{-# LANGUAGE CPP #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE EmptyDataDecls #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE ViewPatterns #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE TypeFamilies #-}

{-# OPTIONS_GHC -fno-warn-missing-signatures #-}
{-# OPTIONS_GHC -fno-warn-orphans #-}

{- |
Module      :  Numeric.LinearAlgebra.Static
Copyright   :  (c) Alberto Ruiz 2014
License     :  BSD3
Stability   :  experimental

Experimental interface with statically checked dimensions.

See code examples at http://dis.um.es/~alberto/hmatrix/static.html.

-}

module Numeric.LinearAlgebra.Static(
    -- * Vector
       ℝ, R,
    vec2, vec3, vec4, (&), (#), split, headTail,
    vector,
    linspace, range, dim,
    -- * Matrix
    L, Sq, build,
    row, col, (|||),(===), splitRows, splitCols,
    unrow, uncol,
    tr,
    eye,
    diag,
    blockAt,
    matrix,
    -- * Complex
    ℂ, C, M, Her, her, 𝑖,
    toComplex,
    fromComplex,
    complex,
    real,
    imag,
    sqMagnitude,
    magnitude,
    -- * Products
    (<>),(#>),(<.>),
    -- * Linear Systems
    linSolve, (<\>),
    -- * Factorizations
    svd, withCompactSVD, svdTall, svdFlat, Eigen(..),
    withNullspace, withOrth, qr, chol,
    -- * Norms
    Normed(..),
    -- * Random arrays
    Seed, RandDist(..),
    randomVector, rand, randn, gaussianSample, uniformSample,
    -- * Misc
    mean, meanCov,
    Disp(..), Domain(..),
    withVector, withMatrix, exactLength, exactDims,
    toRows, toColumns, withRows, withColumns,
    Sized(..), Diag(..), Sym, sym, mTm, unSym, (<·>)
) where


import GHC.TypeLits
import Numeric.LinearAlgebra hiding (
    (<>),(#>),(<.>),Konst(..),diag, disp,(===),(|||),
    row,col,vector,matrix,linspace,toRows,toColumns,
    (<\>),fromList,takeDiag,svd,eig,eigSH,
    eigenvalues,eigenvaluesSH,build,
    qr,size,dot,chol,range,R,C,sym,mTm,unSym,
    randomVector,rand,randn,gaussianSample,uniformSample,meanCov,
    toComplex, fromComplex, complex, real, magnitude
    )
import qualified Numeric.LinearAlgebra as LA
import qualified Numeric.LinearAlgebra.Devel as LA
import Data.Proxy(Proxy(..))
import Internal.Static
import Control.Arrow((***))
import Text.Printf
import Data.Type.Equality ((:~:)(Refl))
import qualified Data.Bifunctor as BF (first)
#if MIN_VERSION_base(4,11,0)
import Prelude hiding ((<>))
#endif

ud1 :: R n -> Vector ℝ
ud1 :: R n -> Vector ℝ
ud1 (R (Dim Vector ℝ
v)) = Vector ℝ
v


infixl 4 &
(&) :: forall n . KnownNat n
    => R n -> ℝ -> R (n+1)
R n
u & :: R n -> ℝ -> R (n + 1)
& ℝ
x = R n
u R n -> R 1 -> R (n + 1)
forall (n :: Nat) (m :: Nat).
(KnownNat n, KnownNat m) =>
R n -> R m -> R (n + m)
# (ℝ -> R 1
forall t s (d :: * -> *). Sized t s d => t -> s
konst ℝ
x :: R 1)

infixl 4 #
(#) :: forall n m . (KnownNat n, KnownNat m)
    => R n -> R m -> R (n+m)
(R Dim n (Vector ℝ)
u) # :: R n -> R m -> R (n + m)
# (R Dim m (Vector ℝ)
v) = Dim (n + m) (Vector ℝ) -> R (n + m)
forall (n :: Nat). Dim n (Vector ℝ) -> R n
R (Dim n (Vector ℝ) -> Dim m (Vector ℝ) -> Dim (n + m) (Vector ℝ)
forall (n :: Nat) (m :: Nat) t.
(KnownNat n, KnownNat m, Numeric t) =>
V n t -> V m t -> V (n + m) t
vconcat Dim n (Vector ℝ)
u Dim m (Vector ℝ)
v)



vec2 :: ℝ -> ℝ -> R 2
vec2 :: ℝ -> ℝ -> R 2
vec2 ℝ
a ℝ
b = Dim 2 (Vector ℝ) -> R 2
forall (n :: Nat). Dim n (Vector ℝ) -> R n
R (ℝ -> ℝ -> Dim 2 (Vector ℝ)
forall t. Storable t => t -> t -> V 2 t
gvec2 ℝ
a ℝ
b)

vec3 :: ℝ -> ℝ -> ℝ -> R 3
vec3 :: ℝ -> ℝ -> ℝ -> R 3
vec3 ℝ
a ℝ
b ℝ
c = Dim 3 (Vector ℝ) -> R 3
forall (n :: Nat). Dim n (Vector ℝ) -> R n
R (ℝ -> ℝ -> ℝ -> Dim 3 (Vector ℝ)
forall t. Storable t => t -> t -> t -> V 3 t
gvec3 ℝ
a ℝ
b ℝ
c)


vec4 :: ℝ -> ℝ -> ℝ -> ℝ -> R 4
vec4 :: ℝ -> ℝ -> ℝ -> ℝ -> R 4
vec4 ℝ
a ℝ
b ℝ
c ℝ
d = Dim 4 (Vector ℝ) -> R 4
forall (n :: Nat). Dim n (Vector ℝ) -> R n
R (ℝ -> ℝ -> ℝ -> ℝ -> Dim 4 (Vector ℝ)
forall t. Storable t => t -> t -> t -> t -> V 4 t
gvec4 ℝ
a ℝ
b ℝ
c ℝ
d)

vector :: KnownNat n => [ℝ] -> R n
vector :: [ℝ] -> R n
vector = [ℝ] -> R n
forall t s (d :: * -> *). Sized t s d => [t] -> s
fromList

matrix :: (KnownNat m, KnownNat n) => [ℝ] -> L m n
matrix :: [ℝ] -> L m n
matrix = [ℝ] -> L m n
forall t s (d :: * -> *). Sized t s d => [t] -> s
fromList

linspace :: forall n . KnownNat n => (ℝ,ℝ) -> R n
linspace :: (ℝ, ℝ) -> R n
linspace (ℝ
a,ℝ
b) = R n
v
  where
    v :: R n
v = Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR (Int -> (ℝ, ℝ) -> Vector ℝ
forall e.
(Fractional e, Container Vector e) =>
Int -> (e, e) -> Vector e
LA.linspace (R n -> IndexOf Vector
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size R n
v) (ℝ
a,ℝ
b))

range :: forall n . KnownNat n => R n
range :: R n
range = R n
v
  where
    v :: R n
v = Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR (Int -> (ℝ, ℝ) -> Vector ℝ
forall e.
(Fractional e, Container Vector e) =>
Int -> (e, e) -> Vector e
LA.linspace Int
d (ℝ
1,Int -> ℝ
forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
d))
    d :: Int
d = R n -> IndexOf Vector
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size R n
v

dim :: forall n . KnownNat n => R n
dim :: R n
dim = R n
v
  where
    v :: R n
v = Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR (ℝ -> Vector ℝ
forall (c :: * -> *) e. Container c e => e -> c e
scalar (Int -> ℝ
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> ℝ) -> Int -> ℝ
forall a b. (a -> b) -> a -> b
$ R n -> IndexOf Vector
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size R n
v))

--------------------------------------------------------------------------------


ud2 :: L m n -> Matrix ℝ
ud2 :: L m n -> Matrix ℝ
ud2 (L (Dim (Dim Matrix ℝ
x))) = Matrix ℝ
x


--------------------------------------------------------------------------------

diag :: KnownNat n => R n -> Sq n
diag :: R n -> Sq n
diag = ℝ -> R n -> Sq n
forall field (vec :: Nat -> *) (mat :: Nat -> Nat -> *) (m :: Nat)
       (n :: Nat) (k :: Nat).
(Domain field vec mat, KnownNat m, KnownNat n, KnownNat k) =>
field -> vec k -> mat m n
diagR ℝ
0

eye :: KnownNat n => Sq n
eye :: Sq n
eye = R n -> Sq n
forall (n :: Nat). KnownNat n => R n -> Sq n
diag R n
1

--------------------------------------------------------------------------------

blockAt :: forall m n . (KnownNat m, KnownNat n) =>  ℝ -> Int -> Int -> Matrix Double -> L m n
blockAt :: ℝ -> Int -> Int -> Matrix ℝ -> L m n
blockAt ℝ
x Int
r Int
c Matrix ℝ
a = L m n
res
  where
    z :: Matrix ℝ
z = ℝ -> Matrix ℝ
forall (c :: * -> *) e. Container c e => e -> c e
scalar ℝ
x
    z1 :: Matrix ℝ
z1 = ℝ -> (Int, Int) -> Matrix ℝ
forall e d (c :: * -> *). Konst e d c => e -> d -> c e
LA.konst ℝ
x (Int
r,Int
c)
    z2 :: Matrix ℝ
z2 = ℝ -> (Int, Int) -> Matrix ℝ
forall e d (c :: * -> *). Konst e d c => e -> d -> c e
LA.konst ℝ
x (Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
0 (Int
m'Int -> Int -> Int
forall a. Num a => a -> a -> a
-(Int
raInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
r)), Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
0 (Int
n'Int -> Int -> Int
forall a. Num a => a -> a -> a
-(Int
caInt -> Int -> Int
forall a. Num a => a -> a -> a
+Int
c)))
    ra :: Int
ra = Int -> Int -> Int
forall a. Ord a => a -> a -> a
min (Matrix ℝ -> Int
forall t. Matrix t -> Int
rows Matrix ℝ
a) (Int -> Int) -> (Int -> Int) -> Int -> Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
0 (Int -> Int) -> Int -> Int
forall a b. (a -> b) -> a -> b
$ Int
m'Int -> Int -> Int
forall a. Num a => a -> a -> a
-Int
r
    ca :: Int
ca = Int -> Int -> Int
forall a. Ord a => a -> a -> a
min (Matrix ℝ -> Int
forall t. Matrix t -> Int
cols Matrix ℝ
a) (Int -> Int) -> (Int -> Int) -> Int -> Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
0 (Int -> Int) -> Int -> Int
forall a b. (a -> b) -> a -> b
$ Int
n'Int -> Int -> Int
forall a. Num a => a -> a -> a
-Int
c
    sa :: Matrix ℝ
sa = (Int, Int) -> (Int, Int) -> Matrix ℝ -> Matrix ℝ
forall a.
Element a =>
(Int, Int) -> (Int, Int) -> Matrix a -> Matrix a
subMatrix (Int
0,Int
0) (Int
ra, Int
ca) Matrix ℝ
a
    (Int
m',Int
n') = L m n -> IndexOf Matrix
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size L m n
res
    res :: L m n
res = Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> L m n) -> Matrix ℝ -> L m n
forall a b. (a -> b) -> a -> b
$ [[Matrix ℝ]] -> Matrix ℝ
forall t. Element t => [[Matrix t]] -> Matrix t
fromBlocks [[Matrix ℝ
z1,Matrix ℝ
z,Matrix ℝ
z],[Matrix ℝ
z,Matrix ℝ
sa,Matrix ℝ
z],[Matrix ℝ
z,Matrix ℝ
z,Matrix ℝ
z2]]

--------------------------------------------------------------------------------


row :: R n -> L 1 n
row :: R n -> L 1 n
row = Matrix ℝ -> L 1 n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> L 1 n) -> (R n -> Matrix ℝ) -> R n -> L 1 n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Vector ℝ -> Matrix ℝ
forall a. Storable a => Vector a -> Matrix a
asRow (Vector ℝ -> Matrix ℝ) -> (R n -> Vector ℝ) -> R n -> Matrix ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. R n -> Vector ℝ
forall (n :: Nat). R n -> Vector ℝ
ud1

--col :: R n -> L n 1
col :: R n -> L n 1
col R n
v = L 1 n -> L n 1
forall m mt. Transposable m mt => m -> mt
tr (L 1 n -> L n 1) -> (R n -> L 1 n) -> R n -> L n 1
forall b c a. (b -> c) -> (a -> b) -> a -> c
. R n -> L 1 n
forall (n :: Nat). R n -> L 1 n
row (R n -> L n 1) -> R n -> L n 1
forall a b. (a -> b) -> a -> b
$ R n
v

unrow :: L 1 n -> R n
unrow :: L 1 n -> R n
unrow = Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR (Vector ℝ -> R n) -> (L 1 n -> Vector ℝ) -> L 1 n -> R n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [Vector ℝ] -> Vector ℝ
forall a. [a] -> a
head ([Vector ℝ] -> Vector ℝ)
-> (L 1 n -> [Vector ℝ]) -> L 1 n -> Vector ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Matrix ℝ -> [Vector ℝ]
forall t. Element t => Matrix t -> [Vector t]
LA.toRows (Matrix ℝ -> [Vector ℝ])
-> (L 1 n -> Matrix ℝ) -> L 1 n -> [Vector ℝ]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. L 1 n -> Matrix ℝ
forall (m :: Nat) (n :: Nat). L m n -> Matrix ℝ
ud2

