From 8a2cedbdc716f171bafaad5735687b186e47626c Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Wed, 31 May 2017 13:59:44 +0930 Subject: [PATCH 01/14] added harmonic, bernoulli, and polylog functions --- examples/bernoulli_ex.jl | 22 ++++ examples/bernoulli_ex.pdf | Bin 0 -> 8484 bytes examples/li_ex.jl | 29 ++++++ src/SpecialFunctions.jl | 10 +- src/bernoulli.jl | 118 ++++++++++++++++++++++ src/harmonic.jl | 48 +++++++++ src/li.jl | 204 ++++++++++++++++++++++++++++++++++++++ test/bernoulli_test.jl | 55 ++++++++++ test/harmonic_test.jl | 37 +++++++ test/li_test.jl | 144 +++++++++++++++++++++++++++ 10 files changed, 666 insertions(+), 1 deletion(-) create mode 100644 examples/bernoulli_ex.jl create mode 100644 examples/bernoulli_ex.pdf create mode 100644 examples/li_ex.jl create mode 100644 src/bernoulli.jl create mode 100644 src/harmonic.jl create mode 100644 src/li.jl create mode 100644 test/bernoulli_test.jl create mode 100644 test/harmonic_test.jl create mode 100644 test/li_test.jl diff --git a/examples/bernoulli_ex.jl b/examples/bernoulli_ex.jl new file mode 100644 index 00000000..92eb161d --- /dev/null +++ b/examples/bernoulli_ex.jl @@ -0,0 +1,22 @@ +# Example of Bernoulli functions +# Replicate (approximately) the plot from http://mathworld.wolfram.com/BernoulliPolynomial.html + +using Plots; plotlyjs() +default(show = true) # shouldn't need display commands + +using SpecialFunctions +const SF = SpecialFunctions + +x = -0.4:0.01:1.0 +plt = plot( x, SF.bernoulli.(1,x), + label="B_1(x)", legendfont=font(20, "Courier"), legend=:bottomright, + xaxis = ("x", font(20, "Courier")), + yaxis = ("", font(20, "Courier"), (-0.15, 0.15), -0.15:0.05:1.5), + size = (1200, 800) + ) + +for i=2:5 + plot!( x, SF.bernoulli.(i,x), label="B_$i(x)") +end + +savefig("bernoulli_ex.pdf") diff --git a/examples/bernoulli_ex.pdf b/examples/bernoulli_ex.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ae0d37dc84981f08aa0df2f64394e2408e02584c GIT binary patch literal 8484 zcmbVy1z1#D)HV&y5Ynj(3JQ`#4lUiC0s;dJFto$arHC{LqJnga(j^^|(jg@sBHb+^ z-++3r_ulV${_pwEnRCvp_pH6wUOVPl>)peoE-lLr=HLS`eb^{C1@M5lK#pcM01*)o zr!vgJ3TX}EMM0VX5D3I63%5hU5U8ge6bX}tnLAp*0AgYQXCwj!wFkH-zIvlpc%y=# z^=Oa9@(tUxG|78ELrj?`7!Nb~`en4Rur+D%l7TG`dFdOLqK3MSPQO>FUSBXa?g>sO z^vm4$o!IU56R)4~JGL61xv*058h2KJ(;shrZ8&tBNcy=L@WW4|U{ajE% zD{huK;9^YtVmnsc%>QD()vw8~!NKnW*H`hBGhx$jIxFIK&)yWhs+6$JSjtg@gJ=hX zCv7qQQt&pXnqY}#c>6Vu5e?Nr`clVtY>6N>J0zVfLh_;P$we`zo6p>t1fM zcD|*4vZ;1wDU{-s!P|MM6)7gar&3I@-gwQp#W%+9OlpX};NoIBxNWY)diWN<6e$|I zF8Dk$g6edyJXRv4QM<-5^`&ok^3m8rzr@pJ4dIc2=bbu%m9hnL6GyMVM;VEw&C?