Refactor LU and LUP decomposition
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@ -2,7 +2,7 @@ module Data.NumIdr.Matrix
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import Data.List
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import Data.Vect
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import Data.Bool.Xor
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import Data.Permutation
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import Data.NumIdr.Multiply
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import public Data.NumIdr.Array
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import Data.NumIdr.Vector
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@ -61,6 +61,11 @@ fromDiag ds o = fromFunction [m,n] (\[i,j] => maybe o (`index` ds) $ i `eq` j)
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eq (FS _) FZ = Nothing
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export
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permutationMatrix : {n : _} -> Num a => Permutation n -> Matrix' n a
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permutationMatrix p = permuteInAxis 0 p (repeatDiag 1 0)
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||| Construct the matrix that scales a vector by the given value.
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export
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scaling : {n : _} -> Num a => a -> Matrix' n a
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@ -117,6 +122,27 @@ minor : Fin (S m) -> Fin (S n) -> Matrix (S m) (S n) a -> Matrix m n a
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minor i j mat = believe_me $ mat!!..[Filter (/=i), Filter (/=j)]
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filterInd : Num a => (Nat -> Nat -> Bool) -> Matrix m n a -> Matrix m n a
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filterInd p mat with (viewShape mat)
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_ | Shape [m,n] = fromFunctionNB [m,n] (\[i,j] => if p i j then mat!#[i,j] else 0)
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export
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upperTriangle : Num a => Matrix m n a -> Matrix m n a
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upperTriangle = filterInd (<=)
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export
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lowerTriangle : Num a => Matrix m n a -> Matrix m n a
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lowerTriangle = filterInd (>=)
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export
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upperTriangleStrict : Num a => Matrix m n a -> Matrix m n a
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upperTriangleStrict = filterInd (<)
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export
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lowerTriangleStrict : Num a => Matrix m n a -> Matrix m n a
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lowerTriangleStrict = filterInd (>)
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--------------------------------------------------------------------------------
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-- Basic operations
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--------------------------------------------------------------------------------
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@ -141,6 +167,23 @@ hstack : {m : _} -> Vect n (Vector m a) -> Matrix m n a
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hstack = stack 1
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export
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swapRows : (i,j : Fin m) -> Matrix m n a -> Matrix m n a
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swapRows = swapInAxis 0
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export
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swapColumns : (i,j : Fin n) -> Matrix m n a -> Matrix m n a
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swapColumns = swapInAxis 1
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export
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permuteRows : Permutation m -> Matrix m n a -> Matrix m n a
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permuteRows = permuteInAxis 0
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export
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permuteColumns : Permutation n -> Matrix m n a -> Matrix m n a
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permuteColumns = permuteInAxis 1
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||| Calculate the outer product of two vectors as a matrix.
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export
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outer : Num a => Vector m a -> Vector n a -> Matrix m n a
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@ -181,14 +224,29 @@ export
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-- LU Decomposition
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public export
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export
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record DecompLU {0 n,a : _} (mat : Matrix' n a) where
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constructor MkLU
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lower, upper : Matrix' n a
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lu : Matrix' n a
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export
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Show a => Show (DecompLU {a} mat) where
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showPrec p (MkLU l u) = showCon p "MkLU" $ showArg l ++ showArg u
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namespace DecompLU
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export
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lower : Num a => DecompLU {n,a} mat -> Matrix' n a
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lower (MkLU lu) with (viewShape lu)
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_ | Shape [n,n] = lowerTriangleStrict lu + identity
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export %inline
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(.lower) : Num a => DecompLU {n,a} mat -> Matrix' n a
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(.lower) = lower
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export
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upper : Num a => DecompLU {n,a} mat -> Matrix' n a
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upper (MkLU lu) = upperTriangle lu
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export %inline
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(.upper) : Num a => DecompLU {n,a} mat -> Matrix' n a
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(.upper) = upper
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iterateN : (n : Nat) -> (Fin n -> a -> a) -> a -> a