--uncol :: L n 1 -> R n
uncol :: L n 1 -> R n
uncol L n 1
v = L 1 n -> R n
forall (n :: Nat). L 1 n -> R n
unrow (L 1 n -> R n) -> (L n 1 -> L 1 n) -> L n 1 -> R n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. L n 1 -> L 1 n
forall m mt. Transposable m mt => m -> mt
tr (L n 1 -> R n) -> L n 1 -> R n
forall a b. (a -> b) -> a -> b
$ L n 1
v


infixl 2 ===
(===) :: (KnownNat r1, KnownNat r2, KnownNat c) => L r1 c -> L r2 c -> L (r1+r2) c
L r1 c
a === :: L r1 c -> L r2 c -> L (r1 + r2) c
=== L r2 c
b = Matrix ℝ -> L (r1 + r2) c
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (L r1 c -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract L r1 c
a Matrix ℝ -> Matrix ℝ -> Matrix ℝ
forall t. Element t => Matrix t -> Matrix t -> Matrix t
LA.=== L r2 c -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract L r2 c
b)


infixl 3 |||
-- (|||) :: (KnownNat r, KnownNat c1, KnownNat c2) => L r c1 -> L r c2 -> L r (c1+c2)
L c r1
a ||| :: L c r1 -> L c r2 -> L c (r1 + r2)
||| L c r2
b = L (r1 + r2) c -> L c (r1 + r2)
forall m mt. Transposable m mt => m -> mt
tr (L c r1 -> L r1 c
forall m mt. Transposable m mt => m -> mt
tr L c r1
a L r1 c -> L r2 c -> L (r1 + r2) c
forall (r1 :: Nat) (r2 :: Nat) (c :: Nat).
(KnownNat r1, KnownNat r2, KnownNat c) =>
L r1 c -> L r2 c -> L (r1 + r2) c
=== L c r2 -> L r2 c
forall m mt. Transposable m mt => m -> mt
tr L c r2
b)


type Sq n  = L n n
--type CSq n = CL n n


type GL = forall n m . (KnownNat n, KnownNat m) => L m n
type GSq = forall n . KnownNat n => Sq n

isKonst :: forall m n . (KnownNat m, KnownNat n) => L m n -> Maybe (ℝ,(Int,Int))
isKonst :: L m n -> Maybe (ℝ, (Int, Int))
isKonst s :: L m n
s@(L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap -> Matrix ℝ
x)
    | Matrix ℝ -> Bool
forall t. Matrix t -> Bool
singleM Matrix ℝ
x = (ℝ, (Int, Int)) -> Maybe (ℝ, (Int, Int))
forall a. a -> Maybe a
Just (Matrix ℝ
x Matrix ℝ -> IndexOf Matrix -> ℝ
forall (c :: * -> *) e. Container c e => c e -> IndexOf c -> e
`atIndex` (Int
0,Int
0), (L m n -> IndexOf Matrix
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size L m n
s))
    | Bool
otherwise = Maybe (ℝ, (Int, Int))
forall a. Maybe a
Nothing


isKonstC :: forall m n . (KnownNat m, KnownNat n) => M m n -> Maybe (ℂ,(Int,Int))
isKonstC :: M m n -> Maybe (ℂ, (Int, Int))
isKonstC s :: M m n
s@(M m n -> Matrix ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap -> Matrix ℂ
x)
    | Matrix ℂ -> Bool
forall t. Matrix t -> Bool
singleM Matrix ℂ
x = (ℂ, (Int, Int)) -> Maybe (ℂ, (Int, Int))
forall a. a -> Maybe a
Just (Matrix ℂ
x Matrix ℂ -> IndexOf Matrix -> ℂ
forall (c :: * -> *) e. Container c e => c e -> IndexOf c -> e
`atIndex` (Int
0,Int
0), (M m n -> IndexOf Matrix
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size M m n
s))
    | Bool
otherwise = Maybe (ℂ, (Int, Int))
forall a. Maybe a
Nothing


infixr 8 <>
(<>) :: forall m k n. (KnownNat m, KnownNat k, KnownNat n) => L m k -> L k n -> L m n
<> :: L m k -> L k n -> L m n
(<>) = L m k -> L k n -> L m n
forall (m :: Nat) (k :: Nat) (n :: Nat).
(KnownNat m, KnownNat k, KnownNat n) =>
L m k -> L k n -> L m n
mulR


infixr 8 #>
(#>) :: (KnownNat m, KnownNat n) => L m n -> R n -> R m
#> :: L m n -> R n -> R m
(#>) = L m n -> R n -> R m
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> R n -> R m
appR


infixr 8 <·>
(<·>) :: KnownNat n => R n -> R n -> ℝ
<·> :: R n -> R n -> ℝ
(<·>) = R n -> R n -> ℝ
forall (n :: Nat). KnownNat n => R n -> R n -> ℝ
dotR

infixr 8 <.>
(<.>) :: KnownNat n => R n -> R n -> ℝ
<.> :: R n -> R n -> ℝ
(<.>) = R n -> R n -> ℝ
forall (n :: Nat). KnownNat n => R n -> R n -> ℝ
dotR

--------------------------------------------------------------------------------

class Diag m d | m -> d
  where
    takeDiag :: m -> d


instance KnownNat n => Diag (L n n) (R n)
  where
    takeDiag :: L n n -> R n
takeDiag L n n
x = Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR (Matrix ℝ -> Vector ℝ
forall t. Element t => Matrix t -> Vector t
LA.takeDiag (L n n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract L n n
x))


instance KnownNat n => Diag (M n n) (C n)
  where
    takeDiag :: M n n -> C n
takeDiag M n n
x = Vector ℂ -> C n
forall (n :: Nat). Vector ℂ -> C n
mkC (Matrix ℂ -> Vector ℂ
forall t. Element t => Matrix t -> Vector t
LA.takeDiag (M n n -> Matrix ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract M n n
x))

--------------------------------------------------------------------------------


toComplex :: KnownNat n => (R n, R n) -> C n
toComplex :: (R n, R n) -> C n
toComplex (R n
r,R n
i) = Vector ℂ -> C n
forall (n :: Nat). Vector ℂ -> C n
mkC (Vector ℂ -> C n) -> Vector ℂ -> C n
forall a b. (a -> b) -> a -> b
$ (Vector ℝ, Vector ℝ) -> Vector ℂ
forall t (c :: * -> *).
(Convert t, Complexable c, RealElement t) =>
(c t, c t) -> c (Complex t)
LA.toComplex (R n -> Vector ℝ
forall (n :: Nat). R n -> Vector ℝ
ud1 R n
r, R n -> Vector ℝ
forall (n :: Nat). R n -> Vector ℝ
ud1 R n
i)

fromComplex :: KnownNat n => C n -> (R n, R n)
fromComplex :: C n -> (R n, R n)
fromComplex (C (Dim Vector ℂ
v)) = let (Vector ℝ
r,Vector ℝ
i) = Vector ℂ -> (Vector ℝ, Vector ℝ)
forall t (c :: * -> *).
(Convert t, Complexable c, RealElement t) =>
c (Complex t) -> (c t, c t)
LA.fromComplex Vector ℂ
v in (Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR Vector ℝ
r, Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR Vector ℝ
i)

complex :: KnownNat n => R n -> C n
complex :: R n -> C n
complex R n
r = Vector ℂ -> C n
forall (n :: Nat). Vector ℂ -> C n
mkC (Vector ℂ -> C n) -> Vector ℂ -> C n
forall a b. (a -> b) -> a -> b
$ (Vector ℝ, Vector ℝ) -> Vector ℂ
forall t (c :: * -> *).
(Convert t, Complexable c, RealElement t) =>
(c t, c t) -> c (Complex t)
LA.toComplex (R n -> Vector ℝ
forall (n :: Nat). R n -> Vector ℝ
ud1 R n
r, ℝ -> Int -> Vector ℝ
forall e d (c :: * -> *). Konst e d c => e -> d -> c e
LA.konst ℝ
0 (R n -> IndexOf Vector
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size R n
r))

real :: KnownNat n => C n -> R n
real :: C n -> R n
real = (R n, R n) -> R n
forall a b. (a, b) -> a
fst ((R n, R n) -> R n) -> (C n -> (R n, R n)) -> C n -> R n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. C n -> (R n, R n)
forall (n :: Nat). KnownNat n => C n -> (R n, R n)
fromComplex

imag :: KnownNat n => C n -> R n 
imag :: C n -> R n
imag = (R n, R n) -> R n
forall a b. (a, b) -> b
snd ((R n, R n) -> R n) -> (C n -> (R n, R n)) -> C n -> R n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. C n -> (R n, R n)
forall (n :: Nat). KnownNat n => C n -> (R n, R n)
fromComplex

sqMagnitude :: KnownNat n => C n -> R n
sqMagnitude :: C n -> R n
sqMagnitude C n
c = let (R n
r,R n
i) = C n -> (R n, R n)
forall (n :: Nat). KnownNat n => C n -> (R n, R n)
fromComplex C n
c in R n
rR n -> R n -> R n
forall a. Floating a => a -> a -> a
**R n
2 R n -> R n -> R n
forall a. Num a => a -> a -> a
+ R n
iR n -> R n -> R n
forall a. Floating a => a -> a -> a
**R n
2

magnitude :: KnownNat n => C n -> R n
magnitude :: C n -> R n
magnitude = R n -> R n
forall a. Floating a => a -> a
sqrt (R n -> R n) -> (C n -> R n) -> C n -> R n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. C n -> R n
forall (n :: Nat). KnownNat n => C n -> R n
sqMagnitude

--------------------------------------------------------------------------------

linSolve :: (KnownNat m, KnownNat n) => L m m -> L m n -> Maybe (L m n)
linSolve :: L m m -> L m n -> Maybe (L m n)
linSolve (L m m -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
a) (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
b) = (Matrix ℝ -> L m n) -> Maybe (Matrix ℝ) -> Maybe (L m n)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> Matrix ℝ -> Maybe (Matrix ℝ)
forall t. Field t => Matrix t -> Matrix t -> Maybe (Matrix t)
LA.linearSolve Matrix ℝ
a Matrix ℝ
b)

(<\>) :: (KnownNat m, KnownNat n, KnownNat r) => L m n -> L m r -> L n r
(L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
a) <\> :: L m n -> L m r -> L n r
<\> (L m r -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
b) = Matrix ℝ -> L n r
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ
a Matrix ℝ -> Matrix ℝ -> Matrix ℝ
forall (c :: * -> *) t.
(LSDiv c, Field t) =>
Matrix t -> c t -> c t
LA.<\> Matrix ℝ
b)

svd :: (KnownNat m, KnownNat n) => L m n -> (L m m, R n, L n n)
svd :: L m n -> (L m m, R n, L n n)
svd (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
m) = (Matrix ℝ -> L m m
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
u, Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR Vector ℝ
s', Matrix ℝ -> L n n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
v)
  where
    (Matrix ℝ
u,Vector ℝ
s,Matrix ℝ
v) = Matrix ℝ -> (Matrix ℝ, Vector ℝ, Matrix ℝ)
forall t. Field t => Matrix t -> (Matrix t, Vector ℝ, Matrix t)
LA.svd Matrix ℝ
m
    s' :: Vector ℝ
s' = [Vector ℝ] -> Vector ℝ
forall t. Storable t => [Vector t] -> Vector t
vjoin [Vector ℝ
s, Vector ℝ
z]
    z :: Vector ℝ
z = ℝ -> Int -> Vector ℝ
forall e d (c :: * -> *). Konst e d c => e -> d -> c e
LA.konst ℝ
0 (Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
0 (Matrix ℝ -> Int
forall t. Matrix t -> Int
cols Matrix ℝ
m Int -> Int -> Int
forall a. Num a => a -> a -> a
- Vector ℝ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℝ
s))


svdTall :: (KnownNat m, KnownNat n, n <= m) => L m n -> (L m n, R n, L n n)
svdTall :: L m n -> (L m n, R n, L n n)
svdTall (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
m) = (Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
u, Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR Vector ℝ
s, Matrix ℝ -> L n n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
v)
  where
    (Matrix ℝ
u,Vector ℝ
s,Matrix ℝ
v) = Matrix ℝ -> (Matrix ℝ, Vector ℝ, Matrix ℝ)
forall t. Field t => Matrix t -> (Matrix t, Vector ℝ, Matrix t)
LA.thinSVD Matrix ℝ
m


svdFlat :: (KnownNat m, KnownNat n, m <= n) => L m n -> (L m m, R m, L n m)
svdFlat :: L m n -> (L m m, R m, L n m)
svdFlat (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
m) = (Matrix ℝ -> L m m
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
u, Vector ℝ -> R m
forall (n :: Nat). Vector ℝ -> R n
mkR Vector ℝ
s, Matrix ℝ -> L n m
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
v)
  where
    (Matrix ℝ
u,Vector ℝ
s,Matrix ℝ
v) = Matrix ℝ -> (Matrix ℝ, Vector ℝ, Matrix ℝ)
forall t. Field t => Matrix t -> (Matrix t, Vector ℝ, Matrix t)
LA.thinSVD Matrix ℝ
m

--------------------------------------------------------------------------------

class Eigen m l v | m -> l, m -> v
  where
    eigensystem :: m -> (l,v)
    eigenvalues :: m -> l

newtype Sym n = Sym (Sq n) deriving Int -> Sym n -> ShowS
[Sym n] -> ShowS
Sym n -> String
(Int -> Sym n -> ShowS)
-> (Sym n -> String) -> ([Sym n] -> ShowS) -> Show (Sym n)
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
forall (n :: Nat). KnownNat n => Int -> Sym n -> ShowS
forall (n :: Nat). KnownNat n => [Sym n] -> ShowS
forall (n :: Nat). KnownNat n => Sym n -> String
showList :: [Sym n] -> ShowS
$cshowList :: forall (n :: Nat). KnownNat n => [Sym n] -> ShowS
show :: Sym n -> String
$cshow :: forall (n :: Nat). KnownNat n => Sym n -> String
showsPrec :: Int -> Sym n -> ShowS
$cshowsPrec :: forall (n :: Nat). KnownNat n => Int -> Sym n -> ShowS
Show


sym :: KnownNat n => Sq n -> Sym n
sym :: Sq n -> Sym n
sym Sq n
m = Sq n -> Sym n
forall (n :: Nat). Sq n -> Sym n
Sym (Sq n -> Sym n) -> Sq n -> Sym n
forall a b. (a -> b) -> a -> b
$ (Sq n
m Sq n -> Sq n -> Sq n
forall a. Num a => a -> a -> a
+ Sq n -> Sq n
forall m mt. Transposable m mt => m -> mt
tr Sq n
m)Sq n -> Sq n -> Sq n
forall a. Fractional a => a -> a -> a
/Sq n
2

mTm :: (KnownNat m, KnownNat n) => L m n -> Sym n
mTm :: L m n -> Sym n
mTm L m n
x = Sq n -> Sym n
forall (n :: Nat). Sq n -> Sym n
Sym (L m n -> L n m
forall m mt. Transposable m mt => m -> mt
tr L m n
x L n m -> L m n -> Sq n
forall (m :: Nat) (k :: Nat) (n :: Nat).
(KnownNat m, KnownNat k, KnownNat n) =>
L m k -> L k n -> L m n
<> L m n
x)

unSym :: Sym n -> Sq n
unSym :: Sym n -> Sq n
unSym (Sym Sq n
x) = Sq n
x


𝑖 :: Sized ℂ s c => s
𝑖 :: s
𝑖 = ℂ -> s
forall t s (d :: * -> *). Sized t s d => t -> s
konst ℂ
iC

newtype Her n = Her (M n n)

her :: KnownNat n => M n n -> Her n
her :: M n n -> Her n
her M n n
m = M n n -> Her n
forall (n :: Nat). M n n -> Her n
Her (M n n -> Her n) -> M n n -> Her n
forall a b. (a -> b) -> a -> b
$ (M n n
m M n n -> M n n -> M n n
forall a. Num a => a -> a -> a
+ M n n -> M n n
forall m mt. Transposable m mt => m -> mt
LA.tr M n n
m)M n n -> M n n -> M n n
forall a. Fractional a => a -> a -> a
/M n n
2


