Numeric.LinearAlgebra.Sparse

Matrix factorizations

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

Applies Givens rotation iteratively to zero out sub-diagonal elements

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

LU factors

Incomplete LU

ilu0 :: SpMatrix Double -> (SpMatrix Double, SpMatrix Double)

used for Incomplete LU : remove entries in m corresponding to zero entries in m2

Condition number

conditionNumberSM :: SpMatrix Double -> Double

uses the R matrix from the QR factorization

Householder reflection

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

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.

Eigensolvers

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

`eigsQR n mm` performs n iterations of the QR algorithm on matrix mm, and returns a SpVector containing all eigenvalues

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

`eigsRayleigh n mm` performs n iterations of the Rayleigh algorithm on matrix mm and returns the eigenpair closest to the initialization. It displays cubic-order convergence, but it also requires an educated guess on the initial eigenpair

Linear solvers

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

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

data LinSolveMethod

Instances

Eq LinSolveMethod

Methods

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

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

Show LinSolveMethod

Methods

showsPrec :: Int -> LinSolveMethod -> ShowS

show :: LinSolveMethod -> String

showList :: [LinSolveMethod] -> ShowS

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

\ : linSolve using the BiCGSTAB method as default

Methods

cgne :: SpMatrix Double -> SpVector Double -> SpVector Double -> CGNE

tfqmr :: SpMatrix Double -> SpVector Double -> SpVector Double -> TFQMR

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

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

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

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

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

_xCgne :: CGNE -> SpVector Double

_xTfq :: TFQMR -> SpVector Double

_xBicgstab :: BICGSTAB -> SpVector Double

_x :: CGS -> SpVector Double

_xBcg :: BCG -> SpVector Double

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

one step of CGS

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

one step of BiCGSTAB

data CGNE

Instances

Eq CGNE

Methods

(==) :: CGNE -> CGNE -> Bool

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

data TFQMR

Instances

Eq TFQMR

Methods

(==) :: TFQMR -> TFQMR -> Bool

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

data BICGSTAB

Instances

Eq BICGSTAB

Methods

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

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

Show BICGSTAB

Methods

showsPrec :: Int -> BICGSTAB -> ShowS

show :: BICGSTAB -> String

showList :: [BICGSTAB] -> ShowS

data CGS

Instances

Eq CGS

Methods

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

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

Show CGS

Methods

showsPrec :: Int -> CGS -> ShowS

show :: CGS -> String

showList :: [CGS] -> ShowS

data BCG

Instances

Eq BCG

Methods

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

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

Show BCG

Methods

showsPrec :: Int -> BCG -> ShowS

show :: BCG -> String

showList :: [BCG] -> ShowS

Matrix partitioning

diagPartitions :: SpMatrix a -> (SpMatrix a, SpMatrix a, SpMatrix a)

Partition a matrix into strictly subdiagonal, diagonal and strictly superdiagonal parts

Random arrays

randArray :: PrimMonad m => Int -> Double -> Double -> m [Double]

Random matrices and vectors

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

Dense SpMatrix

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

Dense SpVector

Sparse "

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

Sparse SpMatrix

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

Sparse SpVector

Sparsify data

sparsifySV :: SpVector Double -> SpVector Double

Sparsify an SpVector

Iteration combinators

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)

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

", NO convergence check

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