-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Numerical computation in native Haskell
--   
--   Currently the library provides iterative linear solvers, matrix
--   decompositions, eigenvalue computations and related utilities. Please
--   see README.md for details
@package sparse-linear-algebra
@version 0.2.1.1

module Numeric.LinearAlgebra.Class
class Functor f => Additive f

-- | Ring zero element
zero :: (Additive f, Num a) => f a

-- | Ring +
(^+^) :: (Additive f, Num a) => f a -> f a -> f a
one :: (Additive f, Num a) => f a
(^*^) :: (Additive f, Num a) => f a -> f a -> f a

-- | negate the values in a functor
negated :: (Num a, Functor f) => f a -> f a

-- | subtract two Additive objects
(^-^) :: (Additive f, Num a) => f a -> f a -> f a
class Additive f => VectorSpace f

-- | multiplication by a scalar
(.*) :: (VectorSpace f, Num a) => a -> f a -> f a

-- | linear interpolation
lerp :: (VectorSpace f, Num a) => a -> f a -> f a -> f a
class VectorSpace f => Hilbert f

-- | inner product
dot :: (Hilbert f, Num a) => f a -> f a -> a

-- | `hilbertDistSq x y = || x - y ||^2`
hilbertDistSq :: (Hilbert f, Num a) => f a -> f a -> a
class Hilbert f => Normed f
norm :: (Normed f, Floating a, Eq a) => a -> f a -> a

-- | Squared 2-norm
normSq :: (Hilbert f, Num a) => f a -> a

-- | L1 norm
norm1 :: (Foldable t, Num a, Functor t) => t a -> a

-- | Euclidean norm
norm2 :: (Hilbert f, Floating a) => f a -> a

-- | Lp norm (p > 0)
normP :: (Foldable t, Functor t, Floating a) => a -> t a -> a

-- | Infinity-norm
normInfty :: (Foldable t, Ord a) => t a -> a

-- | Normalize w.r.t. p-norm (p finite)
normalize :: (Normed f, Floating a, Eq a) => a -> f a -> f a

-- | Lp inner product (p > 0)
dotLp :: (Set t, Foldable t, Floating a) => a -> t a -> t a -> a

-- | Reciprocal
reciprocal :: (Functor f, Fractional b) => f b -> f b

-- | Scale
scale :: (Num b, Functor f) => b -> f b -> f b
class Additive f => FiniteDim f where type FDSize f :: * where {
    type family FDSize f :: *;
}
dim :: FiniteDim f => f a -> FDSize f

-- | unary dimension-checking bracket
withDim :: (FiniteDim f, Show e) => f a -> (FDSize f -> f a -> Bool) -> (f a -> c) -> String -> (f a -> e) -> c

-- | binary dimension-checking bracket
withDim2 :: (FiniteDim f, FiniteDim g, Show e) => f a -> g b -> (FDSize f -> FDSize g -> f a -> g b -> Bool) -> (f a -> g b -> c) -> String -> (f a -> g b -> e) -> c
class Additive f => HasData f a where type HDData f a :: * where {
    type family HDData f a :: *;
}
dat :: HasData f a => f a -> HDData f a
class (FiniteDim f, HasData f a) => Sparse f a
spy :: (Sparse f a, Fractional b) => f a -> b
class Functor f => Set f

-- | union binary lift : apply function on _union_ of two Sets
liftU2 :: Set f => (a -> a -> a) -> f a -> f a -> f a

-- | intersection binary lift : apply function on _intersection_ of two
--   Sets
liftI2 :: Set f => (a -> b -> c) -> f a -> f b -> f c
class Sparse c a => SpContainer c a where type ScIx c :: * where {
    type family ScIx c :: *;
}
scInsert :: SpContainer c a => ScIx c -> a -> c a -> c a
scLookup :: SpContainer c a => c a -> ScIx c -> Maybe a
(@@) :: SpContainer c a => c a -> ScIx c -> a


-- | Testing for values "near" zero
module Numeric.Eps

-- | Provides a test to see if a quantity is near zero.
--   
--   <pre>
--   &gt;&gt;&gt; nearZero (1e-11 :: Double)
--   False
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; nearZero (1e-17 :: Double)
--   True
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; nearZero (1e-5 :: Float)
--   False
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; nearZero (1e-7 :: Float)
--   True
--   </pre>
class Num a => Epsilon a

-- | Determine if a quantity is near zero.
nearZero :: Epsilon a => a -> Bool

-- | Determine if a quantity is near zero.
nearZero :: Epsilon a => a -> Bool

-- | Rounding rule
isNz :: Epsilon a => a -> Bool
roundZero :: Epsilon a => a -> a
roundOne :: Epsilon a => a -> a