instance (KnownNat n) => Disp (Sym n)
  where
    disp :: Int -> Sym n -> IO ()
disp Int
n (Sym Sq n
x) = do
        let a :: Matrix ℝ
a = Sq n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract Sq n
x
        let su :: String
su = Int -> Matrix ℝ -> String
LA.dispf Int
n Matrix ℝ
a
        String -> Int -> IO ()
forall r. PrintfType r => String -> r
printf String
"Sym %d" (Matrix ℝ -> Int
forall t. Matrix t -> Int
cols Matrix ℝ
a) IO () -> IO () -> IO ()
forall (m :: * -> *) a b. Monad m => m a -> m b -> m b
>> String -> IO ()
putStr ((Char -> Bool) -> ShowS
forall a. (a -> Bool) -> [a] -> [a]
dropWhile (Char -> Char -> Bool
forall a. Eq a => a -> a -> Bool
/=Char
'\n') ShowS -> ShowS
forall a b. (a -> b) -> a -> b
$ String
su)

instance (KnownNat n) => Disp (Her n)
  where
    disp :: Int -> Her n -> IO ()
disp Int
n (Her M n n
x) = do
        let a :: Matrix ℂ
a = M n n -> Matrix ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract M n n
x
        let su :: String
su = Int -> Matrix ℂ -> String
LA.dispcf Int
n Matrix ℂ
a
        String -> Int -> IO ()
forall r. PrintfType r => String -> r
printf String
"Her %d" (Matrix ℂ -> Int
forall t. Matrix t -> Int
cols Matrix ℂ
a) IO () -> IO () -> IO ()
forall (m :: * -> *) a b. Monad m => m a -> m b -> m b
>> String -> IO ()
putStr ((Char -> Bool) -> ShowS
forall a. (a -> Bool) -> [a] -> [a]
dropWhile (Char -> Char -> Bool
forall a. Eq a => a -> a -> Bool
/=Char
'\n') ShowS -> ShowS
forall a b. (a -> b) -> a -> b
$ String
su)


instance KnownNat n => Eigen (Sym n) (R n) (L n n)
  where
    eigenvalues :: Sym n -> R n
eigenvalues (Sym (L n n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
m)) =  Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR (Vector ℝ -> R n) -> (Matrix ℝ -> Vector ℝ) -> Matrix ℝ -> R n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Herm ℝ -> Vector ℝ
forall t. Field t => Herm t -> Vector ℝ
LA.eigenvaluesSH (Herm ℝ -> Vector ℝ)
-> (Matrix ℝ -> Herm ℝ) -> Matrix ℝ -> Vector ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Matrix ℝ -> Herm ℝ
forall t. Matrix t -> Herm t
LA.trustSym (Matrix ℝ -> R n) -> Matrix ℝ -> R n
forall a b. (a -> b) -> a -> b
$ Matrix ℝ
m
    eigensystem :: Sym n -> (R n, L n n)
eigensystem (Sym (L n n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
m)) = (Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR Vector ℝ
l, Matrix ℝ -> L n n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
v)
      where
        (Vector ℝ
l,Matrix ℝ
v) = Herm ℝ -> (Vector ℝ, Matrix ℝ)
forall t. Field t => Herm t -> (Vector ℝ, Matrix t)
LA.eigSH (Herm ℝ -> (Vector ℝ, Matrix ℝ))
-> (Matrix ℝ -> Herm ℝ) -> Matrix ℝ -> (Vector ℝ, Matrix ℝ)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Matrix ℝ -> Herm ℝ
forall t. Matrix t -> Herm t
LA.trustSym (Matrix ℝ -> (Vector ℝ, Matrix ℝ))
-> Matrix ℝ -> (Vector ℝ, Matrix ℝ)
forall a b. (a -> b) -> a -> b
$ Matrix ℝ
m

instance KnownNat n => Eigen (Sq n) (C n) (M n n)
  where
    eigenvalues :: Sq n -> C n
eigenvalues (Sq n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
m) = Vector ℂ -> C n
forall (n :: Nat). Vector ℂ -> C n
mkC (Vector ℂ -> C n) -> (Matrix ℝ -> Vector ℂ) -> Matrix ℝ -> C n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Matrix ℝ -> Vector ℂ
forall t. Field t => Matrix t -> Vector ℂ
LA.eigenvalues (Matrix ℝ -> C n) -> Matrix ℝ -> C n
forall a b. (a -> b) -> a -> b
$ Matrix ℝ
m
    eigensystem :: Sq n -> (C n, M n n)
eigensystem (Sq n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
m) = (Vector ℂ -> C n
forall (n :: Nat). Vector ℂ -> C n
mkC Vector ℂ
l, Matrix ℂ -> M n n
forall (m :: Nat) (n :: Nat). Matrix ℂ -> M m n
mkM Matrix ℂ
v)
      where
        (Vector ℂ
l,Matrix ℂ
v) = Matrix ℝ -> (Vector ℂ, Matrix ℂ)
forall t. Field t => Matrix t -> (Vector ℂ, Matrix ℂ)
LA.eig Matrix ℝ
m

chol :: KnownNat n => Sym n -> Sq n
chol :: Sym n -> Sq n
chol (Sq n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract (Sq n -> Matrix ℝ) -> (Sym n -> Sq n) -> Sym n -> Matrix ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Sym n -> Sq n
forall (n :: Nat). Sym n -> Sq n
unSym -> Matrix ℝ
m) = Matrix ℝ -> Sq n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> Sq n) -> Matrix ℝ -> Sq n
forall a b. (a -> b) -> a -> b
$ Herm ℝ -> Matrix ℝ
forall t. Field t => Herm t -> Matrix t
LA.chol (Herm ℝ -> Matrix ℝ) -> Herm ℝ -> Matrix ℝ
forall a b. (a -> b) -> a -> b
$ Matrix ℝ -> Herm ℝ
forall t. Matrix t -> Herm t
LA.trustSym Matrix ℝ
m

--------------------------------------------------------------------------------

withNullspace
    :: forall m n z . (KnownNat m, KnownNat n)
    => L m n
    -> (forall k . (KnownNat k) => L n k -> z)
    -> z
withNullspace :: L m n -> (forall (k :: Nat). KnownNat k => L n k -> z) -> z
withNullspace (Matrix ℝ -> Matrix ℝ
forall t. Field t => Matrix t -> Matrix t
LA.nullspace (Matrix ℝ -> Matrix ℝ) -> (L m n -> Matrix ℝ) -> L m n -> Matrix ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
a) forall (k :: Nat). KnownNat k => L n k -> z
f =
    case Integer -> Maybe SomeNat
someNatVal (Integer -> Maybe SomeNat) -> Integer -> Maybe SomeNat
forall a b. (a -> b) -> a -> b
$ Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Matrix ℝ -> Int
forall t. Matrix t -> Int
cols Matrix ℝ
a of
       Maybe SomeNat
Nothing -> String -> z
forall a. HasCallStack => String -> a
error String
"static/dynamic mismatch"
       Just (SomeNat (Proxy n
_ :: Proxy k)) -> L n n -> z
forall (k :: Nat). KnownNat k => L n k -> z
f (Matrix ℝ -> L n n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
a :: L n k)

withOrth
    :: forall m n z . (KnownNat m, KnownNat n)
    => L m n
    -> (forall k. (KnownNat k) => L n k -> z)
    -> z
withOrth :: L m n -> (forall (k :: Nat). KnownNat k => L n k -> z) -> z
withOrth (Matrix ℝ -> Matrix ℝ
forall t. Field t => Matrix t -> Matrix t
LA.orth (Matrix ℝ -> Matrix ℝ) -> (L m n -> Matrix ℝ) -> L m n -> Matrix ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
a) forall (k :: Nat). KnownNat k => L n k -> z
f =
    case Integer -> Maybe SomeNat
someNatVal (Integer -> Maybe SomeNat) -> Integer -> Maybe SomeNat
forall a b. (a -> b) -> a -> b
$ Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Matrix ℝ -> Int
forall t. Matrix t -> Int
cols Matrix ℝ
a of
       Maybe SomeNat
Nothing -> String -> z
forall a. HasCallStack => String -> a
error String
"static/dynamic mismatch"
       Just (SomeNat (Proxy n
_ :: Proxy k)) -> L n n -> z
forall (k :: Nat). KnownNat k => L n k -> z
f (Matrix ℝ -> L n n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
a :: L n k)

withCompactSVD
    :: forall m n z . (KnownNat m, KnownNat n)
    => L m n
    -> (forall k . (KnownNat k) => (L m k, R k, L n k) -> z)
    -> z
withCompactSVD :: L m n
-> (forall (k :: Nat). KnownNat k => (L m k, R k, L n k) -> z) -> z
withCompactSVD (Matrix ℝ -> (Matrix ℝ, Vector ℝ, Matrix ℝ)
forall t. Field t => Matrix t -> (Matrix t, Vector ℝ, Matrix t)
LA.compactSVD (Matrix ℝ -> (Matrix ℝ, Vector ℝ, Matrix ℝ))
-> (L m n -> Matrix ℝ) -> L m n -> (Matrix ℝ, Vector ℝ, Matrix ℝ)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> (Matrix ℝ
u,Vector ℝ
s,Matrix ℝ
v)) forall (k :: Nat). KnownNat k => (L m k, R k, L n k) -> z
f =
    case Integer -> Maybe SomeNat
someNatVal (Integer -> Maybe SomeNat) -> Integer -> Maybe SomeNat
forall a b. (a -> b) -> a -> b
$ Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Vector ℝ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℝ
s of
       Maybe SomeNat
Nothing -> String -> z
forall a. HasCallStack => String -> a
error String
"static/dynamic mismatch"
       Just (SomeNat (Proxy n
_ :: Proxy k)) -> (L m n, R n, L n n) -> z
forall (k :: Nat). KnownNat k => (L m k, R k, L n k) -> z
f (Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
u :: L m k, Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR Vector ℝ
s :: R k, Matrix ℝ -> L n n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
v :: L n k)

--------------------------------------------------------------------------------

qr :: (KnownNat m, KnownNat n) => L m n -> (L m m, L m n)
qr :: L m n -> (L m m, L m n)
qr (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
x) = (Matrix ℝ -> L m m
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
q, Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
r)
  where
    (Matrix ℝ
q,Matrix ℝ
r) = Matrix ℝ -> (Matrix ℝ, Matrix ℝ)
forall t. Field t => Matrix t -> (Matrix t, Matrix t)
LA.qr Matrix ℝ
x

-- use qrRaw?

--------------------------------------------------------------------------------

split :: forall p n . (KnownNat p, KnownNat n, p<=n) => R n -> (R p, R (n-p))
split :: R n -> (R p, R (n - p))
split (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
v) = ( Vector ℝ -> R p
forall (n :: Nat). Vector ℝ -> R n
mkR (Int -> Int -> Vector ℝ -> Vector ℝ
forall t. Storable t => Int -> Int -> Vector t -> Vector t
subVector Int
0 Int
p' Vector ℝ
v) ,
                         Vector ℝ -> R (n - p)
forall (n :: Nat). Vector ℝ -> R n
mkR (Int -> Int -> Vector ℝ -> Vector ℝ
forall t. Storable t => Int -> Int -> Vector t -> Vector t
subVector Int
p' (Vector ℝ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℝ
v Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
p') Vector ℝ
v) )
  where
    p' :: Int
p' = Integer -> Int
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Integer -> Int) -> (Proxy p -> Integer) -> Proxy p -> Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Proxy p -> Integer
forall (n :: Nat) (proxy :: Nat -> *).
KnownNat n =>
proxy n -> Integer
natVal (Proxy p -> Int) -> Proxy p -> Int
forall a b. (a -> b) -> a -> b
$ (Proxy p
forall a. HasCallStack => a
undefined :: Proxy p) :: Int


headTail :: (KnownNat n, 1<=n) => R n -> (ℝ, R (n-1))
headTail :: R n -> (ℝ, R (n - 1))
headTail = ((Vector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
0) (Vector ℝ -> ℝ) -> (R 1 -> Vector ℝ) -> R 1 -> ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. R 1 -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract (R 1 -> ℝ)
-> (R (n - 1) -> R (n - 1)) -> (R 1, R (n - 1)) -> (ℝ, R (n - 1))
forall (a :: * -> * -> *) b c b' c'.
Arrow a =>
a b c -> a b' c' -> a (b, b') (c, c')
*** R (n - 1) -> R (n - 1)
forall a. a -> a
id) ((R 1, R (n - 1)) -> (ℝ, R (n - 1)))
-> (R n -> (R 1, R (n - 1))) -> R n -> (ℝ, R (n - 1))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. R n -> (R 1, R (n - 1))
forall (p :: Nat) (n :: Nat).
(KnownNat p, KnownNat n, p <= n) =>
R n -> (R p, R (n - p))
split


splitRows :: forall p m n . (KnownNat p, KnownNat m, KnownNat n, p<=m) => L m n -> (L p n, L (m-p) n)
splitRows :: L m n -> (L p n, L (m - p) n)
splitRows (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
x) = ( Matrix ℝ -> L p n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Int -> Matrix ℝ -> Matrix ℝ
forall t. Element t => Int -> Matrix t -> Matrix t
takeRows Int
p' Matrix ℝ
x) ,
                             Matrix ℝ -> L (m - p) n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Int -> Matrix ℝ -> Matrix ℝ
forall t. Element t => Int -> Matrix t -> Matrix t
dropRows Int
p' Matrix ℝ
x) )
  where
    p' :: Int
p' = Integer -> Int
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Integer -> Int) -> (Proxy p -> Integer) -> Proxy p -> Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Proxy p -> Integer
forall (n :: Nat) (proxy :: Nat -> *).
KnownNat n =>
proxy n -> Integer
natVal (Proxy p -> Int) -> Proxy p -> Int
forall a b. (a -> b) -> a -> b
$ (Proxy p
forall a. HasCallStack => a
undefined :: Proxy p) :: Int

splitCols :: forall p m n. (KnownNat p, KnownNat m, KnownNat n, KnownNat (n-p), p<=n) => L m n -> (L m p, L m (n-p))
splitCols :: L m n -> (L m p, L m (n - p))
splitCols = (L p m -> L m p
forall m mt. Transposable m mt => m -> mt
tr (L p m -> L m p)
-> (L (n - p) m -> L m (n - p))
-> (L p m, L (n - p) m)
-> (L m p, L m (n - p))
forall (a :: * -> * -> *) b c b' c'.
Arrow a =>
a b c -> a b' c' -> a (b, b') (c, c')
*** L (n - p) m -> L m (n - p)
forall m mt. Transposable m mt => m -> mt
tr) ((L p m, L (n - p) m) -> (L m p, L m (n - p)))
-> (L m n -> (L p m, L (n - p) m)) -> L m n -> (L m p, L m (n - p))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. L n m -> (L p m, L (n - p) m)
forall (p :: Nat) (m :: Nat) (n :: Nat).
(KnownNat p, KnownNat m, KnownNat n, p <= m) =>
L m n -> (L p n, L (m - p) n)
splitRows (L n m -> (L p m, L (n - p) m))
-> (L m n -> L n m) -> L m n -> (L p m, L (n - p) m)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. L m n -> L n m
forall m mt. Transposable m mt => m -> mt
tr


