module Mcmc.Internal.Gamma
( logGammaG,
)
where
import qualified Data.Vector as VB
import Numeric.Polynomial
mSqrtEps :: RealFloat a => a
mSqrtEps :: a
mSqrtEps = a
1.4901161193847656e-8
mEulerMascheroni :: RealFloat a => a
mEulerMascheroni :: a
mEulerMascheroni = a
0.5772156649015328606065121
logGammaG :: RealFloat a => a -> a
logGammaG :: a -> a
logGammaG a
z
| a
z a -> a -> Bool
forall a. Ord a => a -> a -> Bool
<= a
0 = a
1 a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
0
| a
z a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
forall a. RealFloat a => a
mSqrtEps = a -> a
forall a. Floating a => a -> a
log (a
1 a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
z a -> a -> a
forall a. Num a => a -> a -> a
- a
forall a. RealFloat a => a
mEulerMascheroni)
| a
z a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
0.5 = a -> a -> a
forall a. RealFloat a => a -> a -> a
lgamma1_15 a
z (a
z a -> a -> a
forall a. Num a => a -> a -> a
- a
1) a -> a -> a
forall a. Num a => a -> a -> a
- a -> a
forall a. Floating a => a -> a
log a
z
| a
z a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
1 = a -> a -> a
forall a. RealFloat a => a -> a -> a
lgamma15_2 a
z (a
z a -> a -> a
forall a. Num a => a -> a -> a
- a
1) a -> a -> a
forall a. Num a => a -> a -> a
- a -> a
forall a. Floating a => a -> a
log a
z
| a
z a -> a -> Bool
forall a. Ord a => a -> a -> Bool
<= a
1.5 = a -> a -> a
forall a. RealFloat a => a -> a -> a
lgamma1_15 (a
z a -> a -> a
forall a. Num a => a -> a -> a
- a
1) (a
z a -> a -> a
forall a. Num a => a -> a -> a
- a
2)
| a
z a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
2 = a -> a -> a
forall a. RealFloat a => a -> a -> a
lgamma15_2 (a
z a -> a -> a
forall a. Num a => a -> a -> a
- a
1) (a
z a -> a -> a
forall a. Num a => a -> a -> a
- a
2)
| a
z a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
15 = a -> a
forall a. RealFloat a => a -> a
lgammaSmall a
z
| Bool
otherwise = a -> a
forall a. RealFloat a => a -> a
lanczosApprox a
z
{-# SPECIALIZE logGammaG :: Double -> Double #-}
lgamma1_15 :: RealFloat a => a -> a -> a
lgamma1_15 :: a -> a -> a
lgamma1_15 a
zm1 a
zm2 =
a
r a -> a -> a
forall a. Num a => a -> a -> a
* a
y a -> a -> a
forall a. Num a => a -> a -> a
+ a
r
a -> a -> a
forall a. Num a => a -> a -> a
* ( a -> Vector a -> a
forall (v :: * -> *) a. (Vector v a, Num a) => a -> v a -> a
evaluatePolynomial a
zm1 Vector a
forall a. RealFloat a => Vector a
tableLogGamma_1_15P
a -> a -> a
forall a. Fractional a => a -> a -> a
/ a -> Vector a -> a
forall (v :: * -> *) a. (Vector v a, Num a) => a -> v a -> a
evaluatePolynomial a
zm1 Vector a
forall a. RealFloat a => Vector a
tableLogGamma_1_15Q
)
where
r :: a
r = a
zm1 a -> a -> a
forall a. Num a => a -> a -> a
* a
zm2
y :: a
y = a
0.52815341949462890625
tableLogGamma_1_15P :: RealFloat a => VB.Vector a
tableLogGamma_1_15P :: Vector a
tableLogGamma_1_15P =
[a] -> Vector a
forall a. [a] -> Vector a
VB.fromList
[ a
0.490622454069039543534e-1,
-a
0.969117530159521214579e-1,
-a
0.414983358359495381969e0,
-a
0.406567124211938417342e0,
-a
0.158413586390692192217e0,
-a
0.240149820648571559892e-1,
-a
0.100346687696279557415e-2
]
{-# NOINLINE tableLogGamma_1_15P #-}
tableLogGamma_1_15Q :: RealFloat a => VB.Vector a
tableLogGamma_1_15Q :: Vector a
tableLogGamma_1_15Q =
[a] -> Vector a
forall a. [a] -> Vector a
VB.fromList
[ a
1,
a
0.302349829846463038743e1,
a
0.348739585360723852576e1,
a
0.191415588274426679201e1,
a
0.507137738614363510846e0,
a
0.577039722690451849648e-1,
a
