Merge pull request #92 from fredrikekre/fe/gamma

move [l]gamma, [l]beta and lfact from base
JuliaMath · Jun 13, 2018 · f3ee4e7 · f3ee4e7
2 parents 5ff7cd3 + b98961c
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diff --git a/src/gamma.jl b/src/gamma.jl
@@ -539,3 +539,228 @@ function polygamma(m::Integer, z::Number)
     # There is nothing to fallback to, since this didn't work
     polygamma(m, x)
 end
+
+@static if VERSION >= v"0.7.0-alpha.69"
+
+export gamma, lgamma, beta, lbeta, lfactorial
+
+## from base/special/gamma.jl
+
+gamma(x::Float64) = nan_dom_err(ccall((:tgamma,libm),  Float64, (Float64,), x), x)
+gamma(x::Float32) = nan_dom_err(ccall((:tgammaf,libm),  Float32, (Float32,), x), x)
+
+"""
+    gamma(x)
+
+Compute the gamma function of `x`.
+"""
+gamma(x::Real) = gamma(float(x))
+
+function lgamma_r(x::Float64)
+    signp = Ref{Int32}()
+    y = ccall((:lgamma_r,libm),  Float64, (Float64, Ptr{Int32}), x, signp)
+    return y, signp[]
+end
+function lgamma_r(x::Float32)
+    signp = Ref{Int32}()
+    y = ccall((:lgammaf_r,libm),  Float32, (Float32, Ptr{Int32}), x, signp)
+    return y, signp[]
+end
+lgamma_r(x::Real) = lgamma_r(float(x))
+lgamma_r(x::Number) = lgamma(x), 1 # lgamma does not take abs for non-real x
+"""
+    lgamma_r(x)
+
+Return L,s such that `gamma(x) = s * exp(L)`
+"""
+lgamma_r
+
+"""
+    lfactorial(x)
+
+Compute the logarithmic factorial of a nonnegative integer `x`.
+Equivalent to [`lgamma`](@ref) of `x + 1`, but `lgamma` extends this function
+to non-integer `x`.
+"""
+lfactorial(x::Integer) = x < 0 ? throw(DomainError(x, "`x` must be non-negative.")) : lgamma(x + oneunit(x))
+Base.@deprecate lfact lfactorial
+
+"""
+    lgamma(x)
+
+Compute the logarithm of the absolute value of [`gamma`](@ref) for
+[`Real`](@ref) `x`, while for [`Complex`](@ref) `x` compute the
+principal branch cut of the logarithm of `gamma(x)` (defined for negative `real(x)`
+by analytic continuation from positive `real(x)`).
+"""
+function lgamma end
+
+# asymptotic series for log(gamma(z)), valid for sufficiently large real(z) or |imag(z)|
+@inline function lgamma_asymptotic(z::Complex{Float64})
+    zinv = inv(z)
+    t = zinv*zinv
+    # coefficients are bernoulli[2:n+1] .// (2*(1:n).*(2*(1:n) - 1))
+    return (z-0.5)*log(z) - z + 9.1893853320467274178032927e-01 + # <-- log(2pi)/2
+       zinv*@evalpoly(t, 8.3333333333333333333333368e-02,-2.7777777777777777777777776e-03,
+                         7.9365079365079365079365075e-04,-5.9523809523809523809523806e-04,
+                         8.4175084175084175084175104e-04,-1.9175269175269175269175262e-03,
+                         6.4102564102564102564102561e-03,-2.9550653594771241830065352e-02)
+end
+
+# Compute the logΓ(z) function using a combination of the asymptotic series,
+# the Taylor series around z=1 and z=2, the reflection formula, and the shift formula.
