Refactor LU and LUP decomposition

This commit is contained in:
Kiana Sheibani 2022-09-02 15:01:50 -04:00
parent b74734fbc1
commit c9dada5206
Signed by: toki
GPG key ID: 6CB106C25E86A9F7
2 changed files with 144 additions and 58 deletions

View file

@ -2,7 +2,7 @@ module Data.NumIdr.Matrix
import Data.List
import Data.Vect
import Data.Bool.Xor
import Data.Permutation
import Data.NumIdr.Multiply
import public Data.NumIdr.Array
import Data.NumIdr.Vector
@ -61,6 +61,11 @@ fromDiag ds o = fromFunction [m,n] (\[i,j] => maybe o (`index` ds) $ i `eq` j)
eq (FS _) FZ = Nothing
export
permutationMatrix : {n : _} -> Num a => Permutation n -> Matrix' n a
permutationMatrix p = permuteInAxis 0 p (repeatDiag 1 0)
||| Construct the matrix that scales a vector by the given value.
export
scaling : {n : _} -> Num a => a -> Matrix' n a
@ -117,6 +122,27 @@ minor : Fin (S m) -> Fin (S n) -> Matrix (S m) (S n) a -> Matrix m n a
minor i j mat = believe_me $ mat!!..[Filter (/=i), Filter (/=j)]
filterInd : Num a => (Nat -> Nat -> Bool) -> Matrix m n a -> Matrix m n a
filterInd p mat with (viewShape mat)
_ | Shape [m,n] = fromFunctionNB [m,n] (\[i,j] => if p i j then mat!#[i,j] else 0)
export
upperTriangle : Num a => Matrix m n a -> Matrix m n a
upperTriangle = filterInd (<=)
export
lowerTriangle : Num a => Matrix m n a -> Matrix m n a
lowerTriangle = filterInd (>=)
export
upperTriangleStrict : Num a => Matrix m n a -> Matrix m n a
upperTriangleStrict = filterInd (<)
export
lowerTriangleStrict : Num a => Matrix m n a -> Matrix m n a
lowerTriangleStrict = filterInd (>)
--------------------------------------------------------------------------------
-- Basic operations
--------------------------------------------------------------------------------
@ -141,6 +167,23 @@ hstack : {m : _} -> Vect n (Vector m a) -> Matrix m n a
hstack = stack 1
export
swapRows : (i,j : Fin m) -> Matrix m n a -> Matrix m n a
swapRows = swapInAxis 0
export
swapColumns : (i,j : Fin n) -> Matrix m n a -> Matrix m n a
swapColumns = swapInAxis 1
export
permuteRows : Permutation m -> Matrix m n a -> Matrix m n a
permuteRows = permuteInAxis 0
export
permuteColumns : Permutation n -> Matrix m n a -> Matrix m n a
permuteColumns = permuteInAxis 1
||| Calculate the outer product of two vectors as a matrix.
export
outer : Num a => Vector m a -> Vector n a -> Matrix m n a
@ -181,14 +224,29 @@ export
-- LU Decomposition
public export
export
record DecompLU {0 n,a : _} (mat : Matrix' n a) where
constructor MkLU
lower, upper : Matrix' n a
lu : Matrix' n a
export
Show a => Show (DecompLU {a} mat) where
showPrec p (MkLU l u) = showCon p "MkLU" $ showArg l ++ showArg u
namespace DecompLU
export
lower : Num a => DecompLU {n,a} mat -> Matrix' n a
lower (MkLU lu) with (viewShape lu)
_ | Shape [n,n] = lowerTriangleStrict lu + identity
export %inline
(.lower) : Num a => DecompLU {n,a} mat -> Matrix' n a
(.lower) = lower
export
upper : Num a => DecompLU {n,a} mat -> Matrix' n a
upper (MkLU lu) = upperTriangle lu