toRows :: forall m n . (KnownNat m, KnownNat n) => L m n -> [R n]
toRows :: L m n -> [R n]
toRows (Matrix ℝ -> [Vector ℝ]
forall t. Element t => Matrix t -> [Vector t]
LA.toRows (Matrix ℝ -> [Vector ℝ])
-> (L m n -> Matrix ℝ) -> L m n -> [Vector ℝ]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> [Vector ℝ]
vs) = (Vector ℝ -> R n) -> [Vector ℝ] -> [R n]
forall a b. (a -> b) -> [a] -> [b]
map Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR [Vector ℝ]
vs

withRows
    :: forall n z . KnownNat n
    => [R n]
    -> (forall m . KnownNat m => L m n -> z)
    -> z
withRows :: [R n] -> (forall (m :: Nat). KnownNat m => L m n -> z) -> z
withRows ([Vector ℝ] -> Matrix ℝ
forall t. Element t => [Vector t] -> Matrix t
LA.fromRows ([Vector ℝ] -> Matrix ℝ)
-> ([R n] -> [Vector ℝ]) -> [R n] -> Matrix ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (R n -> Vector ℝ) -> [R n] -> [Vector ℝ]
forall a b. (a -> b) -> [a] -> [b]
map R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
m) forall (m :: Nat). KnownNat m => L m n -> z
f =
    case Integer -> Maybe SomeNat
someNatVal (Integer -> Maybe SomeNat) -> Integer -> Maybe SomeNat
forall a b. (a -> b) -> a -> b
$ Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Matrix ℝ -> Int
forall t. Matrix t -> Int
LA.rows Matrix ℝ
m of
       Maybe SomeNat
Nothing -> String -> z
forall a. HasCallStack => String -> a
error String
"static/dynamic mismatch"
       Just (SomeNat (Proxy n
_ :: Proxy m)) -> L n n -> z
forall (m :: Nat). KnownNat m => L m n -> z
f (Matrix ℝ -> L n n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
m :: L m n)

toColumns :: forall m n . (KnownNat m, KnownNat n) => L m n -> [R m]
toColumns :: L m n -> [R m]
toColumns (Matrix ℝ -> [Vector ℝ]
forall t. Element t => Matrix t -> [Vector t]
LA.toColumns (Matrix ℝ -> [Vector ℝ])
-> (L m n -> Matrix ℝ) -> L m n -> [Vector ℝ]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> [Vector ℝ]
vs) = (Vector ℝ -> R m) -> [Vector ℝ] -> [R m]
forall a b. (a -> b) -> [a] -> [b]
map Vector ℝ -> R m
forall (n :: Nat). Vector ℝ -> R n
mkR [Vector ℝ]
vs

withColumns
    :: forall m z . KnownNat m
    => [R m]
    -> (forall n . KnownNat n => L m n -> z)
    -> z
withColumns :: [R m] -> (forall (n :: Nat). KnownNat n => L m n -> z) -> z
withColumns ([Vector ℝ] -> Matrix ℝ
forall t. Element t => [Vector t] -> Matrix t
LA.fromColumns ([Vector ℝ] -> Matrix ℝ)
-> ([R m] -> [Vector ℝ]) -> [R m] -> Matrix ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (R m -> Vector ℝ) -> [R m] -> [Vector ℝ]
forall a b. (a -> b) -> [a] -> [b]
map R m -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
m) forall (n :: Nat). KnownNat n => L m n -> z
f =
    case Integer -> Maybe SomeNat
someNatVal (Integer -> Maybe SomeNat) -> Integer -> Maybe SomeNat
forall a b. (a -> b) -> a -> b
$ Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Matrix ℝ -> Int
forall t. Matrix t -> Int
LA.cols Matrix ℝ
m of
       Maybe SomeNat
Nothing -> String -> z
forall a. HasCallStack => String -> a
error String
"static/dynamic mismatch"
       Just (SomeNat (Proxy n
_ :: Proxy n)) -> L m n -> z
forall (n :: Nat). KnownNat n => L m n -> z
f (Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
m :: L m n)



--------------------------------------------------------------------------------

build
  :: forall m n . (KnownNat n, KnownNat m)
    => (ℝ -> ℝ -> ℝ)
    -> L m n
build :: (ℝ -> ℝ -> ℝ) -> L m n
build ℝ -> ℝ -> ℝ
f = L m n
r
  where
    r :: L m n
r = Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> L m n) -> Matrix ℝ -> L m n
forall a b. (a -> b) -> a -> b
$ (Int, Int) -> (ℝ -> ℝ -> ℝ) -> Matrix ℝ
forall d f (c :: * -> *) e. Build d f c e => d -> f -> c e
LA.build (L m n -> IndexOf Matrix
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size L m n
r) ℝ -> ℝ -> ℝ
f

--------------------------------------------------------------------------------

withVector
    :: forall z
     . Vector ℝ
    -> (forall n . (KnownNat n) => R n -> z)
    -> z
withVector :: Vector ℝ -> (forall (n :: Nat). KnownNat n => R n -> z) -> z
withVector Vector ℝ
v forall (n :: Nat). KnownNat n => R n -> z
f =
    case Integer -> Maybe SomeNat
someNatVal (Integer -> Maybe SomeNat) -> Integer -> Maybe SomeNat
forall a b. (a -> b) -> a -> b
$ Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Vector ℝ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℝ
v of
       Maybe SomeNat
Nothing -> String -> z
forall a. HasCallStack => String -> a
error String
"static/dynamic mismatch"
       Just (SomeNat (Proxy n
_ :: Proxy m)) -> R n -> z
forall (n :: Nat). KnownNat n => R n -> z
f (Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR Vector ℝ
v :: R m)

-- | Useful for constraining two dependently typed vectors to match each
-- other in length when they are unknown at compile-time.
exactLength
    :: forall n m . (KnownNat n, KnownNat m)
    => R m
    -> Maybe (R n)
exactLength :: R m -> Maybe (R n)
exactLength R m
v = do
    n :~: m
Refl <- Proxy n -> Proxy m -> Maybe (n :~: m)
forall (a :: Nat) (b :: Nat).
(KnownNat a, KnownNat b) =>
Proxy a -> Proxy b -> Maybe (a :~: b)
sameNat (Proxy n
forall k (t :: k). Proxy t
Proxy :: Proxy n) (Proxy m
forall k (t :: k). Proxy t
Proxy :: Proxy m)
    R n -> Maybe (R n)
forall (m :: * -> *) a. Monad m => a -> m a
return (R n -> Maybe (R n)) -> R n -> Maybe (R n)
forall a b. (a -> b) -> a -> b
$ Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR (R m -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap R m
v)

withMatrix
    :: forall z
     . Matrix ℝ
    -> (forall m n . (KnownNat m, KnownNat n) => L m n -> z)
    -> z
withMatrix :: Matrix ℝ
-> (forall (m :: Nat) (n :: Nat).
    (KnownNat m, KnownNat n) =>
    L m n -> z)
-> z
withMatrix Matrix ℝ
a forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> z
f =
    case Integer -> Maybe SomeNat
someNatVal (Integer -> Maybe SomeNat) -> Integer -> Maybe SomeNat
forall a b. (a -> b) -> a -> b
$ Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Matrix ℝ -> Int
forall t. Matrix t -> Int
rows Matrix ℝ
a of
       Maybe SomeNat
Nothing -> String -> z
forall a. HasCallStack => String -> a
error String
"static/dynamic mismatch"
       Just (SomeNat (Proxy n
_ :: Proxy m)) ->
           case Integer -> Maybe SomeNat
someNatVal (Integer -> Maybe SomeNat) -> Integer -> Maybe SomeNat
forall a b. (a -> b) -> a -> b
$ Int -> Integer
forall a b. (Integral a, Num b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Matrix ℝ -> Int
forall t. Matrix t -> Int
cols Matrix ℝ
a of
               Maybe SomeNat
Nothing -> String -> z
forall a. HasCallStack => String -> a
error String
"static/dynamic mismatch"
               Just (SomeNat (Proxy n
_ :: Proxy n)) ->
                  L n n -> z
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> z
f (Matrix ℝ -> L n n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL Matrix ℝ
a :: L m n)

-- | Useful for constraining two dependently typed matrices to match each
-- other in dimensions when they are unknown at compile-time.
exactDims
    :: forall n m j k . (KnownNat n, KnownNat m, KnownNat j, KnownNat k)
    => L m n
    -> Maybe (L j k)
exactDims :: L m n -> Maybe (L j k)
exactDims L m n
m = do
    m :~: j
Refl <- Proxy m -> Proxy j -> Maybe (m :~: j)
forall (a :: Nat) (b :: Nat).
(KnownNat a, KnownNat b) =>
Proxy a -> Proxy b -> Maybe (a :~: b)
sameNat (Proxy m
forall k (t :: k). Proxy t
Proxy :: Proxy m) (Proxy j
forall k (t :: k). Proxy t
Proxy :: Proxy j)
    n :~: k
Refl <- Proxy n -> Proxy k -> Maybe (n :~: k)
forall (a :: Nat) (b :: Nat).
(KnownNat a, KnownNat b) =>
Proxy a -> Proxy b -> Maybe (a :~: b)
sameNat (Proxy n
forall k (t :: k). Proxy t
Proxy :: Proxy n) (Proxy k
forall k (t :: k). Proxy t
Proxy :: Proxy k)
    L j k -> Maybe (L j k)
forall (m :: * -> *) a. Monad m => a -> m a
return (L j k -> Maybe (L j k)) -> L j k -> Maybe (L j k)
forall a b. (a -> b) -> a -> b
$ Matrix ℝ -> L j k
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap L m n
m)

randomVector
    :: forall n . KnownNat n
    => Seed
    -> RandDist
    -> R n
randomVector :: Int -> RandDist -> R n
randomVector Int
s RandDist
d = Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR (Int -> RandDist -> Int -> Vector ℝ
LA.randomVector Int
s RandDist
d
                          (Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy n -> Integer
forall (n :: Nat) (proxy :: Nat -> *).
KnownNat n =>
proxy n -> Integer
natVal (Proxy n
forall k (t :: k). Proxy t
Proxy :: Proxy n)))
                       )

rand
    :: forall m n . (KnownNat m, KnownNat n)
    => IO (L m n)
rand :: IO (L m n)
rand = Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> L m n) -> IO (Matrix ℝ) -> IO (L m n)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Int -> Int -> IO (Matrix ℝ)
LA.rand (Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy m -> Integer
forall (n :: Nat) (proxy :: Nat -> *).
KnownNat n =>
proxy n -> Integer
natVal (Proxy m
forall k (t :: k). Proxy t
Proxy :: Proxy m)))
                       (Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy n -> Integer
forall (n :: Nat) (proxy :: Nat -> *).
KnownNat n =>
proxy n -> Integer
natVal (Proxy n
forall k (t :: k). Proxy t
Proxy :: Proxy n)))

randn
    :: forall m n . (KnownNat m, KnownNat n)
    => IO (L m n)
randn :: IO (L m n)
randn = Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> L m n) -> IO (Matrix ℝ) -> IO (L m n)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Int -> Int -> IO (Matrix ℝ)
LA.randn (Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy m -> Integer
forall (n :: Nat) (proxy :: Nat -> *).
KnownNat n =>
proxy n -> Integer
natVal (Proxy m
forall k (t :: k). Proxy t
Proxy :: Proxy m)))
                         (Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy n -> Integer
forall (n :: Nat) (proxy :: Nat -> *).
KnownNat n =>
proxy n -> Integer
natVal (Proxy n
forall k (t :: k). Proxy t
Proxy :: Proxy n)))

gaussianSample
    :: forall m n . (KnownNat m, KnownNat n)
    => Seed
    -> R n
    -> Sym n
    -> L m n
gaussianSample :: Int -> R n -> Sym n -> L m n
gaussianSample Int
s (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
mu) (Sym (Sq n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
sigma)) =
    Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> L m n) -> Matrix ℝ -> L m n
forall a b. (a -> b) -> a -> b
$ Int -> Int -> Vector ℝ -> Herm ℝ -> Matrix ℝ
LA.gaussianSample Int
s (Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy m -> Integer
forall (n :: Nat) (proxy :: Nat -> *).
KnownNat n =>
proxy n -> Integer
natVal (Proxy m
forall k (t :: k). Proxy t
Proxy :: Proxy m)))
                            Vector ℝ
mu (Matrix ℝ -> Herm ℝ
forall t. Matrix t -> Herm t
LA.trustSym Matrix ℝ
sigma)

uniformSample
    :: forall m n . (KnownNat m, KnownNat n)
    => Seed
    -> R n    -- ^ minimums of each row
    -> R n    -- ^ maximums of each row
    -> L m n
uniformSample :: Int -> R n -> R n -> L m n
uniformSample Int
s (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
mins) (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
maxs) =
    Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> L m n) -> Matrix ℝ -> L m n
forall a b. (a -> b) -> a -> b
$ Int -> Int -> [(ℝ, ℝ)] -> Matrix ℝ
LA.uniformSample Int
s (Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy m -> Integer
forall (n :: Nat) (proxy :: Nat -> *).
KnownNat n =>
proxy n -> Integer
natVal (Proxy m
forall k (t :: k). Proxy t
Proxy :: Proxy m)))
                           ([ℝ] -> [ℝ] -> [(ℝ, ℝ)]
forall a b. [a] -> [b] -> [(a, b)]
zip (Vector ℝ -> [ℝ]
forall a. Storable a => Vector a -> [a]
LA.toList Vector ℝ
mins) (Vector ℝ -> [ℝ]
forall a. Storable a => Vector a -> [a]
LA.toList Vector ℝ
maxs))

meanCov
    :: forall m n . (KnownNat m, KnownNat n, 1 <= m)
    => L m n
    -> (R n, Sym n)
meanCov :: L m n -> (R n, Sym n)
meanCov (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Matrix ℝ
vs) = Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR (Vector ℝ -> R n)
-> (Herm ℝ -> Sym n) -> (Vector ℝ, Herm ℝ) -> (R n, Sym n)
forall (a :: * -> * -> *) b c b' c'.
Arrow a =>
a b c -> a b' c' -> a (b, b') (c, c')
*** (Sq n -> Sym n
forall (n :: Nat). Sq n -> Sym n
Sym (Sq n -> Sym n) -> (Herm ℝ -> Sq n) -> Herm ℝ -> Sym n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Matrix ℝ -> Sq n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> Sq n) -> (Herm ℝ -> Matrix ℝ) -> Herm ℝ -> Sq n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Herm ℝ -> Matrix ℝ
forall t. Herm t -> Matrix t
LA.unSym) ((Vector ℝ, Herm ℝ) -> (R n, Sym n))
-> (Vector ℝ, Herm ℝ) -> (R n, Sym n)
forall a b. (a -> b) -> a -> b
$ Matrix ℝ -> (Vector ℝ, Herm ℝ)
LA.meanCov Matrix ℝ
vs