_yNWfE>)b8zZ^X{X{{+;xmah`q_G^TOT>_; 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Bernoulli numbers of second kind only seem to differ in that B_1 = + 1/2 (instead of -1/2) + +## Arguments +* `n::Integer`: the index into the series, n=0,1,2,3,... + +## Examples +```jldoctest +julia> bernoulli(6) +1 // 42 +``` +""" +function bernoulli(n::Int) + # this just does a lookup -- seemed like it would be easier to code and faster + # for the size of numbers I am working with + if n<0 + warn("n should be > 0") + throw(DomainError) + end + if n>34 + warn("If n > 34, then the numerator needs Int128 at least, and worse, so this code is not the code you want. Try using bernoulli(n, 0.0) to get a floating point approximation to the result.") + throw(DomainError) + end + + # Denominator of Bernoulli number B_n + # http://oeis.org/A027642 + D = [2, 6, 1, 30, 1, 42, 1, 30, 1, 66, 1, 2730, 1, 6, 1, 510, 1, 798, 1, 330, 1, 138, 1, 2730, 1, 6, 1, 870, 1, 14322, 1, 510, 1, 6, 1, 1919190, 1, 6, 1, 13530, 1, 1806, 1, 690, 1, 282, 1, 46410, 1, 66, 1, 1590, 1, 798, 1, 870, 1, 354, 1, 56786730] + + # Numerator of Bernoulli number B_n (storing 62 of these because they are easy) + # http://oeis.org/A027641 + N = [-1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 1] + + if n==0 + return 1 + else + return N[n] // D[n] + end +end + +# get the Bernoulli polynomials from the Hurwitz-zeta function (which is already defined) +# +# +""" + bernoulli(n, x) + + created: Tue May 23 2017 + email: matthew.roughan@adelaide.edu.au + (c) M Roughan, 2017 + + Calculates Bernoulli polynomials from the Hurwitz-zeta function using + ``ζ(-n,x) = -B_{n+1}(x)/(n+1), for Re(x)>0 `` + Its probably not the fastest approach, but we get it almost for free. + + e.g., see + + + https://en.wikipedia.org/wiki/Bernoulli_polynomials + + http://dlmf.nist.gov/24 + +## Arguments +* `n::Integer`: the index into the series, n=0,1,2,3,... +* `x::Real`: the point at which to calculate the polynomial + +## Examples +```jldoctest +julia> bernoulli(6, 1.2) +0.008833523809524069 +``` +""" +function bernoulli(n::Int, x::Real) + if n<0 + warn("n should be > 0") + throw(DomainError) + end + if n == 0 + return 1 # zeta formula doesn't hold for n=0, so return explicit value + elseif n == 1 + return x-0.5 # get some easy cases out of the way quickly + elseif n == 2 + return x^2 - x + 1.0/6.0 + elseif n == 3 + return x^3 - 1.5*x^2 + 0.5*x + elseif n == 4 + return x^4 - 2.0*x^3 + x^2 - 1/30.0 + elseif n == 5 + return x^5 - 2.5*x^4 +(5.0/3.0)*x^3 - x/6.0 + end + + if n <= 34 + # direct summation for reasonably small values of coefficients + k = 0:n + return sum( binomial.(n,k) .* bernoulli.(k) .* x.^(n-k) ) + elseif x > 0 + return -n*zeta(1-n, x) + else + # comments in the gamma.jl note that identity with zeta(s,z) only works for Re(z)>0 + # so exploit symmetries in B_n(x) to compute recursively for x<=0 + return bernoulli(n, x+1) - n*x^(n-1) + end +end diff --git a/src/harmonic.jl b/src/harmonic.jl new file mode 100644 index 00000000..eda23aff --- /dev/null +++ b/src/harmonic.jl @@ -0,0 +1,48 @@ +""" + harmonic(n) + + Calculates harmonic numbers + e.g., see http://mathworld.wolfram.com/HarmonicNumber.html + +## Arguments +* `n::Integer`: index of the Harmonic number to calculate + +## Examples +```jldoctest +julia> harmonic(2) +1.5 +``` +""" +function harmonic(n::Integer) + # γ = euler_mascheroni_const = 0.577215664901532860606512090082402431042 # http://oeis.org/A001620 + if n <=10 + # get exact values for small n + return sum( 1.0./(1:n) ) + else + # numerical approximation for larger n + return γ + digamma(n+1) + end +end + + +""" + harmonic(n,r) + + Calculates generalized harmonic numbers + e.g., see http://mathworld.wolfram.com/HarmonicNumber.html + +## Arguments +* `n::Integer`: index 1 of the Harmonic number to calculate +* `r::Real`: index 2 of the Harmonic number to calculate + +It should be possible to extend this to complex r, but hey. + +## Examples +```jldoctest +julia> harmonic(2,1) +1.5 +``` +""" +function harmonic(n::Integer, r::Real) + sum( 1.0 ./ (1:n).^r ) +end diff --git a/src/li.jl b/src/li.jl new file mode 100644 index 00000000..af3765f7 --- /dev/null +++ b/src/li.jl @@ -0,0 +1,204 @@ +""" + Li(s, z) + + Calculates the Polylogarithm function ``Li_s(z)`` defined by + +```math + L_s = \sum_{n=1}^{\infty} \frac{z^n}{n^s} +``` + + For ideas going into this see + + + Crandall, "Note on fast polylogarithm computation", 2006, + which focusses on the case where s=n (integer and real) + http://www.wolfgang-ehrhardt.de/Polylog.pdf + + + Vepstas, "AN EFFICIENT ALGORITHM FOR ACCELERATING THE CONVERGENCE + OF OSCILLATORY SERIES, USEFUL FOR COMPUTING THE + POLYLOGARITHM AND HURWITZ ZETA FUNCTIONS", 2007 + which treats the general case, but presumes arbitrary precision arithmetic + https://arxiv.org/abs/math/0702243 + + + Wood, "The computation of Plylogarithms", 1992 + which focusses on s=n, integer and real, and which has formatting issues making + it hard to read correctly. + https://www.cs.kent.ac.uk/pubs/1992/110/ + + + Maximon, "The dilogarithm function for complex argument", 2003 + which provides useful test cases for s=2 + + + Zagier, "The dilogarithm function in geometry and number theory", 1989 + similar to Maximon + + Of these the only one that actually specifies a full algorithm is + Crandall, and it also treats special cases more carefully, so this + is the one that I have paid most attention to. However, extending it + for s on the whole complex plane requires some additions, and many + of these are actually most nicely documented on the wikipedia page + + + https://en.wikipedia.org/wiki/Polylogarithm + + With further details at + + + http://mathworld.wolfram.com/Polylogarithm.html + + http://dlmf.nist.gov/25.12#ii + + http://mathworld.wolfram.com/Trilogarithm.html + + http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/ + + The wiki page points out some errors in earlier works, but not all + parts on the page have references, and not all statements seem to + come from any of the listed references? + + The code draws heavily on existing functions, in particular the + Hurwitz-zeta function, which is aliased to zeta(s,q) in Julia. + + Accuracy has been tested using many of the identities known for Li + and relations to known functions as special cases, and by comparison + to `polylog(s, z)` in the `mpmath` arbitrary-precision package in Python. + + http://mpmath.org/doc/current/functions/zeta.html + + The latter shows deviations of the order of + + + 10^{Im(s) - 20} for Im(s) < 0 + + 10^{Im(s) - 20} for Im(s) > 0 + + It isn't clear whether we can do better than this with + double-precision arithmetic. + +## Arguments +* `s::Complex`: the 'fractional' coefficient +* `z::Complex`: the point at which to calculate it +* `accuracy::Real=1.0e-18`: nominal accuracy of calculation, but mainly useful for testing + +## Examples +```jldoctest +julia> Li(-1.0, 0.0) +(0.0,1) +``` +""" +function Li(s::Number, z::Number, accuracy::Real=1.0e-18) + T = 0.5 # threshold at which we change algorithms + if z ≈ 1.0 + if real(s) > 1 + return zeta(s) + else + return Inf + end + elseif z ≈ -1.0 + return -eta(s) + elseif s ≈ 0.0 + return z ./ (1-z) + elseif abs(z) <= T + return Li_direct(s, z, accuracy) + elseif abs(z) >= 1/T && isinteger(s) && real(s) < 0 + # use inversion formula to calculate in terms of Li_n(1/z) + # but note for negative integer s, it collapses to something small + return -(-1.0)^s .*Li_direct(s, 1/z, accuracy) + elseif abs(z) >= 1/T + # use inversion formula to calculate in terms of Li_s(1/z) + G = (2*pi*im).^s .* zeta( 1-s, 0.5 - log(-1/complex(z))/(2*pi*im) ) ./ gamma(s) + F = complex(-1.0)^s .* Li_direct(s, 1/z, accuracy) + A = 2*pi*im*log(complex(z))^(s-1)/(gamma(s)) + if ( isreal(z) && real(z)>=1 ) + Θ = 1 + else + Θ = 0 + end + # println("G = $G, F=$F, Θ=$Θ, A=$A") + return ( G - F - Θ*A ) + else + # power series around mu=0, for z = e^mu + Li_series_mu(s, z, accuracy) + end +end + + +""" + Polylog(s, z) + + This is just an alias of Li(s, z) + +""" +polylog(s::Number, z::Number ) = Li(s, z) + + +#################################################################### +#### these are component functions and aren't exported at this point + +# Dirichlet beta function, for testing results +# https://en.wikipedia.org/wiki/Dirichlet_beta_function +function Dbeta(s::Number) + β = 4.0^(-s) * ( zeta(s,0.25) - zeta(s,0.75) ) +end + +function Li_direct(s::Number, z::Number, accuracy=1.0e-18) + # calculate using direct definition + if abs(z) > 1 || ( abs(z) ≈ 1 && real(s) <= 2) + warn("Should have |z| < 1 or (|z|==1 and Re(s)>2)") + throw(DomainError) + end + if abs(z) > 1/2 + warn("Slow convergence for |z| > 1/2") + end + total = 0.0 + L = ceil(-log10(accuracy)*log2(10)) # summation limit from Crandall, which is conservative + for n=1:L + a = z^n / n^s + total += a + # println(" total = $total") + end + return total +end + +function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) + # calculate using power series around μ = log(z) = 0 + μ = log(complex(z)) + # println("μ = $μ") + if abs(μ) > 2*pi + warn("Should have |μ| < 2*pi, but |μ| = $(abs(μ))") + throw(DomainError) + end + L = Int(ceil(-log10(accuracy)*log2(10))) # summation limit from Crandall, which is conservative + if isinteger(s) + n = Int(round(s)) + if n>0 + # Crandall's 1.4 for s integer + total = μ^(n-1)*(harmonic(n-1) - log(-μ))/gamma(n) + # println(" μ=$μ, total = $total") + for