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@ -196,45 +254,75 @@ iterateN 0 f x = x
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iterateN 1 f x = f FZ x
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iterateN (S n@(S _)) f x = iterateN n (f . FS) $ f FZ x
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gaussStep : (Eq a, Neg a, Fractional a) => Fin n -> Matrix' n a -> Matrix' n a
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gaussStep {n} i lu with (viewShape lu)
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_ | Shape [n,n] =
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if all (==0) $ getColumn i lu then lu else
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let diag = lu!![i,i]
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coeffs = map (/diag) $ lu!!..[StartBound (FS i), One i]
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lu' = indexSetRange [StartBound (FS i), One i]
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coeffs lu
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pivot = lu!!..[One i, StartBound (FS i)]
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offsets = negate $ outer coeffs pivot
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in indexUpdateRange [StartBound (FS i), StartBound (FS i)] (+offsets) lu'
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export
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decompLU : Neg a => Fractional a => (mat : Matrix' n a) -> DecompLU mat
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decompLU : (Eq a, Neg a, Fractional a) => (mat : Matrix' n a) -> DecompLU mat
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decompLU {n} mat with (viewShape mat)
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_ | Shape [n,n] = iterateN n doolittle (MkLU identity mat)
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where
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doolittle : Fin n -> DecompLU mat -> DecompLU mat
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doolittle i (MkLU l u) =
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let v = rewrite rangeLen (S i') n in fromFunctionNB [minus n (S i')]
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(\[x] => u!#[S i' + x,i'] / u!#[i',i'])
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low = indexSetRange [StartBound (FS i), One i] (-v) identity
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in MkLU (indexSetRange [StartBound (FS i), One i] v l) (low *. u)
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where i' : Nat
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i' = cast i
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_ | Shape [n,n] = MkLU $ iterateN n gaussStep mat
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-- LUP Decomposition
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public export
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record DecompLUP {0 n,a : _} (mat : Matrix' n a) where
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constructor MkLUP
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lower, upper, permute : Matrix' n a
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swaps : Nat
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lu : Matrix' n a
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p : Permutation n
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sw : Nat
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namespace DecompLUP
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export
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lower : Num a => DecompLUP {n,a} mat -> Matrix' n a
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lower (MkLUP lu p sw) with (viewShape lu)
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_ | Shape [n,n] = lowerTriangleStrict lu + identity
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export %inline
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(.lower) : Num a => DecompLUP {n,a} mat -> Matrix' n a
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(.lower) = lower
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export
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upper : Num a => DecompLUP {n,a} mat -> Matrix' n a
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upper (MkLUP lu p sw) = upperTriangle lu
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export %inline
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(.upper) : Num a => DecompLUP {n,a} mat -> Matrix' n a
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(.upper) = upper
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export
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permute : DecompLUP {n} mat -> Permutation n
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permute (MkLUP lu p sw) = p
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export %inline
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(.permute) : DecompLUP {n} mat -> Permutation n
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(.permute) = permute
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export
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numSwaps : DecompLUP {n} mat -> Nat
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numSwaps (MkLUP lu p sw) = sw
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export
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Show a => Show (DecompLUP {a} mat) where
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showPrec p (MkLUP l u pr b) = showCon p "MkLUP" $
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showArg l ++ showArg u ++ showArg pr ++ showArg b
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fromLU : DecompLU mat -> DecompLUP mat
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fromLU (MkLU lu) = MkLUP lu identity 0
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export
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fromLU : Num a => DecompLU {n,a} mat -> DecompLUP mat
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fromLU {n} (MkLU l u) with (viewShape l)
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_ | Shape [n,n] = MkLUP l u identity 0
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export
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decompLUP : Ord a => Abs a => Neg a => Fractional a =>
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decompLUP : (Ord a, Abs a, Neg a, Fractional a) =>
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(mat : Matrix' n a) -> DecompLUP mat
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decompLUP {n} mat with (viewShape mat)
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decompLUP {n=0} mat | Shape [0,0] = MkLUP mat mat mat 0
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decompLUP {n=0} mat | Shape [0,0] = MkLUP mat identity 0
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decompLUP {n=S n} mat | Shape [S n,S n] =