-- | Round to respectively 0 or 1
roundZeroOne :: Epsilon a => a -> a
instance Numeric.Eps.Epsilon GHC.Types.Float
instance Numeric.Eps.Epsilon GHC.Types.Double
instance Numeric.Eps.Epsilon Foreign.C.Types.CFloat
instance Numeric.Eps.Epsilon Foreign.C.Types.CDouble

module Data.Sparse.Utils

-- | Wrap a function with a null check, returning in Maybe
harness :: (t -> Bool) -> (t -> a) -> t -> Maybe a

-- | Componentwise tuple operations TODO : use semilattice properties
--   instead
maxTup :: Ord t => (t, t) -> (t, t) -> (t, t)

-- | Componentwise tuple operations TODO : use semilattice properties
--   instead
minTup :: Ord t => (t, t) -> (t, t) -> (t, t)

-- | integer-indexed ziplist
denseIxArray :: [b] -> [(Int, b)]

-- | ", 2d arrays
denseIxArray2 :: Int -> [c] -> [(Int, Int, c)]

-- | foldr over the results of a fmap
foldrMap :: (Foldable t, Functor t) => (a -> b) -> (b -> c -> c) -> c -> t a -> c

-- | strict left fold
foldlStrict :: (a -> b -> a) -> a -> [b] -> a

-- | indexed right fold
ifoldr :: Num i => (a -> b -> b) -> b -> (i -> c -> d -> a) -> c -> [d] -> b
type LB = Int
type UB = Int
inBounds :: LB -> UB -> Int -> Bool
inBounds2 :: (LB, UB) -> (Int, Int) -> Bool
inBounds0 :: UB -> Int -> Bool
inBounds02 :: (UB, UB) -> (Int, Int) -> Bool
head' :: Vector a -> Maybe a
tail' :: Vector a -> Maybe (Vector a)

module Data.Sparse.Types
type Rows = Int
type Cols = Int
type IxRow = Int
type IxCol = Int

module Data.Sparse.IntMap2.IntMap2

-- | Insert an element
insertIM2 :: Key -> Key -> a -> IntMap (IntMap a) -> IntMap (IntMap a)

-- | Lookup a key
lookupIM2 :: Key -> Key -> IntMap (IntMap a) -> Maybe a

-- | Lookup with default 0
lookupWD_IM :: Num a => IntMap (IntMap a) -> (IxRow, IxCol) -> a

-- | Populate an IM2 from a list of (row index, column index, value)
fromListIM2 :: Foldable t => t (Key, Key, a) -> IntMap (IntMap a) -> IntMap (IntMap a)

-- | Indexed left fold over an IM2, with general accumulator
ifoldlIM2' :: (Key -> Key -> a -> b -> b) -> b -> IntMap (IntMap a) -> b

-- | Indexed left fold over an IM2
ifoldlIM2 :: (Key -> Key -> t -> IntMap a -> IntMap a) -> IntMap (IntMap t) -> IntMap a

-- | Left fold over an IM2, with general accumulator
foldlIM2 :: (a -> b -> b) -> b -> IntMap (IntMap a) -> b

-- | Inner indices become outer ones and vice versa. No loss of information
--   because both inner and outer IntMaps are nubbed.
transposeIM2 :: IntMap (IntMap a) -> IntMap (IntMap a)

-- | Map over outer IM and filter all inner IM's
ifilterIM2 :: (Key -> Key -> a -> Bool) -> IntMap (IntMap a) -> IntMap (IntMap a)

-- | Specialized filtering : keep only sub-diagonal elements
filterSubdiag :: IntMap (IntMap a) -> IntMap (IntMap a)
countSubdiagonalNZ :: IntMap (IntMap a) -> Int

-- | List of (row, col) indices of (nonzero) subdiagonal elements
subdiagIndices :: IntMap (IntMap a) -> [(Key, Key)]
rpairs :: (a, [b]) -> [(a, b)]

-- | Map over IM2
mapIM2 :: (a -> b) -> IntMap (IntMap a) -> IntMap (IntMap b)

-- | Indexed map over IM2
imapIM2 :: (Key -> Key -> a -> b) -> IntMap (IntMap a) -> IntMap (IntMap b)

-- | Mapping keys
mapKeysIM2 :: (Key -> Key) -> (Key -> Key) -> IntMap (IntMap a) -> IntMap (IntMap a)
mapColumnIM2 :: (b -> b) -> IntMap (IntMap b) -> Int -> IntMap (IntMap b)
instance Numeric.LinearAlgebra.Class.Set Data.IntMap.Base.IntMap
instance Numeric.LinearAlgebra.Class.Additive Data.IntMap.Base.IntMap
instance Numeric.LinearAlgebra.Class.VectorSpace Data.IntMap.Base.IntMap
instance Numeric.LinearAlgebra.Class.Hilbert Data.IntMap.Base.IntMap
instance Numeric.LinearAlgebra.Class.Normed Data.IntMap.Base.IntMap


module Data.Sparse.SpMatrix
data SpMatrix a
SM :: (Rows, Cols) -> IntMap (IntMap a) -> SpMatrix a
[smDim] :: SpMatrix a -> (Rows, Cols)
[smData] :: SpMatrix a -> IntMap (IntMap a)
sizeStr :: SpMatrix a -> String

