```{- |
Module     : Persistence.Matrix
Maintainer : eben.cowley42@gmail.com
Stability  : experimental

This module contains a variety of matrix utility functions, used in the computation of Betti numbers and simplicial homology groups.

Most importantly, it includes functions for computing the rank, normal form, and kernel of matrices. For the computation of homology groups and Betti numbers, one must perform column operations on one matrix to get it into column echelon form and find its kernel while also performing the inverse row operations on the next matrix to be operated on.

Bool is an instance of Num here (instance given in Util) so that functions can be somewhat generalized to act on both integers and integers modulo 2.

-}

module Persistence.Matrix (
-- * Types
IMatrix
, BMatrix
-- * Utilities
, getDiagonal
, getUnsignedDiagonal
, transposeMat
, transposePar
, multiply
, multiplyPar
-- Int matrices
, rankInt
, rankIntPar
, normalFormInt
, normalFormIntPar
, kernelInt
, kernelIntPar
, imgInKerInt
, imgInKerIntPar
-- Bool matrices
, rankBool
, kernelBool
, imgInKerBool
) where

{--FOR DEVS---------------------------------------------------------------

Matrices are transformed by iterating through each row and selecting a pivot. Zero rows are skipped for finding column eschelon form but a row operation is performed (if possible) if there is a zero row for Smith normal form.

To get the smith normal form, the entire pivot row and column is eliminated before continuing. Also, the pivot is always a diagonal element.

To get column eschelon form, every element in the pivot row after the pivot is eliminated. To get the kernel, all column operations to get the matrix to this form are also performed on the identiy matrix. To get the image of one matrix inside the kernel of the one being put into column eschelon form, perform the inverse row operations on the matrix whose image is needed. See stanford paper or the blog post on simplicial homology.

To get the rank of a matrix, look at the number of non-zero columns in the column eschelon form. To get the kernel, look at the columns of the identity (after all of the same column operations have been performed on it) which correspond to zero columns of the column eschelon form.

Eliminating elements is a slighltly more complicated process since only integer operations are allowed. First, every element that must be eliminated is made divisible by the pivot by using the Bezout coefficients from the extended Euclidean algorithm. Once this is done, integer division and subtraction can be used to eliminate the elements.

Boolean matrices are much easier to work with, they are regular matrices with elements modulo 2. Bool is an instance of Num here and the instance is given in Util.

--}

import Persistence.Util

import Data.List as L
import Data.Vector as V

import Control.Parallel.Strategies

-- * Types

-- | Matrix of integers.
type IMatrix = Vector (Vector Int)

-- | Matrix of integers modulo 2. Alternatively, matrix over the field with 2 elements.
type BMatrix = Vector (Vector Bool)

-- * Utilities

isMatrix :: Vector (Vector a) -> Bool
isMatrix mat =
let rowLen = V.length \$ V.head mat
in V.all (\r -> V.length r == rowLen) mat

-- | Take the transpose a matrix (no fancy optimizations, yet).
transposeMat :: Vector (Vector a) -> Maybe (Vector (Vector a))
transposeMat mat =
if isMatrix mat
then Just \$ V.map (\i -> V.map (\row -> row ! i) mat) \$ 0 `range` ((V.length \$ V.head mat) - 1)
else Nothing

-- | Take the transpose of a matrix using parallel evaluation of rows.
transposePar :: Vector (Vector a) -> Vector (Vector a)
transposePar mat =
parMapVec (\i -> V.map (\row -> row ! i) mat) \$ 0 `range` ((V.length \$ V.head mat) - 1)

-- | Multiply two matrices
multiply :: Num a => Vector (Vector a) -> Vector (Vector a) -> Vector (Vector a)
multiply mat1 mat2 =
let t =
case transposeMat mat2 of
Just m  -> m
Nothing -> error "error in multiply"
in V.map (\row -> V.map (dotProduct row) t) mat1

-- | Multiply matrices, evaluate rows in parallel if processors are available
multiplyPar :: Num a => Vector (Vector a) -> Vector (Vector a) -> Vector (Vector a)
multiplyPar mat1 mat2 = runEval \$ do
let t =
case transposeMat mat2 of
Just m  -> m
Nothing -> error "error in multiplyPar"
rseq t
return \$ parMapVec (\row -> V.map (dotProduct row) t) mat1

-- | Get the diagonal elements.
getDiagonal :: Vector (Vector a) -> [a]
getDiagonal matrix =
if V.null matrix then []
else L.map (\i -> matrix ! i ! i) [0..(min (V.length matrix) (V.length \$ V.head matrix)) - 1]

-- | Get the absolute value of each of the diagonal elements in a list.