+# Many details of these techniques are discussed in D. E. G. Hare,
+# "Computing the principal branch of log-Gamma," J. Algorithms 25, pp. 221-236 (1997),
+# and similar techniques are used (in a somewhat different way) by the
+# SciPy loggamma function.  The key identities are also described
+# at http://functions.wolfram.com/GammaBetaErf/LogGamma/
+function lgamma(z::Complex{Float64})
+    x = real(z)
+    y = imag(z)
+    yabs = abs(y)
+    if !isfinite(x) || !isfinite(y) # Inf or NaN
+        if isinf(x) && isfinite(y)
+            return Complex(x, x > 0 ? (y == 0 ? y : copysign(Inf, y)) : copysign(Inf, -y))
+        elseif isfinite(x) && isinf(y)
+            return Complex(-Inf, y)
+        else
+            return Complex(NaN, NaN)
+        end
+    elseif x > 7 || yabs > 7 # use the Stirling asymptotic series for sufficiently large x or |y|
+        return lgamma_asymptotic(z)
+    elseif x < 0.1 # use reflection formula to transform to x > 0
+        if x == 0 && y == 0 # return Inf with the correct imaginary part for z == 0
+            return Complex(Inf, signbit(x) ? copysign(oftype(x, pi), -y) : -y)
+        end
+        # the 2pi * floor(...) stuff is to choose the correct branch cut for log(sinpi(z))
+        return Complex(1.1447298858494001741434262, # log(pi)
+                       copysign(6.2831853071795864769252842, y) # 2pi
+                       * floor(0.5*x+0.25)) -
+               log(sinpi(z)) - lgamma(1-z)
+    elseif abs(x - 1) + yabs < 0.1
+        # taylor series around zero at z=1
+        # ... coefficients are [-eulergamma; [(-1)^k * zeta(k)/k for k in 2:15]]
+        w = Complex(x - 1, y)
+        return w * @evalpoly(w, -5.7721566490153286060651188e-01,8.2246703342411321823620794e-01,
+                                -4.0068563438653142846657956e-01,2.705808084277845478790009e-01,
+                                -2.0738555102867398526627303e-01,1.6955717699740818995241986e-01,
+                                -1.4404989676884611811997107e-01,1.2550966952474304242233559e-01,
+                                -1.1133426586956469049087244e-01,1.000994575127818085337147e-01,
+                                -9.0954017145829042232609344e-02,8.3353840546109004024886499e-02,
+                                -7.6932516411352191472827157e-02,7.1432946295361336059232779e-02,
+                                -6.6668705882420468032903454e-02)
+    elseif abs(x - 2) + yabs < 0.1
+        # taylor series around zero at z=2
+        # ... coefficients are [1-eulergamma; [(-1)^k * (zeta(k)-1)/k for k in 2:12]]
+        w = Complex(x - 2, y)
+        return w * @evalpoly(w, 4.2278433509846713939348812e-01,3.2246703342411321823620794e-01,
+                               -6.7352301053198095133246196e-02,2.0580808427784547879000897e-02,
+                               -7.3855510286739852662729527e-03,2.8905103307415232857531201e-03,
+                               -1.1927539117032609771139825e-03,5.0966952474304242233558822e-04,
+                               -2.2315475845357937976132853e-04,9.9457512781808533714662972e-05,
+                               -4.4926236738133141700224489e-05,2.0507212775670691553131246e-05)
+    end
+    # use recurrence relation lgamma(z) = lgamma(z+1) - log(z) to shift to x > 7 for asymptotic series
+    shiftprod = Complex(x,yabs)
+    x += 1
+    sb = false # == signbit(imag(shiftprod)) == signbit(yabs)
+    # To use log(product of shifts) rather than sum(logs of shifts),
+    # we need to keep track of the number of + to - sign flips in
+    # imag(shiftprod), as described in Hare (1997), proposition 2.2.