m=0:L + if n - m != 1 + total += μ^m * zeta(n - m) / gamma(m+1) + end + # println(" m=$m, total = $total") + end + # println(" μ=$μ, total = $total") + A = 2*pi*im*log(complex(z))^(s-1)/gamma(n) + if isreal(z) && real(z)>=1 + total -= A + end + # println(" μ=$μ, total = $total") + elseif n==0 + total = z ./ (1-z) + elseif n<0 + # Crandall's 1.5 for s integer + total = factorial(-n) * (-μ)^(n-1) + for k=0:L + total -= μ^k * bernoulli(k-n+1, 0.0) / ( gamma(k+1)*(k-n+1) ) + end + else + error("Should not get this case") + end + else + # equivalent of Crandalls 1.4 for s non-integer + total = gamma(1-s) * (-μ)^(s-1) + for k=0:L + total += μ^k * zeta(s-k)/factorial(Float64(k)) + end + A = 2*pi*im*log(complex(z))^(s-1)/(gamma(s)) + if isreal(z) && real(z)>=1 + total -= A + end + end + return total +end diff --git a/test/bernoulli_test.jl b/test/bernoulli_test.jl new file mode 100644 index 00000000..3be85b1e --- /dev/null +++ b/test/bernoulli_test.jl @@ -0,0 +1,55 @@ +using SpecialFunctions +using Base.Test + +const SF = SpecialFunctions + +@testset "Bernoulli polynomials and numbers" begin + @testset " consistency" begin + for i=0:34 + @test Float64(SF.bernoulli(i)) ≈ SF.bernoulli(i, 0.0) + end + end + + @testset " numbers explicit cases" begin + for i=2:34 + # first two values of zeta don't work + @test Float64(SF.bernoulli(i)) ≈ -i*SF.zeta( 1 - i ) + end + end + + @testset " polynomials explicit cases" begin + X = collect(-2.0 : 0.1 : 3.0) + for i=1:length(X) + x = X[i] + # println(" x = $x") + @test SF.bernoulli(0, x) ≈ 1.0 + @test SF.bernoulli(1, x) ≈ x - 0.5 + @test SF.bernoulli(2, x) ≈ x^2 - x + 1.0/6.0 + @test SF.bernoulli(3, x) ≈ x^3 - 1.5*x^2 + 0.5*x + @test SF.bernoulli(4, x) ≈ x^4 - 2.0*x^3 + x^2 - 1/30.0 + @test SF.bernoulli(5, x) ≈ x^5 - 2.5*x^4 + (5.0/3.0)*x^3 - x/6.0 + @test SF.bernoulli(6, x) ≈ x^6 - 3.0*x^5 + (5.0/2.0)*x^4 - 0.5*x^2 + 1/42.0 + end + end + + @testset " symmetries" begin + X = collect(0.0: 0.1 : 1.0) + for i=1:length(X) + x = X[i] + for n=1:9 + # println(" n = $n, x = $x") + # start to get errors of order 1.0e-14 for n=5, 1.0e-13 around n=10, ... + @test abs( SF.bernoulli(n, 1-x) - (-1.0)^n * SF.bernoulli(n, x) ) < 1.0e-13 + @test abs( SF.bernoulli(n, x+1) - SF.bernoulli(n, x) - n*x^(n-1) ) < 1.0e-13 + @test abs( (-1)^n * SF.bernoulli(n, -x) - SF.bernoulli(n, x) - n*x^(n-1) ) < 1.0e-13 + + # Raabe (1851) + m = 6.0 + k = 0:m-1 + @test abs( sum( SF.bernoulli.(n, x + k/m ) )/m - m^(-n)*SF.bernoulli(n, m*x ) ) < 1.0e-13 + + end + end + end +end + diff --git a/test/harmonic_test.jl b/test/harmonic_test.jl new file mode 100644 index 00000000..3a196a85 --- /dev/null +++ b/test/harmonic_test.jl @@ -0,0 +1,37 @@ +using SpecialFunctions +using Base.Test + +const SF = SpecialFunctions + +@testset "Harmonic numbers" begin + @testset " basics" begin + @test SF.harmonic(1) ≈ 1.0 + @test SF.harmonic(2) ≈ 1.5 + @test SF.harmonic(3) ≈ 11.0/6.0 + @test SF.harmonic(4) ≈ 25.0/12.0 + @test SF.harmonic(5) ≈ 137.0/60.0 + end + + @testset " identities" begin + for n=4:10 + @test SF.harmonic(n) ≈ 1.0/n + SF.harmonic(n-1) + end + + for n=20:30 + @test SF.harmonic(n) ≈ sum( 1.0./(1:n) ) + end + end + + + @testset "Generalized harmonic numbers" begin + for n=1:10 + @test SF.harmonic(n,1.0) ≈ SF.harmonic(n) + @test SF.harmonic(n,0.0) ≈ n + + for r = 3:2:9 + @test SF.harmonic(n,r) ≈ Float64(n)^(-r) + SF.polygamma(r-1,n)/gamma(r) + SF.zeta(r) + end + end + end + +end diff --git a/test/li_test.jl b/test/li_test.jl new file mode 100644 index 00000000..ee975d89 --- /dev/null +++ b/test/li_test.jl @@ -0,0 +1,144 @@ +using Base.Test + +include("li.jl") +ϕ = (sqrt(5)+1)/2.0 + +@testset "Check different input formats" begin + @test Li(-1.0, 0.0) ≈ 0.0 + @test Li(-1, 0.0) ≈ 0.0 + @test Li(-1.0, Complex(0.0)) ≈ 0.0 + 0.0im # don't seem to need imaginary bit here + @test Li(Complex(-1.0), 0.0) ≈ 0.0 + 0.0im + @test Li(Complex(-1.0), 0.0) ≈ 0.0 + @test Li(1, 0) ≈ 0.0 + x = collect(0.0:0.1:0.9) + @test all([Li.(1, x)[i] ≈ Li(1, x[i]) for i=1:length(x)]) + @test Li(Complex(-1.0), Complex(0.3)) ≈ Li(-1.0, 0.3) +end + +@testset "Test s = n (a real integer)" begin + # simple case + @test Li(1, 0.5) ≈ log(2) + + @testset " dilogarithm for real z" begin + @test Li(2,-1.0) ≈ -pi^2/12.0 + @test Li(2, 0.0) ≈ 0.0 + @test Li(2, 0.5) ≈ pi^2/12 - 0.5*log(2)^2 + @test Li(2, 1.0) ≈ pi^2/6.0 + @test Li(2, 2.0) ≈ pi^2/4 - im*pi*log(2) + @test Li(2, -ϕ) ≈ -pi^2/10 - log(ϕ)^2 + @test Li(2, -1/ϕ) ≈ -pi^2/15 + log(ϕ)^2/2 + @test Li(2, 1/ϕ^2) ≈ pi^2/15 - log(ϕ)^2 + @test Li(2, 1/ϕ) ≈ pi^2/10 - log(ϕ)^2 + # wiki has this one, but no ref: + @test Li(2, ϕ) ≈ 11*pi^2/15 + log(Complex(-1/ϕ))^2/2 + @test Li(2, ϕ^2) ≈ -11*pi^2/15 - log(Complex(-ϕ))^2 + + # identities + Z = [3.0 + 0.4im, -3.0 + 0.4im, 3.0 - 0.4im, -3.0 + -0.4im] + for i=1:length(Z) + z = Z[i] + @test Li(2, z) + Li(2, 1/z) ≈ -pi^2/6.0 - log(Complex(-z))^2/2.0 + end + end + + @testset " trilogarithm for real z" begin + @test Li(3,-1.0) ≈ -3*zeta(3)/4 + @test Li(3, 0.0) ≈ 0.0 + @test Li(3, 0.5) ≈ log(2)^3/6.0 - pi^2*log(2)/12.0 + (7.0/8.0)*zeta(3) + @test Li(3, 1.0) ≈ zeta(3) + @test Li(3, ϕ^(-2)) ≈ 4*zeta(3)/5 + 2*log(ϕ)^3/3 - 2*pi^2*log(ϕ)/15 + + # identities + z = 3.0 + 0.4im # test for random complex z + z = 1.5 # test for random complex z + # @test Li(3, z) + Li(3, 1-z) + Li(3, 1 - 1/z) ≈ zeta(3) + log(z)^3/6 + pi^2*log(z)/6 - 0.5*log(z)^2*log(complex(1-z)) + # @test Li(3, z) - Li(3, -z) ≈ 0.25 * Li(3, z^2) + end + + @testset " general case for real z" begin + # X = collect(-0.95:0.1:0.95) + X = collect(-3.0:0.1:3.0) + for i=1:length(X) + x = X[i] + # println(x) + @test Li(1, x) ≈ -log(Complex(1-x)) + @test Li(0, x) ≈ x ./ (1-x) + @test Li(-1, x) ≈ x ./ (1-x).^2 + @test Li(-2, x) ≈ x .* (1+x) ./ (1-x).^3 + @test Li(-3, x) ≈ x .* (1+4*x+x.^2) ./ (1-x).^4 + @test Li(-4, x) ≈ x .* (1+x) .* (1+10*x+x.^2) ./ (1-x).^5 + end + end + + @testset " general case for complex z" begin + X = collect(-3.0:0.5:3.0) + Y = [-1.3, -0.4, 0.4, 1.5] + for i=1:length(X) + for j=1:length(Y) + z = Complex(X[i], Y[j]) + # println(z) + @test Li(1, z) ≈ -log(Complex(1-z)) + @test Li(0, z) ≈ z ./ (1-z) + @test Li(-1, z) ≈ z ./ (1-z).