getUnsignedDiagonal :: Num a => Vector (Vector a) -> [a]
getUnsignedDiagonal matrix =
if V.null matrix then []
else L.map (\i -> abs \$ matrix ! i ! i) [0..(min (V.length matrix) (V.length \$ V.head matrix)) - 1]

--assumes index1 < index2
colOperation :: Int -> Int -> (Int, Int, Int, Int) -> IMatrix -> IMatrix
colOperation index1 index2 (c11, c12, c21, c22) matrix =
let calc row =
let elem1  = row ! index1
elem2  = row ! index2
first  = V.take index1 row
second = V.drop (index1 + 1) (V.take index2 row)
third  = V.drop (index2 + 1) row
in first V.++ (cons (c11*elem1 + c12*elem2) second) V.++ (cons (c22*elem2 + c21*elem1) third)
in V.map calc matrix

colOperationPar :: Int -> Int -> (Int, Int, Int, Int) -> IMatrix -> IMatrix
colOperationPar index1 index2 (c11, c12, c21, c22) matrix =
let calc row =
let elem1  = row ! index1
elem2  = row ! index2
first  = V.take index1 row
second = V.drop (index1 + 1) (V.take index2 row)
third  = V.drop (index2 + 1) row
in first V.++ (cons (c11*elem1 + c12*elem2) second)
V.++ (cons (c22*elem2 + c21*elem1) third)
in parMapVec calc matrix

--assumes index1 < index2
rowOperation :: Int -> Int -> (Int, Int, Int, Int) -> IMatrix -> IMatrix
rowOperation index1 index2 (c11, c12, c21, c22) matrix =
let row1   = matrix ! index1
row2 = matrix ! index2
first  = V.take index1 matrix
second = V.drop (index1 + 1) \$ V.take index2 matrix
third  = V.drop (index2 + 1) matrix
in first V.++ (cons ((c11 `mul` row1) `add` (c12 `mul` row2)) second)
V.++ (cons ((c22 `mul` row2) `add` (c21 `mul` row1)) third)

rowOperationPar :: Int -> Int -> (Int, Int, Int, Int) -> IMatrix -> IMatrix
rowOperationPar index1 index2 (c11, c12, c21, c22) matrix =
let row1   = matrix ! index1
row2 = matrix ! index2
first  = V.take index1 matrix
second = V.drop (index1 + 1) (V.take index2 matrix)
third  = V.drop (index2 + 1) matrix
in runEval \$ do
a <- rpar \$ (c11 `mul` row1) `add` (c12 `mul` row2)
b <- rpar \$ (c21 `mul` row1) `add` (c22 `mul` row2)
rseq (a,b)
return \$ first V.++ (a `cons` second) V.++ (b `cons` third)

-- * Int matrices

--RANK--------------------------------------------------------------------

--finds the pivot in a given row for Gaussian elimination given the index of the pivot row and the matrix
--returns whether or not the row needs to be eliminated with the rearranged matrix and
--the column switch performed (if there was one), returns Nothing if the row is all zeroes
chooseGaussPivotInt :: (Int, Int) -> IMatrix -> Maybe (Bool, IMatrix, Maybe (Int, Int))
chooseGaussPivotInt (rowIndex, colIndex) mat =
let row     = mat ! rowIndex
indices = V.filter (\index -> index > colIndex) \$ V.findIndices (\x -> x /= 0) row
in
if row ! colIndex == 0 then
if V.null indices then Nothing
else
in
if row ! j == 0 then error "Persistence.Matrix.chooseGaussPivotInt. This is a bug. Please email the persistence maintainers."
else Just (V.length indices > 1, V.map (switchElems colIndex j) mat, Just (colIndex, j))
else Just (V.length indices > 0, mat, Nothing)

--does gaussian elimination on the pivot row of an integer matrix
improveRowInt :: (Int, Int) -> Int -> IMatrix -> IMatrix
improveRowInt (rowIndex, colIndex) numCols matrix =
let improve i mat =
if i == numCols then mat
else
let row   = mat ! rowIndex
pivot = row ! colIndex
x     = row ! i
next  = i + 1
in
--boundary operators have lots of zeroes
--better to catch that instead of doing unnecessary %
if pivot == 0 then
if V.all (\a -> a == 0) row then mat
else error "Persistence.Matrix.improveRowInt. This is a bug. Please email the Persistence maintainers."
else
if x == 0 || (x `mod` pivot == 0) then
improve next mat
else
let gcdTriple = extEucAlg pivot x
gcd       = one gcdTriple
in improve next \$ colOperation colIndex i (thr gcdTriple, two gcdTriple, x `div` gcd, -(pivot `div` gcd)) mat

in improve (colIndex + 1) matrix

--given pivot index and pivot paired with matrix whose pivot row has been improved
--eliminates the entries in the pivot row
--the kinds of matrices that the functions work on will have lots of zeroes
--better to catch that with a condition than perform an unnecessary division
elimRowInt :: (Int, Int) -> IMatrix -> IMatrix
elimRowInt (rowIndex, colIndex) elems =
let pCol  = V.map (\row -> row ! colIndex) elems
pivot = pCol ! rowIndex
c1    = colIndex + 1

makeCoeffs i v =
if V.null v then empty
else let x = V.head v; xs = V.tail v in
if x == 0 then makeCoeffs (i + 1) xs
else (i, x `div` pivot) `cons` (makeCoeffs (i + 1) xs)

calc :: IMatrix -> Vector (Int, Int) -> IMatrix
calc mat ops =
if V.null ops then mat
else
let (i, coeff) = V.head ops
in calc (mapWithIndex (\j row ->
replaceElem i ((row ! i) - coeff*(pCol ! j)) row) mat) (V.tail ops)

in
if pivot == 0 then error "Persistence.Matrix.elimRowInt. This is a bug. Please email the Persistence maintainers."
else calc elems \$ makeCoeffs c1 \$ V.drop c1 \$ elems ! rowIndex

-- | Finds the rank of integer matrix (number of linearly independent columns).