export %inline
(.upper) : Num a => DecompLU {n,a} mat -> Matrix' n a
(.upper) = upper
iterateN : (n : Nat) -> (Fin n -> a -> a) -> a -> a
@ -196,45 +254,75 @@ iterateN 0 f x = x
iterateN 1 f x = f FZ x
iterateN (S n@(S _)) f x = iterateN n (f . FS) $ f FZ x
gaussStep : (Eq a, Neg a, Fractional a) => Fin n -> Matrix' n a -> Matrix' n a
gaussStep {n} i lu with (viewShape lu)
_ | Shape [n,n] =
if all (==0) $ getColumn i lu then lu else
let diag = lu!![i,i]
coeffs = map (/diag) $ lu!!..[StartBound (FS i), One i]
lu' = indexSetRange [StartBound (FS i), One i]
coeffs lu
pivot = lu!!..[One i, StartBound (FS i)]
offsets = negate $ outer coeffs pivot
in indexUpdateRange [StartBound (FS i), StartBound (FS i)] (+offsets) lu'
export
decompLU : Neg a => Fractional a => (mat : Matrix' n a) -> DecompLU mat
decompLU : (Eq a, Neg a, Fractional a) => (mat : Matrix' n a) -> DecompLU mat
decompLU {n} mat with (viewShape mat)
_ | Shape [n,n] = iterateN n doolittle (MkLU identity mat)
where
doolittle : Fin n -> DecompLU mat -> DecompLU mat
doolittle i (MkLU l u) =
let v = rewrite rangeLen (S i') n in fromFunctionNB [minus n (S i')]
(\[x] => u!#[S i' + x,i'] / u!#[i',i'])
low = indexSetRange [StartBound (FS i), One i] (-v) identity
in MkLU (indexSetRange [StartBound (FS i), One i] v l) (low *. u)
where i' : Nat
i' = cast i
_ | Shape [n,n] = MkLU $ iterateN n gaussStep mat
-- LUP Decomposition
public export
record DecompLUP {0 n,a : _} (mat : Matrix' n a) where
constructor MkLUP
lower, upper, permute : Matrix' n a
swaps : Nat
lu : Matrix' n a
p : Permutation n
sw : Nat
namespace DecompLUP
export
lower : Num a => DecompLUP {n,a} mat -> Matrix' n a
lower (MkLUP lu p sw) with (viewShape lu)
_ | Shape [n,n] = lowerTriangleStrict lu + identity
export %inline
(.lower) : Num a => DecompLUP {n,a} mat -> Matrix' n a
(.lower) = lower
export
upper : Num a => DecompLUP {n,a} mat -> Matrix' n a
upper (MkLUP lu p sw) = upperTriangle lu
export %inline
(.upper) : Num a => DecompLUP {n,a} mat -> Matrix' n a
(.upper) = upper
export
permute : DecompLUP {n} mat -> Permutation n
permute (MkLUP lu p sw) = p
export %inline
(.permute) : DecompLUP {n} mat -> Permutation n
(.permute) = permute
export
numSwaps : DecompLUP {n} mat -> Nat
numSwaps (MkLUP lu p sw) = sw
export
Show a => Show (DecompLUP {a} mat) where
showPrec p (MkLUP l u pr b) = showCon p "MkLUP" $
showArg l ++ showArg u ++ showArg pr ++ showArg b
fromLU : DecompLU mat -> DecompLUP mat
fromLU (MkLU lu) = MkLUP lu identity 0
export
fromLU : Num a => DecompLU {n,a} mat -> DecompLUP mat
fromLU {n} (MkLU l u) with (viewShape l)
_ | Shape [n,n] = MkLUP l u identity 0
export
decompLUP : Ord a => Abs a => Neg a => Fractional a =>
decompLUP : (Ord a, Abs a, Neg a, Fractional a) =>
(mat : Matrix' n a) -> DecompLUP mat
decompLUP {n} mat with (viewShape mat)
decompLUP {n=0} mat | Shape [0,0] = MkLUP mat mat mat 0
decompLUP {n=0} mat | Shape [0,0] = MkLUP mat identity 0
decompLUP {n=S n} mat | Shape [S n,S n] =