-- | `zeroSM m n` : Empty SpMatrix of size (m, n)
zeroSM :: Rows -> Cols -> SpMatrix a
mkDiagonal :: Int -> [a] -> SpMatrix a

-- | `eye n` : identity matrix of rank <tt>n</tt>
eye :: Num a => Int -> SpMatrix a

-- | Permutation matrix from a (possibly incomplete) list of row swaps
--   starting from row 0 e.g. `permutationSM 5 [1,3]` first swaps rows (0,
--   1) and then rows (1, 3) :
--   
--   <ul>
--   <li><i>0,1,0,0,0</i></li>
--   <li><i>0,0,0,1,0</i></li>
--   <li><i>0,0,1,0,0</i></li>
--   <li><i>1,0,0,0,0</i></li>
--   <li><i>0,0,0,0,1</i></li>
--   </ul>
permutationSM :: Num a => Int -> [IxRow] -> SpMatrix a

-- | Permutation matrix from a (possibly incomplete) list of row pair swaps
--   e.g. `permutPairs 5 [(2,4)]` swaps rows 2 and 4 :
--   
--   <ul>
--   <li><i>1,0,0,0,0</i></li>
--   <li><i>0,1,0,0,0</i></li>
--   <li><i>0,0,0,0,1</i></li>
--   <li><i>0,0,0,1,0</i></li>
--   <li><i>0,0,1,0,0</i></li>
--   </ul>
permutPairsSM :: Num a => Int -> [(IxRow, IxRow)] -> SpMatrix a

-- | `mkSubDiagonal n o xx` creates a square SpMatrix of size <tt>n</tt>
--   with <tt>xx</tt> on the <tt>o</tt>th subdiagonal
mkSubDiagonal :: Int -> Int -> [a] -> SpMatrix a

-- | Insert an element in a preexisting Spmatrix at the specified indices
insertSpMatrix :: IxRow -> IxCol -> a -> SpMatrix a -> SpMatrix a

-- | Add to existing SpMatrix using data from list (row, col, value)
fromListSM' :: Foldable t => t (IxRow, IxCol, a) -> SpMatrix a -> SpMatrix a

-- | Create new SpMatrix using data from list (row, col, value)
fromListSM :: Foldable t => (Int, Int) -> t (IxRow, IxCol, a) -> SpMatrix a

-- | Create new SpMatrix assuming contiguous, 0-based indexing of elements
fromListDenseSM :: Int -> [a] -> SpMatrix a

-- | Populate list with SpMatrix contents and populate missing entries with
--   0
toDenseListSM :: Num t => SpMatrix t -> [(IxRow, IxCol, t)]
lookupSM :: SpMatrix a -> IxRow -> IxCol -> Maybe a

-- | Looks up an element in the matrix with a default (if the element is
--   not found, zero is returned)
lookupWD_SM :: Num a => SpMatrix a -> (IxRow, IxCol) -> a

-- | Zero-default lookup, infix form (no bound checking)
--   
--   Looks up an element in the matrix with a default (if the element is
--   not found, zero is returned)
(@@!) :: Num a => SpMatrix a -> (IxRow, IxCol) -> a

-- | Indexed filtering function
filterSM :: (Key -> Key -> a -> Bool) -> SpMatrix a -> SpMatrix a

-- | Diagonal, subdiagonal, superdiagonal partitions of a SpMatrix (useful
--   for writing preconditioners)
extractDiag :: SpMatrix a -> SpMatrix a

-- | Diagonal, subdiagonal, superdiagonal partitions of a SpMatrix (useful
--   for writing preconditioners)
extractSuperDiag :: SpMatrix a -> SpMatrix a

-- | Diagonal, subdiagonal, superdiagonal partitions of a SpMatrix (useful
--   for writing preconditioners)
extractSubDiag :: SpMatrix a -> SpMatrix a

-- | Extract a submatrix given the specified index bounds, rebalancing keys
--   with the two supplied functions
extractSubmatrixSM :: (Key -> Key) -> (Key -> Key) -> SpMatrix a -> (IxRow, IxRow) -> (IxCol, IxCol) -> SpMatrix a

-- | Extract a submatrix given the specified index bounds NB : subtracts
--   (i1, j1) from the indices
extractSubmatrixRebalanceKeys :: SpMatrix a -> (IxRow, IxRow) -> (IxCol, IxCol) -> SpMatrix a