rankInt :: IMatrix -> Int
rankInt matrix =
let rows     = V.length matrix
cols     = V.length \$ V.head matrix
cols1    = cols - 1

doColOps (rowIndex, colIndex) mat =
if rowIndex == rows || colIndex == cols then mat else
case chooseGaussPivotInt (rowIndex, colIndex) mat of
Just (True, mx, _)  ->
doColOps (rowIndex + 1, colIndex + 1)
\$ elimRowInt (rowIndex, colIndex) \$ improveRowInt (rowIndex, colIndex) cols mx
Just (False, mx, _) -> doColOps (rowIndex + 1, colIndex + 1) mx
Nothing             -> doColOps (rowIndex + 1, colIndex) mat

countNonZeroCols mat =
V.sum \$ V.map (\i -> if V.any (\j -> mat ! j ! i /= 0)
\$ 0 `range` (rows - 1) then 1 else 0) \$ 0 `range` cols1

in countNonZeroCols \$ doColOps (0, 0) matrix

--does gaussian elimination on the pivot row of an integer matrix in parallel
improveRowIntPar :: (Int, Int) -> Int -> IMatrix -> IMatrix
improveRowIntPar (rowIndex, colIndex) numCols matrix =
let improve i mat =
if i == numCols then mat else
let row   = mat ! rowIndex
pivot = row ! colIndex
x     = row ! i
next  = i + 1
in
--boundary operators have lots of zeroes
--better to catch that instead of doing unnecessary %
if x == 0 || (x `mod` pivot == 0) then
improve next mat
else
let gcdTriple = extEucAlg pivot x
gcd       = one gcdTriple
in improve next
\$ colOperationPar colIndex i
(thr gcdTriple, two gcdTriple, x `div` gcd, -(pivot `div` gcd)) mat
in improve (colIndex + 1) matrix

--eliminates a row in parallel
elimRowIntPar :: (Int, Int) -> IMatrix -> IMatrix
elimRowIntPar (rowIndex, colIndex) elems =
let pCol  = V.map (\row -> row ! colIndex) elems
pivot = pCol ! rowIndex
c1    = colIndex + 1

makeCoeffs i v =
if V.null v then empty
else let x = V.head v; xs = V.tail v in
if x == 0 then makeCoeffs (i + 1) xs
else (i, x `div` pivot) `cons` (makeCoeffs (i + 1) xs)

calc :: IMatrix -> Vector (Int, Int) -> IMatrix
calc mat ops =
if V.null ops then mat
else
let (i, coeff) = V.head ops
in
calc (parMapWithIndex
(\j row -> replaceElem i ((row ! i) - coeff*(pCol ! j)) row) mat) (V.tail ops)

in
if pivot == 0
then error "Persistence.Matrix.elimRowIntPar. This is a bug. Please email the Persistence maintainers."
else calc elems \$ makeCoeffs c1 \$ V.drop c1 \$ elems ! rowIndex

-- | Calculates the rank of a matrix by operating on multiple rows in parallel.
rankIntPar :: IMatrix -> Int
rankIntPar matrix =
let rows     = V.length matrix
cols     = V.length \$ V.head matrix
cols1    = cols - 1

doColOps (rowIndex, colIndex) mat =
if rowIndex == rows || colIndex == cols then mat else
case chooseGaussPivotInt (rowIndex, colIndex) mat of
Just (True, mx, _)  ->
doColOps (rowIndex + 1, colIndex + 1) \$ elimRowIntPar (rowIndex, colIndex) \$
improveRowIntPar (rowIndex, colIndex) cols mx
Just (False, mx, _) -> doColOps (rowIndex + 1, colIndex + 1) mx
Nothing             -> doColOps (rowIndex + 1, colIndex) mat

countNonZeroCols mat =
V.sum \$ parMapVec (\i -> if V.any (\j -> mat ! j ! i /= 0) \$
0 `range` (rows - 1) then 1 else 0) \$ 0 `range` cols1

in countNonZeroCols \$ doColOps (0, 0) matrix

--NORMAL FORM-------------------------------------------------------------

--rearranges matrix so that the pivot entry is in the correct position
--returns true if more elimination is necessary
--returns Nothing if there is nothing but zeroes after the current pivot position
chooseRowPivotInt :: (Int, Int) -> Int -> Int -> IMatrix -> Maybe (Bool, IMatrix)
chooseRowPivotInt (rowIndex, colIndex) numRows numCols mat =
let row      = mat ! rowIndex
rIndices = V.toList \$ V.findIndices (\x -> x /= 0) row
in
if 0 == row ! colIndex then
case rIndices of
(i:is)  -> Just ((L.length is) > 0, V.map (switchElems i colIndex) mat)
[]      ->
case V.toList \$ V.findIndices (\x -> x /= 0) \$ V.map (\r -> r ! colIndex) mat of
(i:_)  -> Just (True, switchElems i rowIndex mat)
[]     -> Nothing
else Just ((L.length rIndices) > 1, mat)

--given pivot index and pivot paired with matrix, improves pivot column with row operations
improveColInt :: Int -> Int -> IMatrix -> IMatrix
improveColInt pIndex maxIndex matrix =
let improve i mat =
if i == maxIndex then mat else
let pivot = matrix ! pIndex ! pIndex
x     = matrix ! i ! pIndex
next  = i + 1
in
--boundary operators have lots of zeroes
--better to catch that instead of doing unnecessary %
if x == 0 || (x `mod` pivot == 0) then
improve next mat
else
let gcdTriple = extEucAlg pivot x
gcd       = one gcdTriple
in improve next
\$ rowOperation pIndex i
(thr gcdTriple, two gcdTriple, x `div` gcd, -(pivot `div` gcd)) mat
in improve (pIndex + 1) matrix

--eliminates the pivot column of a matrix to obtain normal form
elimColInt :: (Int, Int) -> IMatrix -> IMatrix
elimColInt (rowIndex, colIndex) elems =
let pRow  = elems ! rowIndex
pivot = pRow ! colIndex
ri1   = rowIndex + 1
makeCoeffs i v =
if V.null v then empty
else let x = V.head v; xs = V.tail v in
if x == 0 then makeCoeffs (i + 1) xs
else (i, x `div` pivot) `cons` (makeCoeffs (i + 1) xs)
calc :: IMatrix -> Vector (Int, Int) -> IMatrix
calc mat ops =
if V.null ops then mat
else let (i, coeff) = V.head ops in
calc (replaceElem i ((mat ! i) `subtr` (coeff `mul` pRow)) mat) (V.tail ops)
in
if pivot == 0
then error "Persistence.Matrix.elimColInt. This is a bug. Please email the Persistence maintainters."
