenvector (Rayleigh iteration)

eigRayleigh :: Int -> SpMatrix Double -> (SpVector Double, Double) -> (SpVector Double, Double)

`eigsRayleigh n mm` performs n iterations of the Rayleigh algorithm on matrix mm. Cubic-order convergence, but it requires a mildly educated guess on the initial eigenpair

Householder vector

hhV :: SpVector Double -> (SpVector Double, Double)

Householder bidiagonalization

SVD

LU factorization

permutAA :: Num b => SpMatrix a -> IxRow -> IxCol -> Maybe (SpMatrix b)

Iterative linear solvers

BCG

bcgStep :: SpMatrix Double -> BCG -> BCG

one step of BCG

data BCG

Constructors

BCG

Fields

• _xBcg, _rBcg, _rHatBcg, _pBcg, _pHatBcg :: SpVector Double

Instances

Eq BCG

Methods

(==) :: BCG -> BCG -> Bool

(/=) :: BCG -> BCG -> Bool

Show BCG

Methods

showsPrec :: Int -> BCG -> ShowS

show :: BCG -> String

showList :: [BCG] -> ShowS

bcg :: SpMatrix Double -> SpVector Double -> SpVector Double -> BCG

CGS

cgsStep :: SpMatrix Double -> SpVector Double -> CGS -> CGS

one step of CGS

data CGS

Constructors

CGS

Fields

• _x, _r, _p, _u :: SpVector Double

Instances

Eq CGS

Methods

(==) :: CGS -> CGS -> Bool

(/=) :: CGS -> CGS -> Bool

Show CGS

Methods

showsPrec :: Int -> CGS -> ShowS

show :: CGS -> String

showList :: [CGS] -> ShowS

cgs :: SpMatrix Double -> SpVector Double -> SpVector Double -> SpVector Double -> CGS

iterate solver until convergence or until max # of iterations is reached

BiCGSTAB

bicgstabStep :: SpMatrix Double -> SpVector Double -> BICGSTAB -> BICGSTAB

one step of BiCGSTAB

data BICGSTAB

Constructors

BICGSTAB

Fields

• _xBicgstab, _rBicgstab, _pBicgstab :: SpVector Double

Instances

Eq BICGSTAB

Methods

(==) :: BICGSTAB -> BICGSTAB -> Bool

(/=) :: BICGSTAB -> BICGSTAB -> Bool

Show BICGSTAB

Methods

showsPrec :: Int -> BICGSTAB -> ShowS

show :: BICGSTAB -> String

showList :: [BICGSTAB] -> ShowS

bicgstab :: SpMatrix Double -> SpVector Double -> SpVector Double -> SpVector Double -> BICGSTAB

iterate solver until convergence or until max # of iterations is reached

Moore-Penrose pseudoinverse

pinv :: SpMatrix Double -> SpVector Double -> SpVector Double

Least-squares approximation of a rectangular system of equaitons. Uses \ for the linear solve

Linear solver interface

data LinSolveMethod

Constructors

BCG_

CGS_

BICGSTAB_

Instances

Eq LinSolveMethod

Methods

(==) :: LinSolveMethod -> LinSolveMethod -> Bool

(/=) :: LinSolveMethod -> LinSolveMethod -> Bool

Show LinSolveMethod

Methods

showsPrec :: Int -> LinSolveMethod -> ShowS

show :: LinSolveMethod -> String

showList :: [LinSolveMethod] -> ShowS

linSolve :: LinSolveMethod -> SpMatrix Double -> SpVector Double -> SpVector Double

Linear solve with _deterministic_ starting vector (every component at 0.1)

(<\>) :: SpMatrix Double -> SpVector Double -> SpVector Double

\ : linSolve using the BiCGSTAB method as default

Control primitives for bounded iteration with convergence check

modifyUntil :: MonadState s m => (s -> Bool) -> (s -> s) -> m s

transform state until a condition is met

loopUntilAcc :: Int -> ([t] -> Bool) -> (t -> t) -> t -> t

Keep a moving window buffer (length 2) of state x to assess convergence, stop when either a condition on that list is satisfied or when max # of iterations is reached

modifyInspectN :: MonadState s m => Int -> ([s] -> Bool) -> (s -> s) -> m s

Keep a moving window buffer (length 2) of state x to assess convergence, stop when either a condition on that list is satisfied or when max # of iterations is reached (i.e. same thing as loopUntilAcc but this one runs in the State monad)

meanl :: (Foldable t, Fractional a) => t a -> a

norm2l :: (Foldable t, Functor t, Floating a) => t a -> a

diffSqL :: Floating a => [a] -> a

untilConverged :: MonadState a m => (a -> SpVector Double) -> (a -> a) -> m a

iterate until convergence is verified or we run out of a fixed iteration budget

normDiffConverged :: (Foldable t, Functor t) => (a -> SpVector Double) -> t a -> Bool

convergence check (FIXME)

runAppendN :: ([t] -> Bool) -> (t -> t) -> Int -> t -> [t]

run niter iterations and append the state x to a list xs, stop when either the xs satisfies a predicate q or when the counter reaches 0

runAppendN' :: (t -> t) -> Int -> t -> [t]

", NO convergence check

Random matrices and vectors

randMat :: PrimMonad m => Int -> m (SpMatrix Double)

Dense SpMatrix

randVec :: PrimMonad m => Int -> m (SpVector Double)

Dense SpVector

randSpMat :: Int -> Int -> IO (SpMatrix Double)

Sparse SpMatrix

randSpVec :: Int -> Int -> IO (SpVector Double)

Sparse SpVector

Pretty printing

showNonZero :: (Show a, Num a, Eq a) => a -> String

toDenseRow :: Num a => SpMatrix a -> Key -> [a]

toDenseRowClip :: (Show a, Num a) => SpMatrix a -> Key -> Int -> String

newline :: IO ()

printDenseSM :: (Show t, Num t) => SpMatrix t -> IO ()

toDenseListClip :: (Show a, Num a) => SpVector a -> Int -> String

printDenseSV :: (Show t, Num t) => SpVector t -> IO ()

Pretty printer typeclass

class PrintDense a where

Minimal complete definition

prd

Methods

prd :: a -> IO ()

Instances

(Show a, Num a) => PrintDense (SpVector a)

Methods

prd :: SpVector a -> IO ()

(Show a, Num a) => PrintDense (SpMatrix a)

Methods

prd :: SpMatrix a -> IO ()

Type synonyms for SpMatrix