+    signflips = 0
+    while x <= 7
+        shiftprod *= Complex(x,yabs)
+        sb′ = signbit(imag(shiftprod))
+        signflips += sb′ & (sb′ != sb)
+        sb = sb′
+        x += 1
+    end
+    shift = log(shiftprod)
+    if signbit(y) # if y is negative, conjugate the shift
+        shift = Complex(real(shift), signflips*-6.2831853071795864769252842 - imag(shift))
+    else
+        shift = Complex(real(shift), imag(shift) + signflips*6.2831853071795864769252842)
+    end
+    return lgamma_asymptotic(Complex(x,y)) - shift
+end
+lgamma(z::Complex{T}) where {T<:Union{Integer,Rational}} = lgamma(float(z))
+lgamma(z::Complex{T}) where {T<:Union{Float32,Float16}} = Complex{T}(lgamma(Complex{Float64}(z)))
+
+gamma(z::Complex) = exp(lgamma(z))
+
+"""
+    beta(x, y)
+
+Euler integral of the first kind ``\\operatorname{B}(x,y) = \\Gamma(x)\\Gamma(y)/\\Gamma(x+y)``.
+"""
+function beta(x::Number, w::Number)
+    yx, sx = lgamma_r(x)
+    yw, sw = lgamma_r(w)
+    yxw, sxw = lgamma_r(x+w)
+    return exp(yx + yw - yxw) * (sx*sw*sxw)
+end
+
+"""
+    lbeta(x, y)
+
+Natural logarithm of the absolute value of the [`beta`](@ref)
+function ``\\log(|\\operatorname{B}(x,y)|)``.
+"""
+lbeta(x::Number, w::Number) = lgamma(x)+lgamma(w)-lgamma(x+w)
+
+## from base/mpfr.jl
+
+# Functions for which NaN results are converted to DomainError, following Base
+function gamma(x::BigFloat)
+    isnan(x) && return x
+    z = BigFloat()
+    ccall((:mpfr_lgamma, :libmpfr), Int32, (Ref{BigFloat}, Ref{BigFloat}, Int32), z, x, ROUNDING_MODE[])
+    isnan(z) && throw(DomainError(x, "NaN result for non-NaN input."))
+    return z
+end
+
+# log of absolute value of gamma function
+function lgamma_r(x::BigFloat)
+    z = BigFloat()
+    lgamma_signp = Ref{Cint}()
+    ccall((:mpfr_lgamma,:libmpfr), Cint, (Ref{BigFloat}, Ref{Cint}, Ref{BigFloat}, Int32), z, lgamma_signp, x, ROUNDING_MODE[])
+    return z, lgamma_signp[]
+end
+
+lgamma(x::BigFloat) = lgamma_r(x)[1]
+
+if Base.MPFR.version() >= v"4.0.0"
+    function beta(y::BigFloat, x::BigFloat)
+        z = BigFloat()
+        ccall((:mpfr_beta, :libmpfr), Int32, (Ref{BigFloat}, Ref{BigFloat}, Ref{BigFloat}, Int32), z, y, x, ROUNDING_MODE[])
+        return z
+    end
+end
+
+## from base/combinatorics.jl'
+
+function gamma(n::Union{Int8,UInt8,Int16,UInt16,Int32,UInt32,Int64,UInt64})
+    n < 0 && throw(DomainError(n, "`n` must not be negative."))
+    n == 0 && return Inf
+    n <= 2 && return 1.0
+    n > 20 && return gamma(Float64(n))
+    @inbounds return Float64(Base._fact_table64[n-1])
+end
+
+## from base/math.jl
+
+@inline lgamma(x::Float64) = nan_dom_err(ccall((:lgamma, libm), Float64, (Float64,), x), x)
+@inline lgamma(x::Float32) = nan_dom_err(ccall((:lgammaf, libm), Float32, (Float32,), x), x)
+@inline lgamma(x::Real) = lgamma(float(x))
+
+## from base/numbers.jl
+# TODO: deprecate instead of doing this type-piracy here?
+Base.factorial(x::Number) = gamma(x + 1) # fallback for x not Integer
+
+else # @static if
+
+import Base: gamma, lgamma, beta, lbeta, lfact
+const lfactorial = lfact
+export lfactorial
+
+end # @static if
diff --git a/test/runtests.jl b/test/runtests.jl
@@ -555,3 +555,84 @@ end
 end
 
 @test sprint(showerror, SF.AmosException(1)) == "AmosException with id 1: input error."