iterateN (S n) doolittle (MkLUP identity mat identity 0)
iterateN (S n) gaussStepSwap (MkLUP mat identity 0)
where
maxIndex : (s,a) -> List (s,a) -> (s,a)
maxIndex x [] = x
@ -243,29 +331,13 @@ decompLUP {n} mat with (viewShape mat)
if abs b < abs d then assert_total $ maxIndex x ((c,d)::xs)
else assert_total $ maxIndex x ((a,b)::xs)
doolittle : Fin (S n) -> DecompLUP mat -> DecompLUP mat
doolittle i (MkLUP l u p sw) =
let (maxi, maxv) = mapFst ((+i') . head)
(maxIndex ([0],0) $ enumerateNB $
u !!.. [StartBound (weaken i), One i])
u' = if maxi == i' then u
else fromFunctionNB _ (\[x,y] =>
if x==i' then u!#[maxi,y]
else if x==maxi then u!#[i',y]
else u!#[x,y])
p' = if maxi == i' then p
else fromFunctionNB _ (\[x,y] =>
if x==i' then p!#[maxi,y]
else if x==maxi then p!#[i',y]
else p!#[x,y])
v = rewrite rangeLen (S i') (S n) in fromFunctionNB [minus n i']
(\[x] => u'!#[S i' + x,i'] / u'!#[i',i'])
low = indexSetRange [StartBound (FS i), One i] (-v) identity
in if maxv == 0 then MkLUP l u p sw else
MkLUP (indexSetRange [StartBound (FS i), One i] v l)
(low *. u') p' (if maxi==i' then S sw else sw)
where i' : Nat
i' = cast i
gaussStepSwap : Fin (S n) -> DecompLUP mat -> DecompLUP mat
gaussStepSwap i (MkLUP lu p sw) =
let (maxi, maxv) = mapFst head
(maxIndex ([0],0) $ enumerate $
indexSetRange [EndBound (weaken i)] 0 $ getColumn i lu)
in if maxi == i then MkLUP (gaussStep i lu) p sw
else MkLUP (gaussStep i $ swapRows maxi i lu) (appendSwap maxi i p) (S sw)
--------------------------------------------------------------------------------
@ -274,12 +346,16 @@ decompLUP {n} mat with (viewShape mat)
export
det : Ord a => Abs a => Neg a => Fractional a =>
Matrix' n a -> a
detWithLUP : (Ord a, Abs a, Neg a, Fractional a) =>
(mat : Matrix' n a) -> DecompLUP mat -> a
detWithLUP {n} mat lup =
(if numSwaps lup `mod` 2 == 0 then 1 else -1)
* product (diagonal lup.lower) * product (diagonal lup.upper)
export
det : (Ord a, Abs a, Neg a, Fractional a) => Matrix' n a -> a
det {n} mat with (viewShape mat)
det {n=0} mat | Shape [0,0] = 1
det {n=1} mat | Shape [1,1] = mat!![0,0]
det {n=2} mat | Shape [2,2] = let [a,b,c,d] = elements mat in a*d - b*c
_ | Shape [n,n] = let MkLUP l u p sw = decompLUP mat
in (if sw `mod` 2 == 0 then 1 else -1)
* product (diagonal l) * product (diagonal u)
_ | Shape [n,n] = detWithLUP mat (decompLUP mat)

View file

@ -15,6 +15,10 @@ export
swap : (i,j : Fin n) -> Permutation n
swap x y = MkPerm [(x,y)]
export
swaps : List (Fin n, Fin n) -> Permutation n
swaps = MkPerm
export
appendSwap : (i,j : Fin n) -> Permutation n -> Permutation n
appendSwap i j (MkPerm a) = MkPerm ((i,j)::a)
@ -51,6 +55,12 @@ export
permuteValues : Permutation n -> Nat -> Nat
permuteValues p = foldMap @{%search} @{mon} (\(i,j) => swapValues i j) p.swaps
export
Show (Permutation n) where
showPrec p (MkPerm a) = showCon p "swaps" $ showArg a
export
Mult (Permutation n) (Permutation n) (Permutation n) where
MkPerm a *. MkPerm b = MkPerm (a ++ b)