^2 + @test Li(-2, z) ≈ z .* (1+z) ./ (1-z).^3 + @test Li(-3, z) ≈ z .* (1+4*z+z.^2) ./ (1-z).^4 + @test Li(-4, z) ≈ z .* (1+z) .* (1+10*z+z.^2) ./ (1-z).^5 + end + end + end +end + +@testset "Particular values |z| == 1" begin + S_r = [2.1 2.5 3.0] + S_i = [-1.3, -1.0, -0.5, 0.0, 0.5, 1.0, 1.3] + for i=1:length(S_r) + for j=1:length(S_i) + s = Complex(S_r[i], S_i[j]) + @test Li(s, -1.0) ≈ -eta(s) + @test Li(s, 1.0) ≈ zeta(s) + @test Li(s, im) ≈ - 2.0.^(-s).*eta.(s) + im*Dbeta.(s) + end + end +end + +@testset "Additional Identities" begin + z = 0.5 + for n=1:5 + @test Li(-n,z) + (-1)^n * Li(-n, 1/z) ≈ 0.0 + end + + # for real s, and real z<1, Li should be real + S = [-1, 0.1, 2] + Z = [-2, -1.0, 0.1, 0.95] + for i=1:length(S) + for j=1:length(Z) + s = S[i] + z = Z[j] + # println("s = $s; z = $z") + @test abs( imag( Li(s, z) ) ) < 1.0e-14 + end + end + + # for real s, and real z>=1, the imaginary part is give + S = [-1.5, 0.1, 2] + Z = [1.05, 3.0] + for i=1:length(S) + for j=1:length(Z) + s = S[i] + z = Z[j] + # println("s = $s; z = $z") + μ = log(z) + @test imag( Li(s, z) ) ≈ -pi*μ^(s-1)/gamma(s) + end + end + + S_r = [2.1 2.5 3.0] + S_i = [-1.3, -1.0, -0.5, 0.0, 0.5, 1.0, 1.3] + z = 3.0 - 0.1im + for i=1:length(S_r) + for j=1:length(S_i) + s = Complex(S_r[i], S_i[j]) + @test Li(s,z) + Li(s,-z) ≈ complex(2)^(1-s) * Li(s, z^2) + end + end +end From a3e193a0bd99989f2cd577206a9c2ca396f93851 Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Thu, 1 Jun 2017 13:44:12 +0930 Subject: [PATCH 02/14] added bernoulli, harmonic, and polylog functions/numbers --- examples/li_ex.jl | 49 +++++---- examples/li_ex.pdf | Bin 0 -> 9125 bytes examples/li_ex2.jl | 35 +++++++ examples/li_ex2.jl~ | 38 +++++++ examples/li_ex2.pdf | Bin 0 -> 190879 bytes src/gamma.jl | 9 ++ src/li.jl | 3 +- test/li_test.jl | 240 ++++++++++++++++++++++---------------------- 8 files changed, 228 insertions(+), 146 deletions(-) create mode 100644 examples/li_ex.pdf create mode 100644 examples/li_ex2.jl create mode 100644 examples/li_ex2.jl~ create mode 100644 examples/li_ex2.pdf diff --git a/examples/li_ex.jl b/examples/li_ex.jl index a1b71b02..be68a1e1 100644 --- a/examples/li_ex.jl +++ b/examples/li_ex.jl @@ -1,29 +1,28 @@ -# do plots of the Li function, similar to those of -# Vepstas, L. (2008). "An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions". Numerical Algorithms. 47 (3): 211–252. arXiv:math.CA/0702243Freely accessible. doi:10.1007/s11075-007-9153-8. - -using PyPlot -include("li.jl") - -# Figure 5 -s = 0.5 + 15*im -step = 0.005 -x = -3.5: step :3.5 -y = -3.5: step :3.5 - -L = zeros(Complex, (length(x), length(y)) ) -for i=1:length(x) - for j=1:length(y) - # println("z = $z") - z = x[i]+im*y[j] - L[i,j] = Li(s,z) - end +# do plots of the Li function, + + + +using Plots; plotlyjs() +default(show = true) # shouldn't need display commands +theme(:solarized_light) + +using SpecialFunctions +const SF = SpecialFunctions + +# replicate (approximately) the plot from http://mathworld.wolfram.com/Polylogarithm.html +x = -2.0 : 0.01 : 1.0 +plt = plot( x, real(SF.Li.