-- | Extract a submatrix given the specified index bounds NB : submatrix
--   indices are _preserved_
extractSubmatrix :: SpMatrix a -> (IxRow, IxRow) -> (IxCol, IxCol) -> SpMatrix a

-- | Extract whole column
extractColSM :: SpMatrix a -> IxCol -> SpMatrix a

-- | Extract column within a row range
extractSubColSM :: SpMatrix a -> IxCol -> (IxRow, IxRow) -> SpMatrix a

-- | Extract column within a row range, rebalance keys
extractSubColSM_RK :: SpMatrix a -> IxCol -> (IxRow, IxRow) -> SpMatrix a

-- | Are the supplied indices within matrix bounds?
isValidIxSM :: SpMatrix a -> (Int, Int) -> Bool

-- | Is the matrix square?
isSquareSM :: SpMatrix a -> Bool

-- | Is the matrix diagonal?
isDiagonalSM :: SpMatrix a -> Bool

-- | Is the matrix lower/upper triangular?
isLowerTriSM :: Eq a => SpMatrix a -> Bool

-- | Is the matrix lower/upper triangular?
isUpperTriSM :: Eq a => SpMatrix a -> Bool

-- | Is the matrix orthogonal? i.e. Q^t ## Q == I
isOrthogonalSM :: (Eq a, Epsilon a) => SpMatrix a -> Bool

-- | Data in internal representation (do not export)
immSM :: SpMatrix t -> IntMap (IntMap t)

-- | (Number of rows, Number of columns)
dimSM :: SpMatrix t -> (Rows, Cols)

-- | Number of rows times number of columns
nelSM :: SpMatrix t -> Int

-- | Number of rows
nrows :: SpMatrix a -> Rows

-- | Number of columns
ncols :: SpMatrix a -> Cols
data SMInfo
SMInfo :: Int -> Double -> SMInfo
[smNz] :: SMInfo -> Int
[smSpy] :: SMInfo -> Double
infoSM :: SpMatrix a -> SMInfo
nzSM :: SpMatrix a -> Int
spySM :: Fractional b => SpMatrix a -> b
nzRow :: SpMatrix a -> Key -> Int
bwMinSM :: SpMatrix a -> Int
bwMaxSM :: SpMatrix a -> Int
bwBoundsSM :: SpMatrix a -> (Int, Int)

-- | Vertical stacking
vertStackSM :: SpMatrix a -> SpMatrix a -> SpMatrix a

-- | Vertical stacking
(-=-) :: SpMatrix a -> SpMatrix a -> SpMatrix a

-- | Horizontal stacking
horizStackSM :: SpMatrix a -> SpMatrix a -> SpMatrix a

-- | Horizontal stacking
(-||-) :: SpMatrix a -> SpMatrix a -> SpMatrix a
ifilterSM :: (Key -> Key -> a -> Bool) -> SpMatrix a -> SpMatrix a

-- | Left fold over SpMatrix
foldlSM :: (a -> b -> b) -> b -> SpMatrix a -> b

-- | Indexed left fold over SpMatrix
ifoldlSM :: (Key -> Key -> a -> b -> b) -> b -> SpMatrix a -> b

-- | Count sub-diagonal nonzeros
countSubdiagonalNZSM :: SpMatrix a -> Int

-- | Filter the index subset that lies below the diagonal (used in the QR
--   decomposition, for example)
subdiagIndicesSM :: SpMatrix a -> [(IxRow, IxCol)]
sparsifyIM2 :: Epsilon a => IntMap (IntMap a) -> IntMap (IntMap a)

-- | Sparsify an SpMatrix
sparsifySM :: Epsilon a => SpMatrix a -> SpMatrix a

-- | Round almost-0 and almost-1 to 0 and 1 respectively
roundZeroOneSM :: Epsilon a => SpMatrix a -> SpMatrix a

-- | Swap two rows of a SpMatrix (bounds not checked)
swapRows :: IxRow -> IxRow -> SpMatrix a -> SpMatrix a

-- | Swap two rows of a SpMatrix (bounds checked)
swapRowsSafe :: IxRow -> IxRow -> SpMatrix a -> SpMatrix a

-- | transposeSM : Matrix transpose
transposeSM :: SpMatrix a -> SpMatrix a
matScale :: Num a => a -> SpMatrix a -> SpMatrix a
normFrobenius :: Floating a => SpMatrix a -> a
matMat :: Num a => SpMatrix a -> SpMatrix a -> SpMatrix a
(##) :: Num a => SpMatrix a -> SpMatrix a -> SpMatrix a

-- | Removes all elements <tt>x</tt> for which `| x | &lt;= eps`)
matMatSparsified :: Epsilon a => SpMatrix a -> SpMatrix a -> SpMatrix a

-- | Removes all elements <tt>x</tt> for which `| x | &lt;= eps`)
(#~#) :: Epsilon a => SpMatrix a -> SpMatrix a -> SpMatrix a