--------------------------------------------------------------------------------

class Domain field vec mat | mat -> vec field, vec -> mat field, field -> mat vec
  where
    mul :: forall m k n. (KnownNat m, KnownNat k, KnownNat n) => mat m k -> mat k n -> mat m n
    app :: forall m n . (KnownNat m, KnownNat n) => mat m n -> vec n -> vec m
    dot :: forall n . (KnownNat n) => vec n -> vec n -> field
    cross :: vec 3 -> vec 3 -> vec 3
    diagR ::  forall m n k . (KnownNat m, KnownNat n, KnownNat k) => field -> vec k -> mat m n
    dvmap :: forall n. KnownNat n => (field -> field) -> vec n -> vec n
    dmmap :: forall n m. (KnownNat m, KnownNat n) => (field -> field) -> mat n m -> mat n m
    outer :: forall n m. (KnownNat m, KnownNat n) => vec n -> vec m -> mat n m
    zipWithVector :: forall n. KnownNat n => (field -> field -> field) -> vec n -> vec n -> vec n
    det :: forall n. KnownNat n => mat n n -> field
    invlndet :: forall n. KnownNat n => mat n n -> (mat n n, (field, field))
    expm :: forall n. KnownNat n => mat n n -> mat n n
    sqrtm :: forall n. KnownNat n => mat n n -> mat n n
    inv :: forall n. KnownNat n => mat n n -> mat n n


instance Domain ℝ R L
  where
    mul :: L m k -> L k n -> L m n
mul = L m k -> L k n -> L m n
forall (m :: Nat) (k :: Nat) (n :: Nat).
(KnownNat m, KnownNat k, KnownNat n) =>
L m k -> L k n -> L m n
mulR
    app :: L m n -> R n -> R m
app = L m n -> R n -> R m
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> R n -> R m
appR
    dot :: R n -> R n -> ℝ
dot = R n -> R n -> ℝ
forall (n :: Nat). KnownNat n => R n -> R n -> ℝ
dotR
    cross :: R 3 -> R 3 -> R 3
cross = R 3 -> R 3 -> R 3
crossR
    diagR :: ℝ -> R k -> L m n
diagR = ℝ -> R k -> L m n
forall (m :: Nat) (n :: Nat) (k :: Nat).
(KnownNat m, KnownNat n, KnownNat k) =>
ℝ -> R k -> L m n
diagRectR
    dvmap :: (ℝ -> ℝ) -> R n -> R n
dvmap = (ℝ -> ℝ) -> R n -> R n
forall (n :: Nat). KnownNat n => (ℝ -> ℝ) -> R n -> R n
mapR
    dmmap :: (ℝ -> ℝ) -> L n m -> L n m
dmmap = (ℝ -> ℝ) -> L n m -> L n m
forall (n :: Nat) (m :: Nat).
(KnownNat n, KnownNat m) =>
(ℝ -> ℝ) -> L n m -> L n m
mapL
    outer :: R n -> R m -> L n m
outer = R n -> R m -> L n m
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
R n -> R m -> L n m
outerR
    zipWithVector :: (ℝ -> ℝ -> ℝ) -> R n -> R n -> R n
zipWithVector = (ℝ -> ℝ -> ℝ) -> R n -> R n -> R n
forall (n :: Nat). KnownNat n => (ℝ -> ℝ -> ℝ) -> R n -> R n -> R n
zipWithR
    det :: L n n -> ℝ
det = L n n -> ℝ
forall (n :: Nat). KnownNat n => L n n -> ℝ
detL
    invlndet :: L n n -> (L n n, (ℝ, ℝ))
invlndet = L n n -> (L n n, (ℝ, ℝ))
forall (n :: Nat). KnownNat n => L n n -> (L n n, (ℝ, ℝ))
invlndetL
    expm :: L n n -> L n n
expm = L n n -> L n n
forall (n :: Nat). KnownNat n => L n n -> L n n
expmL
    sqrtm :: L n n -> L n n
sqrtm = L n n -> L n n
forall (n :: Nat). KnownNat n => L n n -> L n n
sqrtmL
    inv :: L n n -> L n n
inv = L n n -> L n n
forall (n :: Nat). KnownNat n => L n n -> L n n
invL

instance Domain ℂ C M
  where
    mul :: M m k -> M k n -> M m n
mul = M m k -> M k n -> M m n
forall (m :: Nat) (k :: Nat) (n :: Nat).
(KnownNat m, KnownNat k, KnownNat n) =>
M m k -> M k n -> M m n
mulC
    app :: M m n -> C n -> C m
app = M m n -> C n -> C m
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
M m n -> C n -> C m
appC
    dot :: C n -> C n -> ℂ
dot = C n -> C n -> ℂ
forall (n :: Nat). KnownNat n => C n -> C n -> ℂ
dotC
    cross :: C 3 -> C 3 -> C 3
cross = C 3 -> C 3 -> C 3
crossC
    diagR :: ℂ -> C k -> M m n
diagR = ℂ -> C k -> M m n
forall (m :: Nat) (n :: Nat) (k :: Nat).
(KnownNat m, KnownNat n, KnownNat k) =>
ℂ -> C k -> M m n
diagRectC
    dvmap :: (ℂ -> ℂ) -> C n -> C n
dvmap = (ℂ -> ℂ) -> C n -> C n
forall (n :: Nat). KnownNat n => (ℂ -> ℂ) -> C n -> C n
mapC
    dmmap :: (ℂ -> ℂ) -> M n m -> M n m
dmmap = (ℂ -> ℂ) -> M n m -> M n m
forall (n :: Nat) (m :: Nat).
(KnownNat n, KnownNat m) =>
(ℂ -> ℂ) -> M n m -> M n m
mapM'
    outer :: C n -> C m -> M n m
outer = C n -> C m -> M n m
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
C n -> C m -> M n m
outerC
    zipWithVector :: (ℂ -> ℂ -> ℂ) -> C n -> C n -> C n
zipWithVector = (ℂ -> ℂ -> ℂ) -> C n -> C n -> C n
forall (n :: Nat). KnownNat n => (ℂ -> ℂ -> ℂ) -> C n -> C n -> C n
zipWithC
    det :: M n n -> ℂ
det = M n n -> ℂ
forall (n :: Nat). KnownNat n => M n n -> ℂ
detM
    invlndet :: M n n -> (M n n, (ℂ, ℂ))
invlndet = M n n -> (M n n, (ℂ, ℂ))
forall (n :: Nat). KnownNat n => M n n -> (M n n, (ℂ, ℂ))
invlndetM
    expm :: M n n -> M n n
expm = M n n -> M n n
forall (n :: Nat). KnownNat n => M n n -> M n n
expmM
    sqrtm :: M n n -> M n n
sqrtm = M n n -> M n n
forall (n :: Nat). KnownNat n => M n n -> M n n
sqrtmM
    inv :: M n n -> M n n
inv = M n n -> M n n
forall (n :: Nat). KnownNat n => M n n -> M n n
invM

--------------------------------------------------------------------------------

mulR :: forall m k n. (KnownNat m, KnownNat k, KnownNat n) => L m k -> L k n -> L m n

mulR :: L m k -> L k n -> L m n
mulR (L m k -> Maybe (ℝ, (Int, Int))
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> Maybe (ℝ, (Int, Int))
isKonst -> Just (ℝ
a,(Int
_,Int
k))) (L k n -> Maybe (ℝ, (Int, Int))
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> Maybe (ℝ, (Int, Int))
isKonst -> Just (ℝ
b,(Int, Int)
_)) = ℝ -> L m n
forall t s (d :: * -> *). Sized t s d => t -> s
konst (ℝ
a ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
* ℝ
b ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
* Int -> ℝ
forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
k)

mulR (L m k -> Maybe (ℝ, Vector ℝ, (Int, Int))
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> Maybe (ℝ, Vector ℝ, (Int, Int))
isDiag -> Just (ℝ
0,Vector ℝ
a,(Int, Int)
_)) (L k n -> Maybe (ℝ, Vector ℝ, (Int, Int))
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> Maybe (ℝ, Vector ℝ, (Int, Int))
isDiag -> Just (ℝ
0,Vector ℝ
b,(Int, Int)
_)) = ℝ -> R k -> L m n
forall field (vec :: Nat -> *) (mat :: Nat -> Nat -> *) (m :: Nat)
       (n :: Nat) (k :: Nat).
(Domain field vec mat, KnownNat m, KnownNat n, KnownNat k) =>
field -> vec k -> mat m n
diagR ℝ
0 (Vector ℝ -> R k
forall (n :: Nat). Vector ℝ -> R n
mkR Vector ℝ
v :: R k)
  where
    v :: Vector ℝ
v = Vector ℝ
a' Vector ℝ -> Vector ℝ -> Vector ℝ
forall a. Num a => a -> a -> a
* Vector ℝ
b'
    n :: Int
n = Int -> Int -> Int
forall a. Ord a => a -> a -> a
min (Vector ℝ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℝ
a) (Vector ℝ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℝ
b)
    a' :: Vector ℝ
a' = Int -> Int -> Vector ℝ -> Vector ℝ
forall t. Storable t => Int -> Int -> Vector t -> Vector t
subVector Int
0 Int
n Vector ℝ
a
    b' :: Vector ℝ
b' = Int -> Int -> Vector ℝ -> Vector ℝ
forall t. Storable t => Int -> Int -> Vector t -> Vector t
subVector Int
0 Int
n Vector ℝ
b

-- mulR (isDiag -> Just (0,a,_)) (extract -> b) = mkL (asColumn a * takeRows (LA.size a) b)

-- mulR (extract -> a) (isDiag -> Just (0,b,_)) = mkL (takeColumns (LA.size b) a * asRow b)

mulR L m k
a L k n
b = Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (L m k -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract L m k
a Matrix ℝ -> Matrix ℝ -> Matrix ℝ
forall t. Numeric t => Matrix t -> Matrix t -> Matrix t
LA.<> L k n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract L k n
b)


appR :: (KnownNat m, KnownNat n) => L m n -> R n -> R m
appR :: L m n -> R n -> R m
appR (L m n -> Maybe (ℝ, Vector ℝ, (Int, Int))
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> Maybe (ℝ, Vector ℝ, (Int, Int))
isDiag -> Just (ℝ
0, Vector ℝ
w, (Int, Int)
_)) R n
v = Vector ℝ -> R m
forall (n :: Nat). Vector ℝ -> R n
mkR (Vector ℝ
w Vector ℝ -> Vector ℝ -> Vector ℝ
forall a. Num a => a -> a -> a
* Int -> Int -> Vector ℝ -> Vector ℝ
forall t. Storable t => Int -> Int -> Vector t -> Vector t
subVector Int
0 (Vector ℝ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℝ
w) (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract R n
v))
appR L m n
m R n
v = Vector ℝ -> R m
forall (n :: Nat). Vector ℝ -> R n
mkR (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract L m n
m Matrix ℝ -> Vector ℝ -> Vector ℝ
forall t. Numeric t => Matrix t -> Vector t -> Vector t
LA.#> R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract R n
v)


dotR :: KnownNat n => R n -> R n -> ℝ
dotR :: R n -> R n -> ℝ
dotR (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
u) (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
v) = Vector ℝ -> Vector ℝ -> ℝ
forall t. Numeric t => Vector t -> Vector t -> t
LA.dot Vector ℝ
u Vector ℝ
v


crossR :: R 3 -> R 3 -> R 3
crossR :: R 3 -> R 3 -> R 3
crossR (R 3 -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
x) (R 3 -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
y) = ℝ -> ℝ -> ℝ -> R 3
vec3 ℝ
z1 ℝ
z2 ℝ
z3
  where
    z1 :: ℝ
z1 = Vector ℝ
xVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
1ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
*Vector ℝ
yVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
2ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
-Vector ℝ
xVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
2ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
*Vector ℝ
yVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
1
    z2 :: ℝ
z2 = Vector ℝ
xVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
2ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
*Vector ℝ
yVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
0ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
-Vector ℝ
xVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
0ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
*Vector ℝ
yVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
2
    z3 :: ℝ
z3 = Vector ℝ
xVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
0ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
*Vector ℝ
yVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
1ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
-Vector ℝ
xVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
1ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
*Vector ℝ
yVector ℝ -> Int -> ℝ
forall c t. Indexable c t => c -> Int -> t
!Int
0

outerR :: (KnownNat m, KnownNat n) => R n -> R m -> L n m
outerR :: R n -> R m -> L n m
outerR (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
x) (R m -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
y) = Matrix ℝ -> L n m
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Vector ℝ -> Vector ℝ -> Matrix ℝ
forall t. Product t => Vector t -> Vector t -> Matrix t
LA.outer Vector ℝ
x Vector ℝ
y)

mapR :: KnownNat n => (ℝ -> ℝ) -> R n -> R n
mapR :: (ℝ -> ℝ) -> R n -> R n
mapR ℝ -> ℝ
f (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap -> Vector ℝ
v) = Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR ((ℝ -> ℝ) -> Vector ℝ -> Vector ℝ
forall b (c :: * -> *) e.
(Element b, Container c e) =>
(e -> b) -> c e -> c b
LA.cmap ℝ -> ℝ
f Vector ℝ
v)

zipWithR :: KnownNat n => (ℝ -> ℝ -> ℝ) -> R n -> R n -> R n
zipWithR :: (ℝ -> ℝ -> ℝ) -> R n -> R n -> R n
zipWithR ℝ -> ℝ -> ℝ
f (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
x) (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℝ
y) = Vector ℝ -> R n
forall (n :: Nat). Vector ℝ -> R n
mkR ((ℝ -> ℝ -> ℝ) -> Vector ℝ -> Vector ℝ -> Vector ℝ
forall a b c.
(Storable a, Storable b, Storable c) =>
(a -> b -> c) -> Vector a -> Vector b -> Vector c
LA.zipVectorWith ℝ -> ℝ -> ℝ
f Vector ℝ
x Vector ℝ
y)

mapL :: (KnownNat n, KnownNat m) => (ℝ -> ℝ) -> L n m -> L n m
mapL :: (ℝ -> ℝ) -> L n m -> L n m
mapL ℝ -> ℝ
f = (Matrix ℝ -> Matrix ℝ) -> L n m -> L n m
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
(Matrix ℝ -> Matrix ℝ) -> L m n -> L m n
overMatL' ((ℝ -> ℝ) -> Matrix ℝ -> Matrix ℝ
forall b (c :: * -> *) e.
(Element b, Container c e) =>
(e -> b) -> c e -> c b
LA.cmap ℝ -> ℝ
f)

detL :: KnownNat n => Sq n -> ℝ
detL :: Sq n -> ℝ
detL = Matrix ℝ -> ℝ
forall t. Field t => Matrix t -> t
LA.det (Matrix ℝ -> ℝ) -> (Sq n -> Matrix ℝ) -> Sq n -> ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Sq n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap

invlndetL :: KnownNat n => Sq n -> (L n n, (ℝ, ℝ))
invlndetL :: Sq n -> (Sq n, (ℝ, ℝ))
invlndetL = (Matrix ℝ -> Sq n) -> (Matrix ℝ, (ℝ, ℝ)) -> (Sq n, (ℝ, ℝ))
forall (p :: * -> * -> *) a b c.
Bifunctor p =>
(a -> b) -> p a c -> p b c
BF.first Matrix ℝ -> Sq n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL ((Matrix ℝ, (ℝ, ℝ)) -> (Sq n, (ℝ, ℝ)))
-> (Sq n -> (Matrix ℝ, (ℝ, ℝ))) -> Sq n -> (Sq n, (ℝ, ℝ))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Matrix ℝ -> (Matrix ℝ, (ℝ, ℝ))
forall t. Field t => Matrix t -> (Matrix t, (t, t))
LA.invlndet (Matrix ℝ -> (Matrix ℝ, (ℝ, ℝ)))
-> (Sq n -> Matrix ℝ) -> Sq n -> (Matrix ℝ, (ℝ, ℝ))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Sq n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap

expmL :: KnownNat n => Sq n -> Sq n
expmL :: Sq n -> Sq n
expmL = (Matrix ℝ -> Matrix ℝ) -> Sq n -> Sq n
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
(Matrix ℝ -> Matrix ℝ) -> L m n -> L m n
overMatL' Matrix ℝ -> Matrix ℝ
forall t. Field t => Matrix t -> Matrix t
LA.expm

sqrtmL :: KnownNat n => Sq n -> Sq n
sqrtmL :: Sq n -> Sq n
sqrtmL = (Matrix ℝ -> Matrix ℝ) -> Sq n -> Sq n
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
(Matrix ℝ -> Matrix ℝ) -> L m n -> L m n
overMatL' Matrix ℝ -> Matrix ℝ
forall t. Field t => Matrix t -> Matrix t
LA.sqrtm

invL :: KnownNat n => Sq n -> Sq n
invL :: Sq n -> Sq n
invL = (Matrix ℝ -> Matrix ℝ) -> Sq n -> Sq n
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
(Matrix ℝ -> Matrix ℝ) -> L m n -> L m n
overMatL' Matrix ℝ -> Matrix ℝ
forall t. Field t => Matrix t -> Matrix t
LA.inv

--------------------------------------------------------------------------------

mulC :: forall m k n. (KnownNat m, KnownNat k, KnownNat n) => M m k -> M k n -> M m n

mulC :: M m k -> M k n -> M m n
mulC (M m k -> Maybe (ℂ, (Int, Int))
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
M m n -> Maybe (ℂ, (Int, Int))
isKonstC -> Just (ℂ
a,(Int
_,Int
k))) (M k n -> Maybe (ℂ, (Int, Int))
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
M m n -> Maybe (ℂ, (Int, Int))
isKonstC -> Just (ℂ
b,(Int, Int)
_)) = ℂ -> M m n
forall t s (d :: * -> *). Sized t s d => t -> s
konst (ℂ
a ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
* ℂ
b ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
* Int -> ℂ
forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
k)

mulC (M m k -> Maybe (ℂ, Vector ℂ, (Int, Int))
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
M m n -> Maybe (ℂ, Vector ℂ, (Int, Int))
isDiagC -> Just (ℂ
0,Vector ℂ
a,(Int, Int)
_)) (M k n -> Maybe (ℂ, Vector ℂ, (Int, Int))
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
M m n -> Maybe (ℂ, Vector ℂ, (Int, Int))
isDiagC -> Just (ℂ
0,Vector ℂ
b,(Int, Int)
_)) = ℂ -> C k -> M m n
forall field (vec :: Nat -> *) (mat :: Nat -> Nat -> *) (m :: Nat)
       (n :: Nat) (k :: Nat).
(Domain field vec mat, KnownNat m, KnownNat n, KnownNat k) =>
field -> vec k -> mat m n
diagR ℂ
0 (Vector ℂ -> C k
forall (n :: Nat). Vector ℂ -> C n
mkC Vector ℂ
v :: C k)
  where
    v :: Vector ℂ
v = Vector ℂ
a' Vector ℂ -> Vector ℂ -> Vector ℂ
forall a. Num a => a -> a -> a
* Vector ℂ
b'
    n :: Int
n = Int -> Int -> Int
forall a. Ord a => a -> a -> a
min (Vector ℂ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℂ
a) (Vector ℂ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℂ
b)
    a' :: Vector ℂ
a' = Int -> Int -> Vector ℂ -> Vector ℂ
forall t. Storable t => Int -> Int -> Vector t -> Vector t
subVector Int
0 Int
n Vector ℂ
a
    b' :: Vector ℂ
b' = Int -> Int -> Vector ℂ -> Vector ℂ
forall t. Storable t => Int -> Int -> Vector t -> Vector t
subVector Int
0 Int
n Vector ℂ
b

-- mulC (isDiagC -> Just (0,a,_)) (extract -> b) = mkM (asColumn a * takeRows (LA.size a) b)

-- mulC (extract -> a) (isDiagC -> Just (0,b,_)) = mkM (takeColumns (LA.size b) a * asRow b)

mulC M m k
a M k n
b = Matrix ℂ -> M m n
forall (m :: Nat) (n :: Nat). Matrix ℂ -> M m n
mkM (M m k -> Matrix ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract M m k
a Matrix ℂ -> Matrix ℂ -> Matrix ℂ
forall t. Numeric t => Matrix t -> Matrix t -> Matrix t
LA.<> M k n -> Matrix ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract M k n
b)


appC :: (KnownNat m, KnownNat n) => M m n -> C n -> C m
appC :: M m n -> C n -> C m
appC (M m n -> Maybe (ℂ, Vector ℂ, (Int, Int))
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
M m n -> Maybe (ℂ, Vector ℂ, (Int, Int))
isDiagC -> Just (ℂ
0, Vector ℂ
w, (Int, Int)
_)) C n
v = Vector ℂ -> C m
forall (n :: Nat). Vector ℂ -> C n
mkC (Vector ℂ
w Vector ℂ -> Vector ℂ -> Vector ℂ
forall a. Num a => a -> a -> a
* Int -> Int -> Vector ℂ -> Vector ℂ
forall t. Storable t => Int -> Int -> Vector t -> Vector t
subVector Int
0 (Vector ℂ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℂ
w) (C n -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract C n
v))
appC M m n
m C n
v = Vector ℂ -> C m
forall (n :: Nat). Vector ℂ -> C n
mkC (M m n -> Matrix ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract M m n
m Matrix ℂ -> Vector ℂ -> Vector ℂ
forall t. Numeric t => Matrix t -> Vector t -> Vector t
LA.#> C n -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract C n
v)


dotC :: KnownNat n => C n -> C n -> ℂ
dotC :: C n -> C n -> ℂ
dotC (C n -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℂ
u) (C n -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℂ
v) = Vector ℂ -> Vector ℂ -> ℂ
forall t. Numeric t => Vector t -> Vector t -> t
LA.dot Vector ℂ
u Vector ℂ
v


crossC :: C 3 -> C 3 -> C 3
crossC :: C 3 -> C 3 -> C 3
crossC (C 3 -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℂ
x) (C 3 -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℂ
y) = Vector ℂ -> C 3
forall (n :: Nat). Vector ℂ -> C n
mkC ([ℂ] -> Vector ℂ
forall a. Storable a => [a] -> Vector a
LA.fromList [ℂ
z1, ℂ
z2, ℂ
z3])
  where
    z1 :: ℂ
z1 = Vector ℂ
xVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
1ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
*Vector ℂ
yVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
2ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
-Vector ℂ
xVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
2ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
*Vector ℂ
yVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
1
    z2 :: ℂ
z2 = Vector ℂ
xVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
2ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
*Vector ℂ
yVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
0ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
-Vector ℂ
xVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
0ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
*Vector ℂ
yVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
2
    z3 :: ℂ
z3 = Vector ℂ
xVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
0ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
*Vector ℂ
yVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
1ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
-Vector ℂ
xVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
1ℂ -> ℂ -> ℂ
forall a. Num a => a -> a -> a
*Vector ℂ
yVector ℂ -> Int -> ℂ
forall c t. Indexable c t => c -> Int -> t
!Int
0

outerC :: (KnownNat m, KnownNat n) => C n -> C m -> M n m
outerC :: C n -> C m -> M n m
outerC (C n -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℂ
x) (C m -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℂ
y) = Matrix ℂ -> M n m
forall (m :: Nat) (n :: Nat). Matrix ℂ -> M m n
mkM (Vector ℂ -> Vector ℂ -> Matrix ℂ
forall t. Product t => Vector t -> Vector t -> Matrix t
LA.outer Vector ℂ
x Vector ℂ
y)

mapC :: KnownNat n => (ℂ -> ℂ) -> C n -> C n
mapC :: (ℂ -> ℂ) -> C n -> C n
mapC ℂ -> ℂ
f (C n -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap -> Vector ℂ
v) = Vector ℂ -> C n
forall (n :: Nat). Vector ℂ -> C n
mkC ((ℂ -> ℂ) -> Vector ℂ -> Vector ℂ
forall b (c :: * -> *) e.
(Element b, Container c e) =>
(e -> b) -> c e -> c b
LA.cmap ℂ -> ℂ
f Vector ℂ
v)

zipWithC :: KnownNat n => (ℂ -> ℂ -> ℂ) -> C n -> C n -> C n
zipWithC :: (ℂ -> ℂ -> ℂ) -> C n -> C n -> C n
zipWithC ℂ -> ℂ -> ℂ
f (C n -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℂ
x) (C n -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract -> Vector ℂ
y) = Vector ℂ -> C n
forall (n :: Nat). Vector ℂ -> C n
mkC ((ℂ -> ℂ -> ℂ) -> Vector ℂ -> Vector ℂ -> Vector ℂ
forall a b c.
(Storable a, Storable b, Storable c) =>
(a -> b -> c) -> Vector a -> Vector b -> Vector c
LA.zipVectorWith ℂ -> ℂ -> ℂ
f Vector ℂ
x Vector ℂ
y)

mapM' :: (KnownNat n, KnownNat m) => (ℂ -> ℂ) -> M n m -> M n m
mapM' :: (ℂ -> ℂ) -> M n m -> M n m
mapM' ℂ -> ℂ
f = (Matrix ℂ -> Matrix ℂ) -> M n m -> M n m
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
(Matrix ℂ -> Matrix ℂ) -> M m n -> M m n
overMatM' ((ℂ -> ℂ) -> Matrix ℂ -> Matrix ℂ
forall b (c :: * -> *) e.
(Element b, Container c e) =>
(e -> b) -> c e -> c b
LA.cmap ℂ -> ℂ
f)

detM :: KnownNat n => M n n -> ℂ
detM :: M n n -> ℂ
detM = Matrix ℂ -> ℂ
forall t. Field t => Matrix t -> t
LA.det (Matrix ℂ -> ℂ) -> (M n n -> Matrix ℂ) -> M n n -> ℂ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. M n n -> Matrix ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap

invlndetM :: KnownNat n => M n n -> (M n n, (ℂ, ℂ))
invlndetM :: M n n -> (M n n, (ℂ, ℂ))
invlndetM = (Matrix ℂ -> M n n) -> (Matrix ℂ, (ℂ, ℂ)) -> (M n n, (ℂ, ℂ))
forall (p :: * -> * -> *) a b c.
Bifunctor p =>
(a -> b) -> p a c -> p b c
BF.first Matrix ℂ -> M n n
forall (m :: Nat) (n :: Nat). Matrix ℂ -> M m n
mkM ((Matrix ℂ, (ℂ, ℂ)) -> (M n n, (ℂ, ℂ)))
-> (M n n -> (Matrix ℂ, (ℂ, ℂ))) -> M n n -> (M n n, (ℂ, ℂ))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Matrix ℂ -> (Matrix ℂ, (ℂ, ℂ))
forall t. Field t => Matrix t -> (Matrix t, (t, t))
LA.invlndet (Matrix ℂ -> (Matrix ℂ, (ℂ, ℂ)))
-> (M n n -> Matrix ℂ) -> M n n -> (Matrix ℂ, (ℂ, ℂ))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. M n n -> Matrix ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap

expmM :: KnownNat n => M n n -> M n n
expmM :: M n n -> M n n
expmM = (Matrix ℂ -> Matrix ℂ) -> M n n -> M n n
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
(Matrix ℂ -> Matrix ℂ) -> M m n -> M m n
overMatM' Matrix ℂ -> Matrix ℂ
forall t. Field t => Matrix t -> Matrix t
LA.expm

sqrtmM :: KnownNat n => M n n -> M n n
sqrtmM :: M n n -> M n n
sqrtmM = (Matrix ℂ -> Matrix ℂ) -> M n n -> M n n
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
(Matrix ℂ -> Matrix ℂ) -> M m n -> M m n
overMatM' Matrix ℂ -> Matrix ℂ
forall t. Field t => Matrix t -> Matrix t
LA.sqrtm

invM :: KnownNat n => M n n -> M n n
invM :: M n n -> M n n
invM = (Matrix ℂ -> Matrix ℂ) -> M n n -> M n n
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
(Matrix ℂ -> Matrix ℂ) -> M m n -> M m n
overMatM' Matrix ℂ -> Matrix ℂ
forall t. Field t => Matrix t -> Matrix t
LA.inv

