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


-- | Numerical computation in native Haskell
--   
--   <i>Overview</i>
--   
--   The <tt>sparse-linear-algebra</tt> library provides iterative linear
--   solvers, matrix decompositions, eigenvalue computations and related
--   utilities. The user interface is provided by the top-level module
--   <a>Numeric.LinearAlgebra.Sparse</a>:
--   
--   <pre>
--   import           Numeric.LinearAlgebra.Sparse
--   </pre>
--   
--   Please refer to the README file for usage examples.
@package sparse-linear-algebra
@version 0.2.2.0

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 &gt; 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 &gt; 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


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

-- | <a>SpMatrix</a>es form a ring, in that they can be added and possess a
--   zero element

-- | <a>SpMatrix</a>es are maps between finite-dimensional spaces

-- | <a>SpMatrix</a>es are sparse containers too, i.e. any specific
--   component may be missing (so it is assumed to be 0)

-- | `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
takeRows :: IxRow -> SpMatrix a -> SpMatrix a
takeCols :: IxCol -> SpMatrix a -> 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

-- | <a>SpVector</a>s form a vector space because they can be multiplied by
--   a scalar

-- | <a>SpVector</a>s are finite-dimensional vectors

-- | <a>SpVector</a>s are sparse containers too, i.e. any specific
--   component may be missing (so it is assumed to be 0)

-- | <a>SpVector</a>s form a Hilbert space, in that we can define an inner
--   product over them

-- | Since <a>SpVector</a>s form a Hilbert space, we can define a norm for
--   them

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

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

-- | Canonical basis vector in R^n
ei :: Num a => Int -> Key -> 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,
--   where Q is orthogonal and R is upper triangular. Applies Givens
--   rotation iteratively to zero out sub-diagonal elements.
qr :: (Epsilon a, Ord a, Floating a) => SpMatrix a -> (SpMatrix a, SpMatrix a)

-- | Given a matrix A, returns a pair of matrices (L, U) where L is lower
--   triangular and U is upper triangular 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, a vector b 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 :: (Epsilon a, Floating a, Eq a) => SpMatrix a -> SpVector 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
GMRES_ :: LinSolveMethod
CGNE_ :: LinSolveMethod
TFQMR_ :: LinSolveMethod
BCG_ :: LinSolveMethod
CGS_ :: LinSolveMethod
BICGSTAB_ :: LinSolveMethod

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

-- | Least-squares approximation of a rectangular system of equaitons. Uses
--   <a>\</a> for the linear solve
pinv :: 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

-- | 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.CGNE
instance GHC.Show.Show Numeric.LinearAlgebra.Sparse.TFQMR
instance GHC.Show.Show Numeric.LinearAlgebra.Sparse.BCG
instance GHC.Show.Show Numeric.LinearAlgebra.Sparse.CGS
instance GHC.Show.Show Numeric.LinearAlgebra.Sparse.BICGSTAB