-- | A^T B
(#^#) :: Epsilon a => SpMatrix a -> SpMatrix a -> SpMatrix a

-- | A B^T
(##^) :: Epsilon a => SpMatrix a -> SpMatrix a -> SpMatrix a

-- | Contract row <tt>i</tt> of A with column <tt>j</tt> of B up to an
--   index <tt>n</tt>, i.e. summing over repeated indices: Aij Bjk , for j
--   in [0 .. n]
contractSub :: Num a => SpMatrix a -> SpMatrix a -> IxRow -> IxCol -> Int -> a
instance GHC.Show.Show Data.Sparse.SpMatrix.SMInfo
instance GHC.Classes.Eq Data.Sparse.SpMatrix.SMInfo
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Sparse.SpMatrix.SpMatrix a)
instance GHC.Show.Show a => GHC.Show.Show (Data.Sparse.SpMatrix.SpMatrix a)
instance GHC.Base.Functor Data.Sparse.SpMatrix.SpMatrix
instance Numeric.LinearAlgebra.Class.Set Data.Sparse.SpMatrix.SpMatrix
instance Numeric.LinearAlgebra.Class.Additive Data.Sparse.SpMatrix.SpMatrix
instance Numeric.LinearAlgebra.Class.FiniteDim Data.Sparse.SpMatrix.SpMatrix
instance Numeric.LinearAlgebra.Class.HasData Data.Sparse.SpMatrix.SpMatrix a
instance Numeric.LinearAlgebra.Class.Sparse Data.Sparse.SpMatrix.SpMatrix a
instance GHC.Num.Num a => Numeric.LinearAlgebra.Class.SpContainer Data.Sparse.SpMatrix.SpMatrix a


module Data.Sparse.SpVector
data SpVector a
SV :: Int -> IntMap a -> SpVector a
[svDim] :: SpVector a -> Int
[svData] :: SpVector a -> IntMap a

-- | SpVector sparsity
spySV :: Fractional b => SpVector a -> b

-- | Number of nonzeros
nzSV :: SpVector a -> Int
sizeStrSV :: SpVector a -> String

-- | empty sparse vector (length n, no entries)
zeroSV :: Int -> SpVector a

-- | singleton sparse vector (length 1)
singletonSV :: a -> SpVector a

-- | create a sparse vector from an association list while discarding all
--   zero entries
mkSpVector :: Epsilon a => Int -> IntMap a -> SpVector a

-- | ", from logically dense array (consecutive indices)
mkSpVectorD :: Epsilon a => Int -> [a] -> SpVector a
mkSpVector1 :: Int -> IntMap a -> SpVector a

-- | Create new sparse vector, assumin 0-based, contiguous indexing
fromListDenseSV :: Int -> [a] -> SpVector a

-- | Map a function over a range of indices and filter the result (indices
--   and values) to fit in a <tt>n</tt>-long SpVector
spVectorDenseIx :: Epsilon a => (Int -> a) -> UB -> [Int] -> SpVector a

-- | ", using just the integer bounds of the interval
spVectorDenseLoHi :: Epsilon a => (Int -> a) -> UB -> Int -> Int -> SpVector a

-- | one-hot encoding : `oneHotSV n k` produces a SpVector of length n
--   having 1 at the k-th position
oneHotSVU :: Num a => Int -> IxRow -> SpVector a
oneHotSV :: Num a => Int -> IxRow -> SpVector a

-- | DENSE vector of `1`s
onesSV :: Num a => Int -> SpVector a

-- | DENSE vector of `0`s
zerosSV :: Num a => Int -> SpVector a

-- | Populate a SpVector with the contents of a Vector.
fromVector :: Vector a -> SpVector a

-- | Populate a Vector with the entries of a SpVector, discarding the
--   indices (NB: loses sparsity information).
toVector :: SpVector a -> Vector a

-- | <ul>
--   <li>- | Populate a Vector with the entries of a SpVector, replacing
--   the missing entries with 0</li>
--   </ul>
toVectorDense :: Num a => SpVector a -> Vector a

-- | insert element <tt>x</tt> at index <tt>i</tt> in a preexisting
--   SpVector
insertSpVector :: Int -> a -> SpVector a -> SpVector a
fromListSV :: Int -> [(Int, a)] -> SpVector a
toListSV :: SpVector a -> [(Key, a)]

-- | To dense list (default = 0)
toDenseListSV :: Num b => SpVector b -> [b]

-- | Indexed fold over SpVector
ifoldSV :: (Key -> a -> b -> b) -> b -> SpVector a -> b

-- | Lookup an index in a SpVector
lookupSV :: Key -> SpVector a -> Maybe a

-- | Lookup an index, return a default value if lookup fails
lookupDefaultSV :: a -> Key -> SpVector a -> a