+
+@testset "gamma and friends" begin
+    @testset "gamma, lgamma (complex argument)" begin
+        if Base.Math.libm == "libopenlibm"
+            @test gamma.(Float64[1:25;]) == gamma.(1:25)
+        else
+            @test gamma.(Float64[1:25;]) ≈ gamma.(1:25)
+        end
+        for elty in (Float32, Float64)
+            @test gamma(convert(elty,1/2)) ≈ convert(elty,sqrt(π))
+            @test gamma(convert(elty,-1/2)) ≈ convert(elty,-2sqrt(π))
+            @test lgamma(convert(elty,-1/2)) ≈ convert(elty,log(abs(gamma(-1/2))))
+        end
+        @test lgamma(1.4+3.7im) ≈ -3.7094025330996841898 + 2.4568090502768651184im
+        @test lgamma(1.4+3.7im) ≈ log(gamma(1.4+3.7im))
+        @test lgamma(-4.2+0im) ≈ lgamma(-4.2)-5pi*im
+        @test factorial(3.0) == gamma(4.0) == factorial(3)
+        for x in (3.2, 2+1im, 3//2, 3.2+0.1im)
+            @test factorial(x) == gamma(1+x)
+        end
+        @test lfactorial(0) == lfactorial(1) == 0
+        @test lfactorial(2) == lgamma(3)
+        # Ensure that the domain of lfactorial matches that of factorial (issue #21318)
+        @test_throws DomainError lfactorial(-3)
+        @test_throws MethodError lfactorial(1.0)
+    end
+
+    # lgamma test cases (from Wolfram Alpha)
+    @test lgamma(-300im) ≅ -473.17185074259241355733179182866544204963885920016823743 - 1410.3490664555822107569308046418321236643870840962522425im
+    @test lgamma(3.099) ≅ lgamma(3.099+0im) ≅ 0.786413746900558058720665860178923603134125854451168869796
+    @test lgamma(1.15) ≅ lgamma(1.15+0im) ≅ -0.06930620867104688224241731415650307100375642207340564554
+    @test lgamma(0.89) ≅ lgamma(0.89+0im) ≅ 0.074022173958081423702265889979810658434235008344573396963
+    @test lgamma(0.91) ≅ lgamma(0.91+0im) ≅ 0.058922567623832379298241751183907077883592982094770449167
+    @test lgamma(0.01) ≅ lgamma(0.01+0im) ≅ 4.599479878042021722513945411008748087261001413385289652419
+    @test lgamma(-3.4-0.1im) ≅ -1.1733353322064779481049088558918957440847715003659143454 + 12.331465501247826842875586104415980094316268974671819281im
+    @test lgamma(-13.4-0.1im) ≅ -22.457344044212827625152500315875095825738672314550695161 + 43.620560075982291551250251193743725687019009911713182478im
+    @test lgamma(-13.4+0.0im) ≅ conj(lgamma(-13.4-0.0im)) ≅ -22.404285036964892794140985332811433245813398559439824988 - 43.982297150257105338477007365913040378760371591251481493im
+    @test lgamma(-13.4+8im) ≅ -44.705388949497032519400131077242200763386790107166126534 - 22.208139404160647265446701539526205774669649081807864194im
+    @test lgamma(1+exp2(-20)) ≅ lgamma(1+exp2(-20)+0im) ≅ -5.504750066148866790922434423491111098144565651836914e-7
+    @test lgamma(1+exp2(-20)+exp2(-19)*im) ≅ -5.5047799872835333673947171235997541985495018556426e-7 - 1.1009485171695646421931605642091915847546979851020e-6im
+    @test lgamma(-300+2im) ≅ -1419.3444991797240659656205813341478289311980525970715668 - 932.63768120761873747896802932133229201676713644684614785im