else calc elems \$ makeCoeffs ri1 \$ V.drop ri1 \$ V.map (\row -> row ! colIndex) elems

finish :: Int -> IMatrix -> IMatrix
finish diagLen matrix =
let calc i mat =
let i1    = i + 1
row   = mat ! i
entry = row ! i
nextR = mat ! i1
nextE = nextR ! i1
in
if entry == 0 || i1 == diagLen then mat
else if entry `divides` nextE then calc i1 mat
else
let mat'      = replaceElem i (replaceElem i1 nextE row) mat
gcdTriple = extEucAlg entry nextE; gcd = one gcdTriple
improve   =
colOperation i i1
(thr gcdTriple, two gcdTriple, nextE `div` gcd, entry `div` gcd)
cleanup   = \m -> elimColInt (i, i) \$ elimRowInt (i, i) m
in calc i1 \$ cleanup \$ improve mat'

absDiag i mat =
if i == V.length mat
then mat
else absDiag (i + 1)
\$ replaceElem i (replaceElem i (abs \$ mat!i!i) \$ mat!i) mat

filtered = V.partition (\row -> V.any (\x -> x /= 0) row) matrix
in absDiag 0 \$ calc 0 \$ (fst filtered) V.++ (snd filtered)

-- | Get the Smith normal form of an integer matrix.
normalFormInt :: IMatrix -> IMatrix
normalFormInt matrix =
let rows = V.length matrix
cols = V.length \$ V.head matrix
diag = min rows cols

calc (rowIndex, colIndex) mat =
if rowIndex == rows || colIndex == cols then mat
else
case chooseRowPivotInt (rowIndex, colIndex) rows cols mat of
Just (True, mx)  ->
calc (rowIndex + 1, colIndex + 1) \$
elimColInt (rowIndex, colIndex) \$ improveColInt rowIndex rows \$
elimRowInt (rowIndex, colIndex) \$ improveRowInt (rowIndex, colIndex) cols mx
Just (False, mx) ->
calc (rowIndex + 1, colIndex + 1) \$
elimColInt (rowIndex, colIndex) \$ improveColInt rowIndex rows mx
Nothing          -> calc (rowIndex + 1, colIndex) mat

in
if V.null matrix
then empty
else finish diag \$ calc (0, 0) matrix

--improves the pivot column of a matrix in parallel
improveColIntPar :: Int -> Int -> IMatrix -> IMatrix
improveColIntPar pIndex maxIndex matrix =
let improve i mat =
if i == maxIndex then mat else
let col   = V.map (\row -> row ! pIndex) mat
pivot = col ! pIndex
x     = col ! i
next  = i + 1
in
--boundary operators have lots of zeroes
--better to catch that instead of doing unnecessary %
if x == 0 || (x `mod` pivot == 0) then
improve next mat
else
let gcdTriple = extEucAlg pivot x
gcd       = one gcdTriple
in improve next
\$ rowOperationPar pIndex i
(thr gcdTriple, two gcdTriple, x `div` gcd, -(pivot `div` gcd)) mat
in improve (pIndex + 1) matrix

--NEEDS TO BE PARALLELIZED
--eliminates pivot column in parallel
elimColIntPar :: (Int, Int) -> IMatrix -> IMatrix
elimColIntPar (rowIndex, colIndex) elems =
let pRow  = elems ! rowIndex
pivot = pRow ! colIndex
ri1   = rowIndex + 1

makeCoeffs i v =
if V.null v then empty
else let x = V.head v; xs = V.tail v in
if x == 0 then makeCoeffs (i + 1) xs
else (i, x `div` pivot) `cons` (makeCoeffs (i + 1) xs)

calc :: IMatrix -> Vector (Int, Int) -> IMatrix
calc mat ops =
if V.null ops then mat
else let (i, coeff) = V.head ops in
calc (replaceElem i ((mat ! i) `subtr` (coeff `mul` pRow)) mat) (V.tail ops)

in calc elems \$ makeCoeffs ri1 \$ V.drop ri1 \$ V.map (\row -> row ! colIndex) elems

-- | Gets the Smith normal form of a matrix, uses lots of parallelism if processors are available.