-- | Lookup an index in a SpVector, returns 0 if lookup fails
lookupDenseSV :: Num a => Key -> SpVector a -> a

-- | Tail elements
tailSV :: SpVector a -> SpVector a

-- | Head element
headSV :: Num a => SpVector a -> a

-- | Keep the first n components of the SpVector (like <a>take</a> for
--   lists)
takeSV :: Int -> SpVector a -> SpVector a

-- | Discard the first n components of the SpVector and rebalance the keys
--   (like <a>drop</a> for lists)
--   
--   Keep the first n components of the SpVector (like <a>take</a> for
--   lists)
dropSV :: Int -> SpVector a -> SpVector a

-- | Concatenate two sparse vectors
concatSV :: SpVector a -> SpVector a -> SpVector a

-- | Filter
filterSV :: (a -> Bool) -> SpVector a -> SpVector a

-- | Indexed filter
ifilterSV :: (Int -> a -> Bool) -> SpVector a -> SpVector a

-- | Generate an arbitrary (not random) vector <tt>u</tt> such that `v dot
--   u = 0`
orthogonalSV :: Fractional a => SpVector a -> SpVector a
instance GHC.Classes.Eq a => GHC.Classes.Eq (Data.Sparse.SpVector.SpVector a)
instance GHC.Base.Functor Data.Sparse.SpVector.SpVector
instance Numeric.LinearAlgebra.Class.Set Data.Sparse.SpVector.SpVector
instance Data.Foldable.Foldable Data.Sparse.SpVector.SpVector
instance Numeric.LinearAlgebra.Class.Additive Data.Sparse.SpVector.SpVector
instance Numeric.LinearAlgebra.Class.VectorSpace Data.Sparse.SpVector.SpVector
instance Numeric.LinearAlgebra.Class.FiniteDim Data.Sparse.SpVector.SpVector
instance Numeric.LinearAlgebra.Class.HasData Data.Sparse.SpVector.SpVector a
instance Numeric.LinearAlgebra.Class.Sparse Data.Sparse.SpVector.SpVector a
instance GHC.Num.Num a => Numeric.LinearAlgebra.Class.SpContainer Data.Sparse.SpVector.SpVector a
instance Numeric.LinearAlgebra.Class.Hilbert Data.Sparse.SpVector.SpVector
instance Numeric.LinearAlgebra.Class.Normed Data.Sparse.SpVector.SpVector
instance GHC.Show.Show a => GHC.Show.Show (Data.Sparse.SpVector.SpVector a)


module Data.Sparse.Common

-- | Insert row , using the provided row index transformation function
insertRowWith :: (IxCol -> IxCol) -> SpMatrix a -> SpVector a -> Key -> SpMatrix a

-- | Insert row
insertRow :: SpMatrix a -> SpVector a -> Key -> SpMatrix a

-- | Insert column, using the provided row index transformation function
insertColWith :: (IxRow -> IxRow) -> SpMatrix a -> SpVector a -> IxCol -> SpMatrix a

-- | Insert column
insertCol :: SpMatrix a -> SpVector a -> IxCol -> SpMatrix a

-- | Fill the diagonal of a SpMatrix with the components of a SpVector
diagonalSM :: SpVector a -> SpMatrix a

-- | Outer product (all-with-all matrix)
outerProdSV :: Num a => SpVector a -> SpVector a -> SpMatrix a

-- | Outer product (all-with-all matrix)
(><) :: Num a => SpVector a -> SpVector a -> SpMatrix a

-- | Demote (n x 1) or (1 x n) SpMatrix to SpVector
toSV :: SpMatrix a -> SpVector a

-- | promote a SV to SM
svToSM :: SpVector a -> SpMatrix a

-- | Lookup a row in a SpMatrix; returns an SpVector with the row, if this
--   is non-empty
lookupRowSM :: SpMatrix a -> IxRow -> Maybe (SpVector a)

-- | Extract jth column
extractCol :: SpMatrix a -> IxCol -> SpVector a

-- | Extract ith row
extractRow :: SpMatrix a -> IxRow -> SpVector a

-- | Generic extraction function
extractVectorDenseWith :: Num a => (Int -> (IxRow, IxCol)) -> SpMatrix a -> SpVector a

-- | Extract ith row (dense)
extractRowDense :: Num a => SpMatrix a -> IxRow -> SpVector a

-- | Extract jth column
extractColDense :: Num a => SpMatrix a -> IxCol -> SpVector a