0.195768102601107189171e-2
]
{-# NOINLINE tableLogGamma_1_15Q #-}
lgamma15_2 :: RealFloat a => a -> a -> a
lgamma15_2 :: a -> a -> a
lgamma15_2 a
zm1 a
zm2 =
a
r a -> a -> a
forall a. Num a => a -> a -> a
* a
y a -> a -> a
forall a. Num a => a -> a -> a
+ a
r
a -> a -> a
forall a. Num a => a -> a -> a
* ( a -> Vector a -> a
forall (v :: * -> *) a. (Vector v a, Num a) => a -> v a -> a
evaluatePolynomial (- a
zm2) Vector a
forall a. RealFloat a => Vector a
tableLogGamma_15_2P
a -> a -> a
forall a. Fractional a => a -> a -> a
/ a -> Vector a -> a
forall (v :: * -> *) a. (Vector v a, Num a) => a -> v a -> a
evaluatePolynomial (- a
zm2) Vector a
forall a. RealFloat a => Vector a
tableLogGamma_15_2Q
)
where
r :: a
r = a
zm1 a -> a -> a
forall a. Num a => a -> a -> a
* a
zm2
y :: a
y = a
0.452017307281494140625
tableLogGamma_15_2P :: RealFloat a => VB.Vector a
tableLogGamma_15_2P :: Vector a
tableLogGamma_15_2P =
[a] -> Vector a
forall a. [a] -> Vector a
VB.fromList
[ -a
0.292329721830270012337e-1,
a
0.144216267757192309184e0,
-a
0.142440390738631274135e0,
a
0.542809694055053558157e-1,
-a
0.850535976868336437746e-2,
a
0.431171342679297331241e-3
]
{-# NOINLINE tableLogGamma_15_2P #-}
tableLogGamma_15_2Q :: RealFloat a => VB.Vector a
tableLogGamma_15_2Q :: Vector a
tableLogGamma_15_2Q =
[a] -> Vector a
forall a. [a] -> Vector a
VB.fromList
[ a
1,
-a
0.150169356054485044494e1,
a
0.846973248876495016101e0,
-a
0.220095151814995745555e0,
a
0.25582797155975869989e-1,
-a
0.100666795539143372762e-2,
-a
0.827193521891290553639e-6
]
{-# NOINLINE tableLogGamma_15_2Q #-}
lgammaSmall :: RealFloat a => a -> a
lgammaSmall :: a -> a
lgammaSmall = a -> a -> a
forall a. RealFloat a => a -> a -> a
go a
0
where
go :: t -> t -> t
go t
acc t
z
| t
z t -> t -> Bool
forall a. Ord a => a -> a -> Bool
< t
3 = t
acc t -> t -> t
forall a. Num a => a -> a -> a
+ t -> t
forall a. RealFloat a => a -> a
lgamma2_3 t
z
| Bool
otherwise = t -> t -> t
go (t
acc t -> t -> t
forall a. Num a => a -> a -> a
+ t -> t
forall a. Floating a => a -> a
log t
zm1) t
zm1
where
zm1 :: t
zm1 = t
z t -> t -> t
forall a. Num a => a -> a -> a
- t
1
lgamma2_3 :: RealFloat a => a -> a
lgamma2_3 :: a -> a
lgamma2_3 a
z =
a
r a -> a -> a
forall a. Num a => a -> a -> a
* a
y a -> a -> a
forall a. Num a => a -> a -> a
+ a
r
a -> a -> a
forall a. Num a => a -> a -> a
* ( a -> Vector a -> a
forall (v :: * -> *) a. (Vector v a, Num a) => a -> v a -> a
evaluatePolynomial a
zm2 Vector a
forall a. RealFloat a => Vector a
tableLogGamma_2_3P
a -> a -> a
forall a. Fractional a => a -> a -> a
/ a -> Vector a -> a
forall (v :: * -> *) a. (Vector v a, Num a) => a -> v a -> a
evaluatePolynomial a
zm2 Vector a
forall a. RealFloat a => Vector a
tableLogGamma_2_3Q
)
where
r :: a
r = a
zm2 a -> a -> a
forall a. Num a => a -> a -> a
* (a
z a -> a -> a
forall a. Num a => a -> a -> a
+ a
1)
zm2 :: a
zm2 = a
z a -> a -> a
forall a. Num a => a -> a -> a
- a
2
y :: a
y = a
0.158963680267333984375e0
tableLogGamma_2_3P :: RealFloat a => VB.Vector a
tableLogGamma_2_3P :: Vector a
tableLogGamma_2_3P =
[a] -> Vector a
forall a. [a] -> Vector a
VB.fromList
[ -a
0.180355685678449379109e-1,
a
0.25126649619989678683e-1,
a
0.494103151567532234274e-1,
a
0.172491608709613993966e-1,
-a
0.259453563205438108893e-3,
-a
0.541009869215204396339e-3,
-a
0.324588649825948492091e-4
]
{-# NOINLINE tableLogGamma_2_3P #-}
tableLogGamma_2_3Q :: RealFloat a => VB.Vector a
tableLogGamma_2_3Q :: Vector a
tableLogGamma_2_3Q =
[a] -> Vector a
forall a. [a] -> Vector a
VB.fromList