normalFormIntPar :: IMatrix -> IMatrix
normalFormIntPar matrix =
let rows = V.length matrix
cols = V.length \$ V.head matrix
diag = min rows cols

calc (rowIndex, colIndex) mat =
if rowIndex == rows || colIndex == cols then mat
else
case chooseRowPivotInt (rowIndex, colIndex) rows cols mat of
Just (True, mx)  ->
calc (rowIndex + 1, colIndex + 1) \$
elimColIntPar (rowIndex, colIndex) \$ improveColIntPar rowIndex rows \$
elimRowIntPar (rowIndex, colIndex) \$ improveRowIntPar (rowIndex, colIndex) cols mx
Just (False, mx) ->
calc (rowIndex + 1, colIndex + 1) \$
elimColIntPar (rowIndex, colIndex) \$ improveColIntPar rowIndex rows mx
Nothing          -> calc (rowIndex + 1, colIndex) mat

in
if V.null matrix
then empty
else finish diag \$ calc (0, 0) matrix

--KERNEL------------------------------------------------------------------

--improves the pivot row of an integer matrix
--performs the same column operations on the identity
improveRowIntWithId :: (Int, Int) -> Int -> IMatrix -> IMatrix -> (IMatrix, Int, IMatrix)
improveRowIntWithId (rowIndex, colIndex) numCols elems identity =
let improve i mat ide =
if i == numCols then (mat, mat ! rowIndex ! colIndex, ide) else
let row   = mat ! rowIndex
pivot = row ! colIndex
x     = row ! i
next  = i + 1
in
if x == 0 || (x `mod` pivot == 0) then
improve next mat ide
else
let gcdTriple = extEucAlg pivot x
gcd       = one gcdTriple
transform =
colOperation colIndex i
(thr gcdTriple, two gcdTriple, x `div` gcd, -(pivot `div` gcd))
in improve next (transform mat) (transform ide)

in improve (colIndex + 1) elems identity

--eliminates all the entries in the pivot row that come after the pivot
--after the matrix has been improved
--returns the new matrix (fst) and
--the identity with whatever column operations were performed (snd)
elimRowIntWithId :: (Int, Int) -> Int -> (IMatrix, Int, IMatrix) -> (IMatrix, IMatrix)
elimRowIntWithId (rowIndex, colIndex) numCols (elems, pivot, identity) =
let row            = elems ! rowIndex
elim i mat ide =
if i == numCols then (mat, ide)
else
let coeff     = (row ! i) `div` pivot
transform =
V.map (\r -> (V.take i r)
V.++ (cons ((r ! i) - coeff*(r ! colIndex)) (V.drop (i + 1) r)))
in elim (i + 1) (transform mat) (transform ide)
in elim (colIndex + 1) elems identity

-- | Finds a basis for the kernel of a matrix, arranges the basis vectors into the rows of a matrix.
kernelInt :: IMatrix -> IMatrix
kernelInt matrix =
let rows     = V.length matrix
cols     = V.length \$ V.head matrix
cols1    = cols - 1
identity = V.map (\i -> (V.replicate i 0)
V.++ (cons 1 (V.replicate (cols1 - i) 0))) \$ 0 `range` cols1

doColOps (rowIndex, colIndex) (elems, ide) =
if rowIndex == rows || colIndex == cols then (elems, ide) else
case chooseGaussPivotInt (rowIndex, colIndex) elems of
Just (True, mx, Just (i, j))  ->
doColOps (rowIndex + 1, colIndex + 1) \$ elimRowIntWithId (rowIndex, colIndex) cols \$
improveRowIntWithId (rowIndex, colIndex) cols mx \$ V.map (switchElems i j) ide
Just (True, _, Nothing)       ->
doColOps (rowIndex + 1, colIndex + 1) \$ elimRowIntWithId (rowIndex, colIndex) cols \$
improveRowIntWithId (rowIndex, colIndex) cols elems ide
Just (False, mx, Just (i, j)) ->
doColOps (rowIndex + 1, colIndex + 1) (mx, V.map (switchElems i j) ide)
Just (False, _, _)            -> doColOps (rowIndex + 1, colIndex + 1) (elems, ide)
Nothing                       -> doColOps (rowIndex + 1, colIndex) (elems, ide)

result   = doColOps (0, 0) (matrix, identity)
elems    = fst result
ide      = snd result
in V.map (\i -> V.map (\row -> row ! i) ide)
\$ V.filter (\i -> V.all (\row -> row ! i == 0) elems) \$ 0 `range` cols1

--improves row in parallel and does the same thing to the identity matrix in parallel
improveRowIntWithIdPar :: (Int, Int) -> Int -> IMatrix -> IMatrix -> (IMatrix, Int, IMatrix)
improveRowIntWithIdPar (rowIndex, colIndex) numCols elems identity =
let improve i mat ide =
if i == numCols then (mat, mat ! rowIndex ! colIndex, ide) else
let row   = mat ! rowIndex
pivot = row ! colIndex
x     = row ! i
next  = i + 1
in
--boundary operators have lots of zeroes
--better to catch that instead of doing unnecessary %
if x == 0 || (x `mod` pivot == 0) then
improve next mat ide
else
let gcdTriple = extEucAlg pivot x
gcd       = one gcdTriple
transform = colOperationPar colIndex i
(thr gcdTriple, two gcdTriple, x `div` gcd, -(pivot `div` gcd))
in improve next (transform mat) (transform ide)
in improve (colIndex + 1) elems identity

--eliminates entries in the pivot row in parallel and does the same to the identity
elimRowIntWithIdPar :: (Int, Int) -> Int -> (IMatrix, Int, IMatrix) -> (IMatrix, IMatrix)
elimRowIntWithIdPar (rowIndex, colIndex) numCols (elems, pivot, identity) =
let row            = elems ! rowIndex
elim i mat ide =
if i == numCols then (mat, ide)
else
let coeff     = (row ! i) `div` pivot
transform = parMapVec (\r -> (V.take i r)
V.++ (cons ((r ! i) - coeff*(r ! colIndex)) (V.drop (i + 1) r)))
in elim (i + 1) (transform mat) (transform ide)
in elim (colIndex + 1) elems identity

-- | Computes basis vectors for the kernel of an integer matrix and arranges them into the rows of a matrix using lots of parallelism if processors are available.