-- | Extract the diagonal
extractDiagDense :: Num a => SpMatrix a -> SpVector a

-- | extract row interval (all entries between columns j1 and j2, INCLUDED,
--   are returned) extractSubRow :: SpMatrix a -&gt; IxRow -&gt; (IxCol,
--   IxCol) -&gt; SpVector a extractSubRow m i (j1, j2) = case lookupRowSM
--   m i of Nothing -&gt; zeroSV (ncols m) Just rv -&gt; ifilterSV (j _
--   -&gt; j &gt;= j1 &amp;&amp; j &lt;= j2) rv
--   
--   ", returning in Maybe extractSubRow :: SpMatrix a -&gt; IxRow -&gt;
--   (Int, Int) -&gt; Maybe (SpVector a) extractSubRow m i (j1, j2) =
--   resizeSV (j2 - j1) . ifilterSV (j _ -&gt; j &gt;= j1 &amp;&amp; j
--   <a>j2) &lt;$</a> lookupRowSM m i
--   
--   Extract an interval of SpVector components, changing accordingly the
--   resulting SpVector size. Keys are _not_ rebalanced, i.e. components
--   are still labeled according with respect to the source matrix.
extractSubRow :: SpMatrix a -> IxRow -> (Int, Int) -> SpVector a

-- | extract column interval
extractSubCol :: SpMatrix a -> IxCol -> (IxRow, IxRow) -> SpVector a

-- | extract row interval, rebalance keys by subtracting lowest one
extractSubRow_RK :: SpMatrix a -> IxRow -> (IxCol, IxCol) -> SpVector a

-- | extract column interval, rebalance keys by subtracting lowest one
extractSubCol_RK :: SpMatrix a -> IxCol -> (IxRow, IxRow) -> SpVector a

-- | Matrix-on-vector
matVec :: Num a => SpMatrix a -> SpVector a -> SpVector a

-- | Matrix-on-vector
(#>) :: Num a => SpMatrix a -> SpVector a -> SpVector a

-- | Vector-on-matrix (FIXME : transposes matrix: more costly than
--   <a>matVec</a>, I think)
vecMat :: Num a => SpVector a -> SpMatrix a -> SpVector a

-- | Vector-on-matrix (FIXME : transposes matrix: more costly than
--   <a>matVec</a>, I think)
(<#) :: Num a => SpVector a -> SpMatrix a -> SpVector a

-- | Pack a V.Vector of SpVectors as columns of an SpMatrix
fromCols :: Vector (SpVector a) -> SpMatrix a
prd :: PrintDense a => a -> IO ()
instance (GHC.Show.Show a, GHC.Num.Num a) => Data.Sparse.Common.PrintDense (Data.Sparse.SpVector.SpVector a)
instance (GHC.Show.Show a, GHC.Num.Num a) => Data.Sparse.Common.PrintDense (Data.Sparse.SpMatrix.SpMatrix a)

module Numeric.LinearAlgebra.Sparse

-- | Given a matrix A, returns a pair of matrices (Q, R) such that Q R = A,
--   Q is orthogonal and R is upper triangular. Applies Givens rotation
--   iteratively to zero out sub-diagonal elements
qr :: (Epsilon a, Floating a, Real a) => SpMatrix a -> (SpMatrix a, SpMatrix a)

-- | Given a matrix A, returns a pair of matrices (L, U) such that L U = A
lu :: (Epsilon a, Fractional a, Real a) => SpMatrix a -> (SpMatrix a, SpMatrix a)

-- | Given a positive semidefinite matrix A, returns a lower-triangular
--   matrix L such that L L^T = A
chol :: (Epsilon a, Real a, Floating a) => SpMatrix a -> SpMatrix a

-- | uses the R matrix from the QR factorization
conditionNumberSM :: (Epsilon a, RealFloat a) => SpMatrix a -> a
hhMat :: Num a => a -> SpVector a -> SpMatrix a

-- | a vector <tt>x</tt> uniquely defines an orthogonal plane; the
--   Householder operator reflects any point <tt>v</tt> with respect to
--   this plane: v' = (I - 2 x &gt;&lt; x) v
hhRefl :: Num a => SpVector a -> SpMatrix a

-- | 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.
--   
--   NB: the current version is quite inefficient in that: 1. the Givens'
--   matrix <tt>G_i</tt> is different from Identity only in 4 entries 2. at
--   each iteration <tt>i</tt> we multiply <tt>G_i</tt> by the previous
--   partial result <tt>M</tt>. Since this corresponds to a rotation, and
--   the <a>givensCoef</a> function already computes the value of the
--   resulting non-zero component (output <tt>r</tt>), `G_i ## M` can be
--   simplified by just changing two entries of <tt>M</tt> (i.e. zeroing
--   one out and changing the other into <tt>r</tt>).
givens :: (Floating a, Epsilon a, Ord a) => SpMatrix a -> IxRow -> IxCol -> SpMatrix a