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iterateN (S n) doolittle (MkLUP identity mat identity 0)
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iterateN (S n) gaussStepSwap (MkLUP mat identity 0)
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where
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maxIndex : (s,a) -> List (s,a) -> (s,a)
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maxIndex x [] = x
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@ -243,29 +331,13 @@ decompLUP {n} mat with (viewShape mat)
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if abs b < abs d then assert_total $ maxIndex x ((c,d)::xs)
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else assert_total $ maxIndex x ((a,b)::xs)
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doolittle : Fin (S n) -> DecompLUP mat -> DecompLUP mat
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doolittle i (MkLUP l u p sw) =
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let (maxi, maxv) = mapFst ((+i') . head)
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(maxIndex ([0],0) $ enumerateNB $
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u !!.. [StartBound (weaken i), One i])
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u' = if maxi == i' then u
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else fromFunctionNB _ (\[x,y] =>
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if x==i' then u!#[maxi,y]
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else if x==maxi then u!#[i',y]
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else u!#[x,y])
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p' = if maxi == i' then p
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else fromFunctionNB _ (\[x,y] =>
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if x==i' then p!#[maxi,y]
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else if x==maxi then p!#[i',y]
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else p!#[x,y])
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v = rewrite rangeLen (S i') (S n) in fromFunctionNB [minus n i']
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(\[x] => u'!#[S i' + x,i'] / u'!#[i',i'])
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low = indexSetRange [StartBound (FS i), One i] (-v) identity
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in if maxv == 0 then MkLUP l u p sw else
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MkLUP (indexSetRange [StartBound (FS i), One i] v l)
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(low *. u') p' (if maxi==i' then S sw else sw)
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where i' : Nat
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i' = cast i
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gaussStepSwap : Fin (S n) -> DecompLUP mat -> DecompLUP mat
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gaussStepSwap i (MkLUP lu p sw) =
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let (maxi, maxv) = mapFst head
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(maxIndex ([0],0) $ enumerate $
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indexSetRange [EndBound (weaken i)] 0 $ getColumn i lu)
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in if maxi == i then MkLUP (gaussStep i lu) p sw
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else MkLUP (gaussStep i $ swapRows maxi i lu) (appendSwap maxi i p) (S sw)
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--------------------------------------------------------------------------------
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@ -274,12 +346,16 @@ decompLUP {n} mat with (viewShape mat)
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export
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det : Ord a => Abs a => Neg a => Fractional a =>
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Matrix' n a -> a
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detWithLUP : (Ord a, Abs a, Neg a, Fractional a) =>
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(mat : Matrix' n a) -> DecompLUP mat -> a
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detWithLUP {n} mat lup =
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(if numSwaps lup `mod` 2 == 0 then 1 else -1)
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* product (diagonal lup.lower) * product (diagonal lup.upper)
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export
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det : (Ord a, Abs a, Neg a, Fractional a) => Matrix' n a -> a
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det {n} mat with (viewShape mat)
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det {n=0} mat | Shape [0,0] = 1
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det {n=1} mat | Shape [1,1] = mat!![0,0]
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det {n=2} mat | Shape [2,2] = let [a,b,c,d] = elements mat in a*d - b*c
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_ | Shape [n,n] = let MkLUP l u p sw = decompLUP mat
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in (if sw `mod` 2 == 0 then 1 else -1)
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* product (diagonal l) * product (diagonal u)
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_ | Shape [n,n] = detWithLUP mat (decompLUP mat)
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@ -15,6 +15,10 @@ export
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swap : (i,j : Fin n) -> Permutation n
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swap x y = MkPerm [(x,y)]
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export
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swaps : List (Fin n, Fin n) -> Permutation n
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swaps = MkPerm
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export
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appendSwap : (i,j : Fin n) -> Permutation n -> Permutation n
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appendSwap i j (MkPerm a) = MkPerm ((i,j)::a)
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@ -51,6 +55,12 @@ export
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permuteValues : Permutation n -> Nat -> Nat
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permuteValues p = foldMap @{%search} @{mon} (\(i,j) => swapValues i j) p.swaps
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export
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Show (Permutation n) where
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showPrec p (MkPerm a) = showCon p "swaps" $ showArg a
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export
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Mult (Permutation n) (Permutation n) (Permutation n) where
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MkPerm a *. MkPerm b = MkPerm (a ++ b)
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