kernelIntPar :: IMatrix -> IMatrix
kernelIntPar matrix =
let rows     = V.length matrix
cols     = V.length \$ V.head matrix
cols1    = cols - 1
identity = V.map (\i -> (V.replicate i 0)
V.++ (cons 1 (V.replicate (cols1 - i) 0))) \$ 0 `range` cols1

doColOps (rowIndex, colIndex) (elems, ide) =
if rowIndex == rows || colIndex == cols then (elems, ide) else
case chooseGaussPivotInt (rowIndex, colIndex) elems of
Just (True, mx, Just (i, j))  ->
doColOps (rowIndex + 1, colIndex + 1)
\$ elimRowIntWithIdPar (rowIndex, colIndex) cols
\$ improveRowIntWithIdPar (rowIndex, colIndex) cols mx
\$ V.map (switchElems i j) ide
Just (True, _, Nothing)       ->
doColOps (rowIndex + 1, colIndex + 1)
\$ elimRowIntWithIdPar (rowIndex, colIndex) cols
\$ improveRowIntWithIdPar (rowIndex, colIndex) cols elems ide
Just (False, mx, Just (i, j)) ->
doColOps (rowIndex + 1, colIndex + 1) (mx, V.map (switchElems i j) ide)
Just (False, _, _)            -> doColOps (rowIndex + 1, colIndex + 1) (elems, ide)
Nothing                       -> doColOps (rowIndex + 1, colIndex) (elems, ide)

result   = doColOps (0, 0) (matrix, identity)
elems    = fst result
ide      = snd result
in V.map (\i -> V.map (\row -> row ! i) ide)
\$ V.filter (\i -> V.all (\row -> row ! i == 0) elems) \$ 0 `range` cols1

--FIND IMAGE IN BASIS OF KERNEL-------------------------------------------

--improves the row of the first matrix
--and performs the inverse column operations on the second matrix
improveRowIntWithInv :: (Int, Int) -> Int -> IMatrix -> IMatrix -> (IMatrix, Int, IMatrix)
improveRowIntWithInv (rowIndex, colIndex) numCols kernel image =
let improve i ker img =
if i == numCols then (ker, ker ! rowIndex ! colIndex, img) else
let row   = ker ! rowIndex
pivot = row ! colIndex
x     = row ! i
next  = i + 1
in
if x == 0 || (x `mod` pivot == 0) then
improve next ker img
else
let gcdTriple  = extEucAlg pivot x
gcd        = one gcdTriple
q1         = pivot `div` gcd
q2         = x `div` gcd
transform1 =
colOperationPar colIndex i (thr gcdTriple, two gcdTriple, q2, -q1)
transform2 =
rowOperationPar colIndex i (-q1, -(two gcdTriple), -q2, thr gcdTriple)
in improve next (transform1 ker) (transform2 img)
in improve (colIndex + 1) kernel image

--eliminates the row if the first matrix
--performs inverse column operations on the second matrix
elimRowIntWithInv :: (Int, Int) -> Int -> (IMatrix, Int, IMatrix) -> (IMatrix, IMatrix)
elimRowIntWithInv (rowIndex, colIndex) numCols (kernel, pivot, image) =
let elim i ker img
| i == numCols            = (ker, img)
| row ! i == 0 = elim (i + 1) ker img
| otherwise               =
let coeff      = (row ! i) `div` pivot
transform1 = V.map (\r -> replaceElem i ((r ! i) - coeff*(r ! colIndex)) r)
transform2 = \mat -> replaceElem colIndex
((coeff `mul` (mat ! i)) `add` (mat ! colIndex)) mat
in elim (i + 1) (transform1 ker) (transform2 img)
where row = ker ! rowIndex
in elim (colIndex + 1) kernel image

-- | Calculates the image of the second matrix represented in the basis of the kernel of the first matrix.