--------------------------------------------------------------------------------

diagRectR :: forall m n k . (KnownNat m, KnownNat n, KnownNat k) => ℝ -> R k -> L m n
diagRectR :: ℝ -> R k -> L m n
diagRectR ℝ
x R k
v
    | Int
m' Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1 = Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (ℝ -> Vector ℝ -> Int -> Int -> Matrix ℝ
forall t. Storable t => t -> Vector t -> Int -> Int -> Matrix t
LA.diagRect ℝ
x Vector ℝ
ev Int
m' Int
n')
    | Int
m'Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
n' Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
0 = L m n
r
    | Bool
otherwise = [ℝ] -> L m n
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
[ℝ] -> L m n
matrix []
  where
    r :: L m n
r = Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Vector ℝ -> Matrix ℝ
forall a. Storable a => Vector a -> Matrix a
asRow ([Vector ℝ] -> Vector ℝ
forall t. Storable t => [Vector t] -> Vector t
vjoin [ℝ -> Vector ℝ
forall (c :: * -> *) e. Container c e => e -> c e
scalar ℝ
x, Vector ℝ
ev, Vector ℝ
zeros]))
    ev :: Vector ℝ
ev = R k -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract R k
v
    zeros :: Vector ℝ
zeros = ℝ -> Int -> Vector ℝ
forall e d (c :: * -> *). Konst e d c => e -> d -> c e
LA.konst ℝ
x (Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
0 ((Int -> Int -> Int
forall a. Ord a => a -> a -> a
min Int
m' Int
n') Int -> Int -> Int
forall a. Num a => a -> a -> a
- Vector ℝ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℝ
ev))
    (Int
m',Int
n') = L m n -> IndexOf Matrix
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size L m n
r


diagRectC :: forall m n k . (KnownNat m, KnownNat n, KnownNat k) => ℂ -> C k -> M m n
diagRectC :: ℂ -> C k -> M m n
diagRectC ℂ
x C k
v
    | Int
m' Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1 = Matrix ℂ -> M m n
forall (m :: Nat) (n :: Nat). Matrix ℂ -> M m n
mkM (ℂ -> Vector ℂ -> Int -> Int -> Matrix ℂ
forall t. Storable t => t -> Vector t -> Int -> Int -> Matrix t
LA.diagRect ℂ
x Vector ℂ
ev Int
m' Int
n')
    | Int
m'Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
n' Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
0 = M m n
r
    | Bool
otherwise = [ℂ] -> M m n
forall t s (d :: * -> *). Sized t s d => [t] -> s
fromList []
  where
    r :: M m n
r = Matrix ℂ -> M m n
forall (m :: Nat) (n :: Nat). Matrix ℂ -> M m n
mkM (Vector ℂ -> Matrix ℂ
forall a. Storable a => Vector a -> Matrix a
asRow ([Vector ℂ] -> Vector ℂ
forall t. Storable t => [Vector t] -> Vector t
vjoin [ℂ -> Vector ℂ
forall (c :: * -> *) e. Container c e => e -> c e
scalar ℂ
x, Vector ℂ
ev, Vector ℂ
zeros]))
    ev :: Vector ℂ
ev = C k -> Vector ℂ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract C k
v
    zeros :: Vector ℂ
zeros = ℂ -> Int -> Vector ℂ
forall e d (c :: * -> *). Konst e d c => e -> d -> c e
LA.konst ℂ
x (Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
0 ((Int -> Int -> Int
forall a. Ord a => a -> a -> a
min Int
m' Int
n') Int -> Int -> Int
forall a. Num a => a -> a -> a
- Vector ℂ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size Vector ℂ
ev))
    (Int
m',Int
n') = M m n -> IndexOf Matrix
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size M m n
r

--------------------------------------------------------------------------------

mean :: (KnownNat n, 1<=n) => R n -> ℝ
mean :: R n -> ℝ
mean R n
v = R n
v R n -> R n -> ℝ
forall (n :: Nat). KnownNat n => R n -> R n -> ℝ
<·> (R n
1R n -> R n -> R n
forall a. Fractional a => a -> a -> a
/R n
forall (n :: Nat). KnownNat n => R n
dim)

test :: (Bool, IO ())
test :: (Bool, IO ())
test = (Bool
ok,IO ()
info)
  where
    ok :: Bool
ok =   Sq 5 -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract (Sq 5
forall (n :: Nat). KnownNat n => Sq n
eye :: Sq 5) Matrix ℝ -> Matrix ℝ -> Bool
forall a. Eq a => a -> a -> Bool
== Int -> Matrix ℝ
forall a. (Num a, Element a) => Int -> Matrix a
ident Int
5
           Bool -> Bool -> Bool
&& (Sq 3 -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap (Sq 3 -> Matrix ℝ) -> (Sym 3 -> Sq 3) -> Sym 3 -> Matrix ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
.Sym 3 -> Sq 3
forall (n :: Nat). Sym n -> Sq n
unSym) (Sq 3 -> Sym 3
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> Sym n
mTm Sq 3
forall (n :: Nat). KnownNat n => Sq n
sm :: Sym 3) Matrix ℝ -> Matrix ℝ -> Bool
forall a. Eq a => a -> a -> Bool
== Matrix ℝ -> Matrix ℝ
forall m mt. Transposable m mt => m -> mt
tr ((Int
3Int -> Int -> [ℝ] -> Matrix ℝ
forall a. Storable a => Int -> Int -> [a] -> Matrix a
><Int
3)[ℝ
1..]) Matrix ℝ -> Matrix ℝ -> Matrix ℝ
forall t. Numeric t => Matrix t -> Matrix t -> Matrix t
LA.<> (Int
3Int -> Int -> [ℝ] -> Matrix ℝ
forall a. Storable a => Int -> Int -> [a] -> Matrix a
><Int
3)[ℝ
1..]
           Bool -> Bool -> Bool
&& L 3 5 -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
unwrap (L 3 5
GL
tm :: L 3 5) Matrix ℝ -> Matrix ℝ -> Bool
forall a. Eq a => a -> a -> Bool
== Int -> [ℝ] -> Matrix ℝ
LA.matrix Int
5 [ℝ
1..ℝ
15]
           Bool -> Bool -> Bool
&& ℝ
thingS ℝ -> ℝ -> Bool
forall a. Eq a => a -> a -> Bool
== ℝ
thingD
           Bool -> Bool -> Bool
&& ℝ
precS ℝ -> ℝ -> Bool
forall a. Eq a => a -> a -> Bool
== ℝ
precD
           Bool -> Bool -> Bool
&& Vector ℝ -> (forall (n :: Nat). KnownNat n => R n -> ℝ) -> ℝ
forall z.
Vector ℝ -> (forall (n :: Nat). KnownNat n => R n -> z) -> z
withVector ([ℝ] -> Vector ℝ
LA.vector [ℝ
1..ℝ
15]) forall (n :: Nat). KnownNat n => R n -> ℝ
sumV ℝ -> ℝ -> Bool
forall a. Eq a => a -> a -> Bool
== Vector ℝ -> ℝ
forall (c :: * -> *) e. Container c e => c e -> e
sumElements ([ℝ] -> Vector ℝ
forall a. Storable a => [a] -> Vector a
LA.fromList [ℝ
1..ℝ
15])

    info :: IO ()
info = do
        R 2 -> IO ()
forall a. Show a => a -> IO ()
print (R 2 -> IO ()) -> R 2 -> IO ()
forall a b. (a -> b) -> a -> b
$ R 2
u
        R 3 -> IO ()
forall a. Show a => a -> IO ()
print (R 3 -> IO ()) -> R 3 -> IO ()
forall a b. (a -> b) -> a -> b
$ R 3
v
        Sq 3 -> IO ()
forall a. Show a => a -> IO ()
print (Sq 3
forall (n :: Nat). KnownNat n => Sq n
eye :: Sq 3)
        ℝ -> IO ()
forall a. Show a => a -> IO ()
print (ℝ -> IO ()) -> ℝ -> IO ()
forall a b. (a -> b) -> a -> b
$ ((R 2
u R 2 -> ℝ -> R (2 + 1)
forall (n :: Nat). KnownNat n => R n -> ℝ -> R (n + 1)
& ℝ
5) R 3 -> R 3 -> R 3
forall a. Num a => a -> a -> a
+ R 3
1) R 3 -> R 3 -> ℝ
forall (n :: Nat). KnownNat n => R n -> R n -> ℝ
<·> R 3
v
        L 2 5 -> IO ()
forall a. Show a => a -> IO ()
print (L 2 5
GL
tm :: L 2 5)
        L 2 3 -> IO ()
forall a. Show a => a -> IO ()
print (L 2 3
GL
tm L 2 3 -> Sq 3 -> L 2 3
forall (m :: Nat) (k :: Nat) (n :: Nat).
(KnownNat m, KnownNat k, KnownNat n) =>
L m k -> L k n -> L m n
<> Sq 3
forall (n :: Nat). KnownNat n => Sq n
sm :: L 2 3)
        ℝ -> IO ()
forall a. Show a => a -> IO ()
print ℝ
thingS
        ℝ -> IO ()
forall a. Show a => a -> IO ()
print ℝ
thingD
        ℝ -> IO ()
forall a. Show a => a -> IO ()
print ℝ
precS
        ℝ -> IO ()
forall a. Show a => a -> IO ()
print ℝ
precD
        ℝ -> IO ()
forall a. Show a => a -> IO ()
print (ℝ -> IO ()) -> ℝ -> IO ()
forall a b. (a -> b) -> a -> b
$ Vector ℝ -> (forall (n :: Nat). KnownNat n => R n -> ℝ) -> ℝ
forall z.
Vector ℝ -> (forall (n :: Nat). KnownNat n => R n -> z) -> z
withVector ([ℝ] -> Vector ℝ
LA.vector [ℝ
1..ℝ
15]) forall (n :: Nat). KnownNat n => R n -> ℝ
sumV
        IO ()
splittest

    sumV :: R n -> ℝ
sumV R n
w = R n
w R n -> R n -> ℝ
forall (n :: Nat). KnownNat n => R n -> R n -> ℝ
<·> ℝ -> R n
forall t s (d :: * -> *). Sized t s d => t -> s
konst ℝ
1

    u :: R 2
u = ℝ -> ℝ -> R 2
vec2 ℝ
3 ℝ
5

    𝕧 :: ℝ -> R 1
𝕧 ℝ
x = [ℝ] -> R 1
forall (n :: Nat). KnownNat n => [ℝ] -> R n
vector [ℝ
x] :: R 1

    v :: R (2 + 1)
v = ℝ -> R 1
𝕧 ℝ
2 R 1 -> ℝ -> R (1 + 1)
forall (n :: Nat). KnownNat n => R n -> ℝ -> R (n + 1)
& ℝ
4 R 2 -> ℝ -> R (2 + 1)
forall (n :: Nat). KnownNat n => R n -> ℝ -> R (n + 1)
& ℝ
7

    tm :: GL
    tm :: L m n
tm = ℝ -> [ℝ] -> L m n
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
ℝ -> [ℝ] -> L m n
lmat ℝ
0 [ℝ
1..]

    lmat :: forall m n . (KnownNat m, KnownNat n) => ℝ -> [ℝ] -> L m n
    lmat :: ℝ -> [ℝ] -> L m n
lmat ℝ
z [ℝ]
xs = L m n
r
      where
        r :: L m n
r = Matrix ℝ -> L m n
forall (m :: Nat) (n :: Nat). Matrix ℝ -> L m n
mkL (Matrix ℝ -> L m n) -> ([ℝ] -> Matrix ℝ) -> [ℝ] -> L m n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Int -> Vector ℝ -> Matrix ℝ
forall t. Storable t => Int -> Vector t -> Matrix t
reshape Int
n' (Vector ℝ -> Matrix ℝ) -> ([ℝ] -> Vector ℝ) -> [ℝ] -> Matrix ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [ℝ] -> Vector ℝ
forall a. Storable a => [a] -> Vector a
LA.fromList ([ℝ] -> Vector ℝ) -> ([ℝ] -> [ℝ]) -> [ℝ] -> Vector ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Int -> [ℝ] -> [ℝ]
forall a. Int -> [a] -> [a]
take (Int
m'Int -> Int -> Int
forall a. Num a => a -> a -> a
*Int
n') ([ℝ] -> L m n) -> [ℝ] -> L m n
forall a b. (a -> b) -> a -> b
$ [ℝ]
xs [ℝ] -> [ℝ] -> [ℝ]
forall a. [a] -> [a] -> [a]
++ ℝ -> [ℝ]
forall a. a -> [a]
repeat ℝ
z
        (Int
m',Int
n') = L m n -> IndexOf Matrix
forall t s (d :: * -> *). Sized t s d => s -> IndexOf d
size L m n
r

    sm :: GSq
    sm :: Sq n
sm = ℝ -> [ℝ] -> Sq n
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
ℝ -> [ℝ] -> L m n
lmat ℝ
0 [ℝ
1..]

    thingS :: ℝ
thingS = (R 2
u R 2 -> ℝ -> R (2 + 1)
forall (n :: Nat). KnownNat n => R n -> ℝ -> R (n + 1)
& ℝ
1) R 3 -> R 3 -> ℝ
forall (n :: Nat). KnownNat n => R n -> R n -> ℝ
<·> L 10 3 -> L 3 10
forall m mt. Transposable m mt => m -> mt
tr L 10 3
q L 3 10 -> R 10 -> R 3
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> R n -> R m
#> L 10 3
q L 10 3 -> R 3 -> R 10
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> R n -> R m
#> R 3
v
      where
        q :: L 10 3
q = L 10 3
GL
tm :: L 10 3

    thingD :: ℝ
thingD = [Vector ℝ] -> Vector ℝ
forall t. Storable t => [Vector t] -> Vector t
vjoin [R 2 -> Vector ℝ
forall (n :: Nat). R n -> Vector ℝ
ud1 R 2
u, Vector ℝ
1] Vector ℝ -> Vector ℝ -> ℝ
forall t. Numeric t => Vector t -> Vector t -> t
LA.<.> Matrix ℝ -> Matrix ℝ
forall m mt. Transposable m mt => m -> mt
tr Matrix ℝ
m Matrix ℝ -> Vector ℝ -> Vector ℝ
forall t. Numeric t => Matrix t -> Vector t -> Vector t
LA.#> Matrix ℝ
m Matrix ℝ -> Vector ℝ -> Vector ℝ
forall t. Numeric t => Matrix t -> Vector t -> Vector t
LA.#> R 3 -> Vector ℝ
forall (n :: Nat). R n -> Vector ℝ
ud1 R 3
v
      where
        m :: Matrix ℝ
m = Int -> [ℝ] -> Matrix ℝ
LA.matrix Int
3 [ℝ
1..ℝ
30]

    precS :: ℝ
precS = (ℝ
1::Double) ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
+ (ℝ
2::Double) ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
* ((R 3
1 :: R 3) R 3 -> R 3 -> R 3
forall a. Num a => a -> a -> a
* (R 2
u R 2 -> ℝ -> R (2 + 1)
forall (n :: Nat). KnownNat n => R n -> ℝ -> R (n + 1)
& ℝ
6)) R 3 -> R 3 -> ℝ
forall (n :: Nat). KnownNat n => R n -> R n -> ℝ
<·> ℝ -> Sq 3
forall t s (d :: * -> *). Sized t s d => t -> s
konst ℝ
2 Sq 3 -> R 3 -> R 3
forall (m :: Nat) (n :: Nat).
(KnownNat m, KnownNat n) =>
L m n -> R n -> R m
#> R 3
v
    precD :: ℝ
precD = ℝ
1 ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
+ ℝ
2 ℝ -> ℝ -> ℝ
forall a. Num a => a -> a -> a
* [Vector ℝ] -> Vector ℝ
forall t. Storable t => [Vector t] -> Vector t
vjoin[R 2 -> Vector ℝ
forall (n :: Nat). R n -> Vector ℝ
ud1 R 2
u, Vector ℝ
6] Vector ℝ -> Vector ℝ -> ℝ
forall t. Numeric t => Vector t -> Vector t -> t
LA.<.> ℝ -> (Int, Int) -> Matrix ℝ
forall e d (c :: * -> *). Konst e d c => e -> d -> c e
LA.konst ℝ
2 (Vector ℝ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size (R 2 -> Vector ℝ
forall (n :: Nat). R n -> Vector ℝ
ud1 R 2
u) Int -> Int -> Int
forall a. Num a => a -> a -> a
+Int
1, Vector ℝ -> IndexOf Vector
forall (c :: * -> *) t. Container c t => c t -> IndexOf c
LA.size (R 3 -> Vector ℝ
forall (n :: Nat). R n -> Vector ℝ
ud1 R 3
v)) Matrix ℝ -> Vector ℝ -> Vector ℝ
forall t. Numeric t => Matrix t -> Vector t -> Vector t
LA.#> R 3 -> Vector ℝ
forall (n :: Nat). R n -> Vector ℝ
ud1 R 3
v