+    @test lgamma(300+2im) ≅ 1409.19538972991765122115558155209493891138852121159064304 + 11.4042446282102624499071633666567192538600478241492492652im
+    @test lgamma(1-6im) ≅ -7.6099596929506794519956058191621517065972094186427056304 - 5.5220531255147242228831899544009162055434670861483084103im
+    @test lgamma(1-8im) ≅ -10.607711310314582247944321662794330955531402815576140186 - 9.4105083803116077524365029286332222345505790217656796587im
+    @test lgamma(1+6.5im) ≅ conj(lgamma(1-6.5im)) ≅ -8.3553365025113595689887497963634069303427790125048113307 + 6.4392816159759833948112929018407660263228036491479825744im
+    @test lgamma(1+1im) ≅ conj(lgamma(1-1im)) ≅ -0.6509231993018563388852168315039476650655087571397225919 - 0.3016403204675331978875316577968965406598997739437652369im
+    @test lgamma(-pi*1e7 + 6im) ≅ -5.10911758892505772903279926621085326635236850347591e8 - 9.86959420047365966439199219724905597399295814979993e7im
+    @test lgamma(-pi*1e7 + 8im) ≅ -5.10911765175690634449032797392631749405282045412624e8 - 9.86959074790854911974415722927761900209557190058925e7im
+    @test lgamma(-pi*1e14 + 6im) ≅ -1.0172766411995621854526383224252727000270225301426e16 - 9.8696044010873714715264929863618267642124589569347e14im
+    @test lgamma(-pi*1e14 + 8im) ≅ -1.0172766411995628137711690403794640541491261237341e16 - 9.8696044010867038531027376655349878694397362250037e14im
+    @test lgamma(2.05 + 0.03im) ≅ conj(lgamma(2.05 - 0.03im)) ≅ 0.02165570938532611215664861849215838847758074239924127515 + 0.01363779084533034509857648574107935425251657080676603919im
+    @test lgamma(2+exp2(-20)+exp2(-19)*im) ≅ 4.03197681916768997727833554471414212058404726357753e-7 + 8.06398296652953575754782349984315518297283664869951e-7im
+
+    @testset "lgamma for non-finite arguments" begin
+        @test lgamma(Inf + 0im) === Inf + 0im
+        @test lgamma(Inf - 0.0im) === Inf - 0.0im
+        @test lgamma(Inf + 1im) === Inf + Inf*im
+        @test lgamma(Inf - 1im) === Inf - Inf*im
+        @test lgamma(-Inf + 0.0im) === -Inf - Inf*im
+        @test lgamma(-Inf - 0.0im) === -Inf + Inf*im
+        @test lgamma(Inf*im) === -Inf + Inf*im
+        @test lgamma(-Inf*im) === -Inf - Inf*im
+        @test lgamma(Inf + Inf*im) === lgamma(NaN + 0im) === lgamma(NaN*im) === NaN + NaN*im
+    end
+end
+
+@testset "beta, lbeta" begin
+    @test beta(3/2,7/2) ≈ 5π/128
+    @test beta(3,5) ≈ 1/105
+    @test lbeta(5,4) ≈ log(beta(5,4))
+    @test beta(5,4) ≈ beta(4,5)
+    @test beta(-1/2, 3) ≈ beta(-1/2 + 0im, 3 + 0im) ≈ -16/3
+    @test lbeta(-1/2, 3) ≈ log(16/3)
+    @test beta(Float32(5),Float32(4)) == beta(Float32(4),Float32(5))
+    @test beta(3,5) ≈ beta(3+0im,5+0im)
+    @test(beta(3.2+0.1im,5.3+0.3im) ≈ exp(lbeta(3.2+0.1im,5.3+0.3im)) ≈
+          0.00634645247782269506319336871208405439180447035257028310080 -
+          0.00169495384841964531409376316336552555952269360134349446910im)
+
+    @test beta(big(1.0),big(1.2)) ≈ beta(1.0,1.2) rtol=4*eps()
+end