-- | Given a matrix A and a positive integer <tt>n</tt>, this procedure
--   finds the basis of an order <tt>n</tt> Krylov subspace (as the columns
--   of matrix Q), along with an upper Hessenberg matrix H, such that A =
--   Q^T H Q. At the i`th iteration, it finds (i + 1) coefficients (the
--   i`th column of the Hessenberg matrix H) and the (i + 1)`th Krylov
--   vector.
arnoldi :: (Floating a, Eq a) => SpMatrix a -> Int -> (SpMatrix a, SpMatrix a)

-- | `eigsQR n mm` performs <tt>n</tt> iterations of the QR algorithm on
--   matrix <tt>mm</tt>, and returns a SpVector containing all eigenvalues
eigsQR :: (Epsilon a, Real a, Floating a) => Int -> SpMatrix a -> SpVector a

-- | `eigsRayleigh n mm` performs <tt>n</tt> iterations of the Rayleigh
--   algorithm on matrix <tt>mm</tt> and returns the eigenpair closest to
--   the initialization. It displays cubic-order convergence, but it also
--   requires an educated guess on the initial eigenpair eigRayleigh :: Int
--   -- max # iterations -&gt; SpMatrix Double -- matrix -&gt; (SpVector
--   Double, Double) -- initial guess of (eigenvector, eigenvalue) -&gt;
--   (SpVector Double, Double) -- final estimate of (eigenvector,
--   eigenvalue)
eigRayleigh :: Int -> SpMatrix Double -> (SpVector Double, Double) -> (SpVector Double, Double)

-- | Linear solve with _deterministic_ starting vector (every component at
--   0.1)
linSolve :: LinSolveMethod -> SpMatrix Double -> SpVector Double -> SpVector Double
data LinSolveMethod

-- | <a>\</a> : linSolve using the BiCGSTAB method as default
(<\>) :: SpMatrix Double -> SpVector Double -> SpVector Double

-- | Direct solver based on a triangular factorization of the system
--   matrix.
luSolve :: (Fractional a, Eq a, Epsilon a) => SpMatrix a -> SpMatrix a -> SpVector a -> SpVector a
cgne :: SpMatrix Double -> SpVector Double -> SpVector Double -> CGNE
tfqmr :: SpMatrix Double -> SpVector Double -> SpVector Double -> TFQMR

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

-- | one step of CGS
cgsStep :: SpMatrix Double -> SpVector Double -> CGS -> CGS

-- | one step of BiCGSTAB
bicgstabStep :: SpMatrix Double -> SpVector Double -> BICGSTAB -> BICGSTAB
data CGNE
data TFQMR
data BICGSTAB
data CGS
data BCG

-- | used for Incomplete LU : remove entries in <tt>m</tt> corresponding to
--   zero entries in <tt>m2</tt>
ilu0 :: (Epsilon a, Real a, Fractional a) => SpMatrix a -> (SpMatrix a, SpMatrix a)

-- | `mSsor aa omega` : if `omega = 1` it returns the symmetric
--   Gauss-Seidel preconditioner
mSsor :: Fractional a => SpMatrix a -> a -> (SpMatrix a, SpMatrix a)

-- | Partition a matrix into strictly subdiagonal, diagonal and strictly
--   superdiagonal parts
diagPartitions :: SpMatrix a -> (SpMatrix a, SpMatrix a, SpMatrix a)
randArray :: PrimMonad m => Int -> Double -> Double -> m [Double]

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

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

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

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

-- | Sparsify an SpVector
sparsifySV :: Epsilon a => SpVector a -> SpVector a

-- | Keep a moving window buffer (length 2) of state <tt>x</tt> 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
--   <a>loopUntilAcc</a> but this one runs in the State monad)
modifyInspectN :: MonadState s m => Int -> ([s] -> Bool) -> (s -> s) -> m s

-- | ", NO convergence check
runAppendN' :: (t -> t) -> Int -> t -> [t]

-- | iterate until convergence is verified or we run out of a fixed
--   iteration budget
untilConverged :: MonadState a m => (a -> SpVector Double) -> (a -> a) -> m a
diffSqL :: Floating a => [a] -> a
instance GHC.Show.Show Numeric.LinearAlgebra.Sparse.LinSolveMethod
instance GHC.Classes.Eq Numeric.LinearAlgebra.Sparse.LinSolveMethod
instance GHC.Classes.Eq Numeric.LinearAlgebra.Sparse.BICGSTAB
instance GHC.Classes.Eq Numeric.LinearAlgebra.Sparse.CGS
instance GHC.Classes.Eq Numeric.LinearAlgebra.Sparse.BCG
instance GHC.Classes.Eq Numeric.LinearAlgebra.Sparse.TFQMR
instance GHC.Classes.Eq Numeric.LinearAlgebra.Sparse.CGNE
instance GHC.Show.Show Numeric.LinearAlgebra.Sparse.BCG
instance GHC.Show.Show Numeric.LinearAlgebra.Sparse.CGS
instance GHC.Show.Show Numeric.LinearAlgebra.Sparse.BICGSTAB