[ a
1,
a
0.196202987197795200688e1,
a
0.148019669424231326694e1,
a
0.541391432071720958364e0,
a
0.988504251128010129477e-1,
a
0.82130967464889339326e-2,
a
0.224936291922115757597e-3,
-a
0.223352763208617092964e-6
]
{-# NOINLINE tableLogGamma_2_3Q #-}
lanczosApprox :: RealFloat a => a -> a
lanczosApprox :: a -> a
lanczosApprox a
z =
(a -> a
forall a. Floating a => a -> a
log (a
z a -> a -> a
forall a. Num a => a -> a -> a
+ a
g a -> a -> a
forall a. Num a => a -> a -> a
- a
0.5) a -> a -> a
forall a. Num a => a -> a -> a
- a
1) a -> a -> a
forall a. Num a => a -> a -> a
* (a
z a -> a -> a
forall a. Num a => a -> a -> a
- a
0.5)
a -> a -> a
forall a. Num a => a -> a -> a
+ a -> a
forall a. Floating a => a -> a
log (Vector (a, a) -> a -> a
forall a. RealFloat a => Vector (a, a) -> a -> a
evalRatio Vector (a, a)
forall a. RealFloat a => Vector (a, a)
tableLanczos a
z)
where
g :: a
g = a
6.024680040776729583740234375
tableLanczos :: RealFloat a => VB.Vector (a, a)
tableLanczos :: Vector (a, a)
tableLanczos =
[(a, a)] -> Vector (a, a)
forall a. [a] -> Vector a
VB.fromList
[ (a
56906521.91347156388090791033559122686859, a
0),
(a
103794043.1163445451906271053616070238554, a
39916800),
(a
86363131.28813859145546927288977868422342, a
120543840),
(a
43338889.32467613834773723740590533316085, a
150917976),
(a
14605578.08768506808414169982791359218571, a
105258076),
(a
3481712.15498064590882071018964774556468, a
45995730),
(a
601859.6171681098786670226533699352302507, a
13339535),
(a
75999.29304014542649875303443598909137092, a
2637558),
(a
6955.999602515376140356310115515198987526, a
357423),
(a
449.9445569063168119446858607650988409623, a
32670),
(a
19.51992788247617482847860966235652136208, a
1925),
(a
0.5098416655656676188125178644804694509993, a
66),
(a
0.006061842346248906525783753964555936883222, a
1)
]
{-# NOINLINE tableLanczos #-}
data L a = L !a !a
evalRatio :: RealFloat a => VB.Vector (a, a) -> a -> a
evalRatio :: Vector (a, a) -> a -> a
evalRatio Vector (a, a)
coef a
x
| a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
> a
1 = L a -> a
forall a. Fractional a => L a -> a
fini (L a -> a) -> L a -> a
forall a b. (a -> b) -> a -> b
$ (L a -> (a, a) -> L a) -> L a -> Vector (a, a) -> L a
forall a b. (a -> b -> a) -> a -> Vector b -> a
VB.foldl' L a -> (a, a) -> L a
stepL (a -> a -> L a
forall a. a -> a -> L a
L a
0 a
0) Vector (a, a)
coef
| Bool
otherwise = L a -> a
forall a. Fractional a => L a -> a
fini (L a -> a) -> L a -> a
forall a b. (a -> b) -> a -> b
$ ((a, a) -> L a -> L a) -> L a -> Vector (a, a) -> L a
forall a b. (a -> b -> b) -> b -> Vector a -> b
VB.foldr' (a, a) -> L a -> L a
stepR (a -> a -> L a
forall a. a -> a -> L a
L a
0 a
0) Vector (a, a)
coef
where
fini :: L a -> a
fini (L a
num a
den) = a
num a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
den
stepR :: (a, a) -> L a -> L a
stepR (a
a, a
b) (L a
num a
den) = a -> a -> L a
forall a. a -> a -> L a
L (a
num a -> a -> a
forall a. Num a => a -> a -> a
* a
x a -> a -> a
forall a. Num a => a -> a -> a
+ a
a) (a
den a -> a -> a
forall a. Num a => a -> a -> a
* a
x a -> a -> a
forall a. Num a => a -> a -> a
+ a
b)
stepL :: L a -> (a, a) -> L a
stepL (L a
num a
den) (a
a, a
b) = a -> a -> L a
forall a. a -> a -> L a
L (a
num a -> a -> a
forall a. Num a => a -> a -> a
* a
rx a -> a -> a
forall a. Num a => a -> a -> a
+ a
a) (a
den a -> a -> a
forall a. Num a => a -> a -> a
* a
rx a -> a -> a
forall a. Num a => a -> a -> a
+ a
b)
rx :: a
rx = a -> a
forall a. Fractional a => a -> a
recip a
x