imgInKerInt :: IMatrix -> IMatrix -> IMatrix
imgInKerInt toColEsch toImage =
let rows     = V.length toColEsch
cols     = V.length \$ V.head toColEsch

doColOps (rowIndex, colIndex) (ker, img) =
if rowIndex == rows || colIndex == cols then (ker, img)
else case chooseGaussPivotInt (rowIndex, colIndex) ker of
Just (True, _, Nothing)       ->
doColOps (rowIndex + 1, colIndex + 1) \$ elimRowIntWithInv (rowIndex, colIndex) cols \$
improveRowIntWithInv (rowIndex, colIndex) cols ker img
Just (True, mx, Just (i, j))  ->
doColOps (rowIndex + 1, colIndex + 1) \$ elimRowIntWithInv (rowIndex, colIndex) cols \$
improveRowIntWithInv (rowIndex, colIndex) cols mx \$ switchElems i j img
Just (False, mx, Just (i, j)) ->
doColOps (rowIndex + 1, colIndex + 1) (mx, switchElems i j img)
Just (False, _, _)            -> doColOps (rowIndex + 1, colIndex + 1) (ker, img)
Nothing                       -> doColOps (rowIndex + 1, colIndex) (ker, img)

result  = doColOps (0, 0) (toColEsch, toImage)
colEsch = fst result
image   = snd result
in V.map (\i -> image ! i)
\$ V.filter (\i -> V.all (\row -> row ! i == 0) colEsch) \$ 0 `range` (cols - 1)

--improves row and does inverse operations in parallel
improveRowIntWithInvPar :: (Int, Int) -> Int -> IMatrix -> IMatrix -> (IMatrix, Int, IMatrix)
improveRowIntWithInvPar (rowIndex, colIndex) numCols kernel image =
let improve i ker img =
if i == numCols then (ker, ker ! rowIndex ! colIndex, img) else
let row   = ker ! rowIndex
pivot = row ! colIndex
x     = row ! i
next  = i + 1
in
if x == 0 || (x `mod` pivot == 0) then
improve next ker img
else
let gcdTriple  = extEucAlg pivot x
gcd        = one gcdTriple
q1         = pivot `div` gcd
q2         = x `div` gcd
transform1 =
colOperationPar colIndex i (thr gcdTriple, two gcdTriple, q2, -q1)
transform2 =
rowOperationPar colIndex i (-q1, -(two gcdTriple), -q2, thr gcdTriple)
in improve next (transform1 ker) (transform2 img)
in improve (colIndex + 1) kernel image

--eliminates row in parallel
--INVERSE OPERATIONS NEED TO BE PARALLELIZED
elimRowIntWithInvPar :: (Int, Int) -> Int -> (IMatrix, Int, IMatrix) -> (IMatrix, IMatrix)
elimRowIntWithInvPar (rowIndex, colIndex) numCols (kernel, pivot, image) =
let elim i ker img
| i == numCols            = (ker, img)
| row ! i == 0 = elim (i + 1) ker img
| otherwise               =
let coeff      = (row ! i) `div` pivot
transform1 = parMapVec (\r -> replaceElem i ((r ! i) - coeff*(r ! colIndex)) r)
transform2 = \mat -> replaceElem colIndex
((coeff `mul` (mat ! i)) `add` (mat ! colIndex)) mat
in elim (i + 1) (transform1 ker) (transform2 img)
where row = ker ! rowIndex
in elim (colIndex + 1) kernel image

-- | Calculates the image of the second matrix represented in the basis of the kernel of the first matrix. Uses lots of parallelism if processors are available.
imgInKerIntPar :: IMatrix -> IMatrix -> IMatrix
imgInKerIntPar toColEsch toImage =
let rows     = V.length toColEsch
cols     = V.length \$ V.head toColEsch

doColOps (rowIndex, colIndex) (ker, img) =
if rowIndex == rows || colIndex == cols then (ker, img)
else case chooseGaussPivotInt (rowIndex, colIndex) ker of
Just (True, _, Nothing)       ->
doColOps (rowIndex + 1, colIndex + 1) \$ elimRowIntWithInvPar (rowIndex, colIndex) cols \$
improveRowIntWithInvPar (rowIndex, colIndex) cols ker img
Just (True, mx, Just (i, j))  ->
doColOps (rowIndex + 1, colIndex + 1) \$ elimRowIntWithInvPar (rowIndex, colIndex) cols \$
improveRowIntWithInvPar (rowIndex, colIndex) cols mx \$ switchElems i j img
Just (False, mx, Just (i, j)) ->
doColOps (rowIndex + 1, colIndex + 1) (mx, switchElems i j img)
Just (False, _, _)            -> doColOps (rowIndex + 1, colIndex + 1) (ker, img)
Nothing                       -> doColOps (rowIndex + 1, colIndex) (ker, img)

result = doColOps (0, 0) (toColEsch, toImage)
ker    = fst result
img    = snd result
in V.map (\i -> img ! i)
\$ V.filter (\i -> V.all (\row -> row ! i == 0) ker) \$ 0 `range` (cols - 1)

-- * Bool matrices

--RANK--------------------------------------------------------------------

--given the index of the pivot row and the matrix
--determines whether there is a non-zero element in the row, does necessary rearranging
--and returns the column operation that was performed if there was one
--returns Nothing if the entire row is zero
chooseGaussPivotBool :: (Int, Int) -> BMatrix -> Maybe (Bool, BMatrix, Maybe (Int, Int))
chooseGaussPivotBool (rowIndex, colIndex) mat =
let row     = mat ! rowIndex
indices = V.filter (\index -> index > colIndex) \$ V.findIndices id row
in
if not \$ row ! colIndex then
if V.null indices then Nothing
else
in Just (V.length indices > 0, V.map (switchElems colIndex j) mat, Just (colIndex, j))
else Just (V.length indices > 0, mat, Nothing)

--eliminates pivot row of a boolean matrix
elimRowBool :: (Int, Int) -> Int -> BMatrix -> BMatrix
elimRowBool (rowIndex, colIndex) numCols elems =
let row = elems ! rowIndex
elim i mat
| i == numCols  = mat
| not \$ row ! i = elim (i + 1) mat
| otherwise     = elim (i + 1)
\$ V.map (\row -> replaceElem i ((row ! i) + (row ! colIndex)) row) mat
in elim (colIndex + 1) elems

-- | Find the rank of a mod 2 matrix (number of linearly independent columns).