splittest :: IO ()
splittest
    = do
    let v :: R 7
v = R 7
forall (n :: Nat). KnownNat n => R n
range :: R 7
        a :: R 4
a = (R 3, R 4) -> R 4
forall a b. (a, b) -> b
snd (R 7 -> (R 3, R (7 - 3))
forall (p :: Nat) (n :: Nat).
(KnownNat p, KnownNat n, p <= n) =>
R n -> (R p, R (n - p))
split R 7
v) :: R 4
    R 4 -> IO ()
forall a. Show a => a -> IO ()
print (R 4 -> IO ()) -> R 4 -> IO ()
forall a b. (a -> b) -> a -> b
$ R 4
a
    R 5 -> IO ()
forall a. Show a => a -> IO ()
print (R 5 -> IO ()) -> R 5 -> IO ()
forall a b. (a -> b) -> a -> b
$ (ℝ, R 5) -> R 5
forall a b. (a, b) -> b
snd ((ℝ, R 5) -> R 5) -> (R 7 -> (ℝ, R 5)) -> R 7 -> R 5
forall b c a. (b -> c) -> (a -> b) -> a -> c
. R 6 -> (ℝ, R 5)
forall (n :: Nat). (KnownNat n, 1 <= n) => R n -> (ℝ, R (n - 1))
headTail (R 6 -> (ℝ, R 5)) -> (R 7 -> R 6) -> R 7 -> (ℝ, R 5)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (ℝ, R 6) -> R 6
forall a b. (a, b) -> b
snd ((ℝ, R 6) -> R 6) -> (R 7 -> (ℝ, R 6)) -> R 7 -> R 6
forall b c a. (b -> c) -> (a -> b) -> a -> c
. R 7 -> (ℝ, R 6)
forall (n :: Nat). (KnownNat n, 1 <= n) => R n -> (ℝ, R (n - 1))
headTail (R 7 -> R 5) -> R 7 -> R 5
forall a b. (a -> b) -> a -> b
$ R 7
v
    ℝ -> IO ()
forall a. Show a => a -> IO ()
print (ℝ -> IO ()) -> ℝ -> IO ()
forall a b. (a -> b) -> a -> b
$ R 3 -> ℝ
forall (n :: Nat). (KnownNat n, (1 <=? n) ~ 'True) => R n -> ℝ
first (ℝ -> ℝ -> ℝ -> R 3
vec3 ℝ
1 ℝ
2 ℝ
3)
    ℝ -> IO ()
forall a. Show a => a -> IO ()
print (ℝ -> IO ()) -> ℝ -> IO ()
forall a b. (a -> b) -> a -> b
$ R 3 -> ℝ
forall (n :: Nat).
(KnownNat (n - 1), KnownNat n, (1 <=? n) ~ 'True,
 (1 <=? (n - 1)) ~ 'True) =>
R n -> ℝ
second (ℝ -> ℝ -> ℝ -> R 3
vec3 ℝ
1 ℝ
2 ℝ
3)
    ℝ -> IO ()
forall a. Show a => a -> IO ()
print (ℝ -> IO ()) -> ℝ -> IO ()
forall a b. (a -> b) -> a -> b
$ R 3 -> ℝ
forall (n :: Nat).
(KnownNat ((n - 1) - 1), KnownNat (n - 1), KnownNat n,
 (1 <=? ((n - 1) - 1)) ~ 'True, (1 <=? n) ~ 'True,
 (1 <=? (n - 1)) ~ 'True) =>
R n -> ℝ
third (ℝ -> ℝ -> ℝ -> R 3
vec3 ℝ
1 ℝ
2 ℝ
3)
    L 4 6 -> IO ()
forall a. Show a => a -> IO ()
print (L 4 6 -> IO ()) -> L 4 6 -> IO ()
forall a b. (a -> b) -> a -> b
$ ((L 2 6, L 4 6) -> L 4 6
forall a b. (a, b) -> b
snd ((L 2 6, L 4 6) -> L 4 6) -> (L 2 6, L 4 6) -> L 4 6
forall a b. (a -> b) -> a -> b
$ L 6 6 -> (L 2 6, L (6 - 2) 6)
forall (p :: Nat) (m :: Nat) (n :: Nat).
(KnownNat p, KnownNat m, KnownNat n, p <= m) =>
L m n -> (L p n, L (m - p) n)
splitRows L 6 6
forall (n :: Nat). KnownNat n => Sq n
eye :: L 4 6)
 where
    first :: R n -> ℝ
first R n
v = (ℝ, R (n - 1)) -> ℝ
forall a b. (a, b) -> a
fst ((ℝ, R (n - 1)) -> ℝ) -> (R n -> (ℝ, R (n - 1))) -> R n -> ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. R n -> (ℝ, R (n - 1))
forall (n :: Nat). (KnownNat n, 1 <= n) => R n -> (ℝ, R (n - 1))
headTail (R n -> ℝ) -> R n -> ℝ
forall a b. (a -> b) -> a -> b
$ R n
v
    second :: R n -> ℝ
second R n
v = R (n - 1) -> ℝ
forall (n :: Nat). (KnownNat n, (1 <=? n) ~ 'True) => R n -> ℝ
first (R (n - 1) -> ℝ) -> (R n -> R (n - 1)) -> R n -> ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (ℝ, R (n - 1)) -> R (n - 1)
forall a b. (a, b) -> b
snd ((ℝ, R (n - 1)) -> R (n - 1))
-> (R n -> (ℝ, R (n - 1))) -> R n -> R (n - 1)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. R n -> (ℝ, R (n - 1))
forall (n :: Nat). (KnownNat n, 1 <= n) => R n -> (ℝ, R (n - 1))
headTail (R n -> ℝ) -> R n -> ℝ
forall a b. (a -> b) -> a -> b
$ R n
v
    third :: R n -> ℝ
third R n
v = R ((n - 1) - 1) -> ℝ
forall (n :: Nat). (KnownNat n, (1 <=? n) ~ 'True) => R n -> ℝ
first (R ((n - 1) - 1) -> ℝ) -> (R n -> R ((n - 1) - 1)) -> R n -> ℝ
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (ℝ, R ((n - 1) - 1)) -> R ((n - 1) - 1)
forall a b. (a, b) -> b
snd ((ℝ, R ((n - 1) - 1)) -> R ((n - 1) - 1))
-> (R n -> (ℝ, R ((n - 1) - 1))) -> R n -> R ((n - 1) - 1)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. R (n - 1) -> (ℝ, R ((n - 1) - 1))
forall (n :: Nat). (KnownNat n, 1 <= n) => R n -> (ℝ, R (n - 1))
headTail (R (n - 1) -> (ℝ, R ((n - 1) - 1)))
-> (R n -> R (n - 1)) -> R n -> (ℝ, R ((n - 1) - 1))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (ℝ, R (n - 1)) -> R (n - 1)
forall a b. (a, b) -> b
snd ((ℝ, R (n - 1)) -> R (n - 1))
-> (R n -> (ℝ, R (n - 1))) -> R n -> R (n - 1)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. R n -> (ℝ, R (n - 1))
forall (n :: Nat). (KnownNat n, 1 <= n) => R n -> (ℝ, R (n - 1))
headTail (R n -> ℝ) -> R n -> ℝ
forall a b. (a -> b) -> a -> b
$ R n
v


instance (KnownNat n', KnownNat m') => Testable (L n' m')
  where
    checkT :: L n' m' -> (Bool, IO ())
checkT L n' m'
_ = (Bool, IO ())
test

--------------------------------------------------------------------------------

instance KnownNat n => Normed (R n)
  where
    norm_0 :: R n -> ℝ
norm_0 R n
v = Vector ℝ -> ℝ
forall a. Normed a => a -> ℝ
norm_0 (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract R n
v)
    norm_1 :: R n -> ℝ
norm_1 R n
v = Vector ℝ -> ℝ
forall a. Normed a => a -> ℝ
norm_1 (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract R n
v)
    norm_2 :: R n -> ℝ
norm_2 R n
v = Vector ℝ -> ℝ
forall a. Normed a => a -> ℝ
norm_2 (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract R n
v)
    norm_Inf :: R n -> ℝ
norm_Inf R n
v = Vector ℝ -> ℝ
forall a. Normed a => a -> ℝ
norm_Inf (R n -> Vector ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract R n
v)

instance (KnownNat m, KnownNat n) => Normed (L m n)
  where
    norm_0 :: L m n -> ℝ
norm_0 L m n
m = Matrix ℝ -> ℝ
forall a. Normed a => a -> ℝ
norm_0 (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract L m n
m)
    norm_1 :: L m n -> ℝ
norm_1 L m n
m = Matrix ℝ -> ℝ
forall a. Normed a => a -> ℝ
norm_1 (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract L m n
m)
    norm_2 :: L m n -> ℝ
norm_2 L m n
m = Matrix ℝ -> ℝ
forall a. Normed a => a -> ℝ
norm_2 (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract L m n
m)
    norm_Inf :: L m n -> ℝ
norm_Inf L m n
m = Matrix ℝ -> ℝ
forall a. Normed a => a -> ℝ
norm_Inf (L m n -> Matrix ℝ
forall t s (d :: * -> *). Sized t s d => s -> d t
extract L m n
m)

mkSym :: (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
f = Sq n -> Sym n
forall (n :: Nat). Sq n -> Sym n
Sym (Sq n -> Sym n) -> (Sym n -> Sq n) -> Sym n -> Sym n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Sq n -> Sq n
f (Sq n -> Sq n) -> (Sym n -> Sq n) -> Sym n -> Sq n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Sym n -> Sq n
forall (n :: Nat). Sym n -> Sq n
unSym
mkSym2 :: (Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
mkSym2 Sq n -> Sq n -> Sq n
f Sym n
x Sym n
y = Sq n -> Sym n
forall (n :: Nat). Sq n -> Sym n
Sym (Sq n -> Sq n -> Sq n
f (Sym n -> Sq n
forall (n :: Nat). Sym n -> Sq n
unSym Sym n
x) (Sym n -> Sq n
forall (n :: Nat). Sym n -> Sq n
unSym Sym n
y))

instance KnownNat n =>  Num (Sym n)
  where
    + :: Sym n -> Sym n -> Sym n
(+) = (Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat) (n :: Nat).
(Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
mkSym2 Sq n -> Sq n -> Sq n
forall a. Num a => a -> a -> a
(+)
    * :: Sym n -> Sym n -> Sym n
(*) = (Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat) (n :: Nat).
(Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
mkSym2 Sq n -> Sq n -> Sq n
forall a. Num a => a -> a -> a
(*)
    (-) = (Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat) (n :: Nat).
(Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
mkSym2 (-)
    abs :: Sym n -> Sym n
abs = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Num a => a -> a
abs
    signum :: Sym n -> Sym n
signum = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Num a => a -> a
signum
    negate :: Sym n -> Sym n
negate = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Num a => a -> a
negate
    fromInteger :: Integer -> Sym n
fromInteger = Sq n -> Sym n
forall (n :: Nat). Sq n -> Sym n
Sym (Sq n -> Sym n) -> (Integer -> Sq n) -> Integer -> Sym n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Integer -> Sq n
forall a. Num a => Integer -> a
fromInteger

instance KnownNat n => Fractional (Sym n)
  where
    fromRational :: Rational -> Sym n
fromRational = Sq n -> Sym n
forall (n :: Nat). Sq n -> Sym n
Sym (Sq n -> Sym n) -> (Rational -> Sq n) -> Rational -> Sym n
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Rational -> Sq n
forall a. Fractional a => Rational -> a
fromRational
    / :: Sym n -> Sym n -> Sym n
(/) = (Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat) (n :: Nat).
(Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
mkSym2 Sq n -> Sq n -> Sq n
forall a. Fractional a => a -> a -> a
(/)

instance KnownNat n => Floating (Sym n)
  where
    sin :: Sym n -> Sym n
sin   = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
sin
    cos :: Sym n -> Sym n
cos   = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
cos
    tan :: Sym n -> Sym n
tan   = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
tan
    asin :: Sym n -> Sym n
asin  = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
asin
    acos :: Sym n -> Sym n
acos  = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
acos
    atan :: Sym n -> Sym n
atan  = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
atan
    sinh :: Sym n -> Sym n
sinh  = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
sinh
    cosh :: Sym n -> Sym n
cosh  = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
cosh
    tanh :: Sym n -> Sym n
tanh  = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
tanh
    asinh :: Sym n -> Sym n
asinh = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
asinh
    acosh :: Sym n -> Sym n
acosh = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
acosh
    atanh :: Sym n -> Sym n
atanh = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
atanh
    exp :: Sym n -> Sym n
exp   = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
exp
    log :: Sym n -> Sym n
log   = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
log
    sqrt :: Sym n -> Sym n
sqrt  = (Sq n -> Sq n) -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat). (Sq n -> Sq n) -> Sym n -> Sym n
mkSym Sq n -> Sq n
forall a. Floating a => a -> a
sqrt
    ** :: Sym n -> Sym n -> Sym n
(**)  = (Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
forall (n :: Nat) (n :: Nat) (n :: Nat).
(Sq n -> Sq n -> Sq n) -> Sym n -> Sym n -> Sym n
mkSym2 Sq n -> Sq n -> Sq n
forall a. Floating a => a -> a -> a
(**)
    pi :: Sym n
pi    = Sq n -> Sym n
forall (n :: Nat). Sq n -> Sym n
Sym Sq n
forall a. Floating a => a
pi

instance KnownNat n => Additive (Sym n) where
    add :: Sym n -> Sym n -> Sym n
add = Sym n -> Sym n -> Sym n
forall a. Num a => a -> a -> a
(+)

instance KnownNat n => Transposable (Sym n) (Sym n) where
    tr :: Sym n -> Sym n
tr  = Sym n -> Sym n
forall a. a -> a
id
    tr' :: Sym n -> Sym n
tr' = Sym n -> Sym n
forall a. a -> a
id

instance KnownNat n => Transposable (Her n) (Her n) where
    tr :: Her n -> Her n
tr          = Her n -> Her n
forall a. a -> a
id
    tr' :: Her n -> Her n
tr' (Her M n n
m) = M n n -> Her n
forall (n :: Nat). M n n -> Her n
Her (M n n -> M n n
forall m mt. Transposable m mt => m -> mt
tr' M n n
m)