rankBool :: BMatrix -> Int
rankBool matrix =
let rows  = V.length matrix
cols  = V.length \$ V.head matrix
cols1 = cols - 1

doColOps (rowIndex, colIndex) mat =
if rowIndex == rows || colIndex == cols then mat else
case chooseGaussPivotBool (rowIndex, colIndex) mat of
Just (True, mx, _)  ->
doColOps (rowIndex + 1, colIndex + 1) \$ elimRowBool (rowIndex, colIndex) cols mx
Just (False, mx, _) -> doColOps (rowIndex + 1, colIndex + 1) mat
Nothing             -> doColOps (rowIndex + 1, colIndex) mat

countNonZeroCols mat =
V.sum \$ V.map (\i ->
if V.any (\j -> mat ! j ! i /= 0) (0 `range` (rows - 1)) then 1 else 0) \$ 0 `range` cols1
in countNonZeroCols \$ doColOps (0, 0) matrix

--KERNEL------------------------------------------------------------------

--eliminates all the entries in the pivot row that come after the pivot
--after the matrix has been improved
--returns the new matrix (fst) paired
--and the identity with whatever column operations were performed (snd)
elimRowBoolWithId :: (Int, Int) -> Int -> BMatrix -> BMatrix -> (BMatrix, BMatrix)
elimRowBoolWithId (rowIndex, colIndex) numCols elems identity =
let row = elems ! rowIndex
elim i mat ide
| i == numCols  = (mat, ide)
| not \$ row ! i = elim (i + 1) mat ide
| otherwise     =
let transform = V.map (\row -> replaceElem i ((row ! i) + (row ! colIndex)) row)
in elim (i + 1) (transform mat) (transform ide)
in elim (colIndex + 1) elems identity

-- | Finds the basis of the kernel of a matrix, arranges the basis vectors into the rows of a matrix.
kernelBool :: BMatrix -> BMatrix
kernelBool matrix =
let rows     = V.length matrix
cols     = V.length \$ V.head matrix
cols1    = cols - 1
identity = V.map (\i -> (V.replicate i False)
V.++ (cons True (V.replicate (cols1 - i) False))) \$ 0 `range` cols1

doColOps (rowIndex, colIndex) (ker, ide) =
if rowIndex == rows || colIndex == cols then (ker, ide)
else
case chooseGaussPivotBool (rowIndex, colIndex) ker of
Just (True, _, Nothing)      ->
doColOps (rowIndex + 1, colIndex + 1) \$
elimRowBoolWithId (rowIndex, colIndex) cols ker ide
Just (True, mx, Just (i, j)) ->
doColOps (rowIndex + 1, colIndex + 1) \$
elimRowBoolWithId (rowIndex, colIndex) cols mx \$ V.map (switchElems i j) ide
Just (False, _, Just (i, j)) ->
doColOps (rowIndex + 1, colIndex + 1) (ker, V.map (switchElems i j) ide)
Just (False, _, _)           -> doColOps (rowIndex + 1, colIndex + 1) (ker, ide)
Nothing                      -> doColOps (rowIndex + 1, colIndex) (ker, ide)

result = doColOps (0, 0) (matrix, identity)
ker    = fst result
img    = snd result
in V.map (\i -> img ! i) \$ V.filter (\i -> V.all (\row -> not \$ row ! i) ker) \$ 0 `range` cols1

--IMAGE IN BASIS OF KERNEL------------------------------------------------

elimRowBoolWithInv :: (Int, Int) -> Int -> BMatrix -> BMatrix -> (BMatrix, BMatrix)
elimRowBoolWithInv (rowIndex, colIndex) numCols toColEch toImage =
let row = toColEch ! rowIndex
elim i ech img
| i == numCols  = (ech, img)
| not \$ row ! i = elim (i + 1) ech img
| otherwise     =
let transform1 = V.map (\r -> replaceElem i ((r ! i) + (r ! colIndex)) r)
transform2 = \mat -> replaceElem colIndex ((mat ! i) `add` (mat ! colIndex)) mat
in elim (i + 1) (transform1 ech) (transform2 img)
in elim (colIndex + 1) toColEch toImage

-- | Calculates the image of the second matrix represented in the basis of the kernel of the first matrix.
imgInKerBool :: BMatrix -> BMatrix -> BMatrix
imgInKerBool toColEch toImage =
let rows  = V.length toColEch
cols  = V.length \$ V.head toColEch
cols1 = cols - 1

doColOps (rowIndex, colIndex) (ech, img) =
if rowIndex == rows || colIndex == cols then (ech, img)
else
case chooseGaussPivotBool (rowIndex, colIndex) ech of
Just (True, _, Nothing)       ->
doColOps (rowIndex + 1, colIndex + 1) \$
elimRowBoolWithInv (rowIndex, colIndex) cols ech img
Just (True, mx, Just (i, j))  ->
doColOps (rowIndex + 1, colIndex + 1) \$
elimRowBoolWithInv (rowIndex, colIndex) cols mx \$ switchElems i j img
Just (False, mx, Just (i, j)) ->
doColOps (rowIndex + 1, colIndex + 1) (mx, switchElems i j img)
Just (False, _, _)            -> doColOps (rowIndex + 1, colIndex + 1) (ech, img)
Nothing                       -> doColOps (rowIndex + 1, colIndex) (ech, img)

result = doColOps (0, 0) (toColEch, toImage)
ker    = fst result
img    = snd result
in V.map (\i -> img ! i) \$ V.filter (\i -> V.all (\row -> not \$ row ! i) ker) \$ 0 `range` cols1```