(-2,x)), + label="Li_{-2}(x)", legendfont=font(20, "Courier"), legend=:bottomright, + xaxis = ("x", font(20, "Courier")), + yaxis = ("", font(20, "Courier"), (-1, 1), -1.0:0.25:1.0), + size = (1200, 800) + ) + +for i=-1:2 + plot!( x, real(SF.Li.(i,x)), label="Li_{$i}(x)") 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(-sign(real(L[i]))+1.0)/2.0 +j = abs(L) .< 1.0e-8 +a[j] = NaN +heatmap(xs, ys, a, + xaxis = ("real", font(20, "Courier")), + yaxis = ("imag", font(20, "Courier")), + size = (800, 800), + color = :Spectral) + +sleep(1) +savefig("li_ex2.pdf") diff --git a/examples/li_ex2.jl~ b/examples/li_ex2.jl~ new file mode 100644 index 00000000..8f7bab99 --- /dev/null +++ b/examples/li_ex2.jl~ @@ -0,0 +1,38 @@ +# do plots of the Li function, + + + +using Plots; plotlyjs() +default(show = true) # shouldn't need display commands + +using SpecialFunctions +const SF = SpecialFunctions + +# replicate (approximately) the plot from http://mathworld.wolfram.com/Polylogarithm.html +x = -2.0 : 0.01 : 1.0 +plt = plot( x, real(SF.Li.(-2,x)), + label="Li_{-2}(x)", legendfont=font(20, "Courier"), legend=:bottomright, + xaxis = ("x", font(20, "Courier")), + yaxis = ("", font(20, "Courier"), (-1, 1), -1.0:0.25:1.0), + size = (1200, 800) + ) + +for i=-1:2 + plot!( x, real(SF.Li.(i,x)), label="Li_{$i}(x)") +end + 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z?xwsf1x)EQSAQ2US@xe-Td_zmYsoo@3|eP2H+T&2Xi%#?u0T5FhyWUeAqs%H;Nu5{ z$KgO$d;|r@00NPS2PYW%puCr3XfzfefRczahDJlK6=w_!;6PYb#c+5KN%j~(z+xfz zIAcTt8XOX5OdBA8)5#tK&joaaIb(n}fk;^0hc z9cMW#dTm`S_+El)Syx*NzY2SRKp^7RlmjtX z(+3f=1{VSm3uS9__CdsHuf~)JBE7m_2vRnz7<4k|q|#xPY(YIvfvzj)d0|eYF~GEL lSUN%b&{GKo-!JeT=*J+_8Ss?|5YZR_5ZJL}-yuVRe*;s+$YcNj literal 0 HcmV?d00001 diff --git a/src/gamma.jl b/src/gamma.jl index 116a1fce..157b2205 100644 --- a/src/gamma.jl +++ b/src/gamma.jl @@ -317,6 +317,15 @@ function zeta(s::ComplexOrReal{Float64}, z::ComplexOrReal{Float64}) return ζ end + +""" + hurwitz_zeta(s, z) + + An alias for zeta(s, z) +""" +hurwitz_zeta(s::ComplexOrReal{Float64}, z::ComplexOrReal{Float64}) = zeta(s, z) + + """ polygamma(m, x) diff --git a/src/li.jl b/src/li.jl index af3765f7..c5ef356e 100644 --- a/src/li.jl +++ b/src/li.jl @@ -31,7 +31,7 @@ similar to Maximon Of these the only one that actually specifies a full algorithm is - Crandall, and it also treats special cases more carefully, so this + Crandall, and he also treats special cases more carefully, so this is the one that I have paid most attention to. However, extending it for s on the whole complex plane requires some additions, and many of these are actually most nicely documented on the wikipedia page @@ -128,6 +128,7 @@ polylog(s::Number, z::Number ) = Li(s, z) # Dirichlet beta function, for testing results # https://en.wikipedia.org/wiki/Dirichlet_beta_function +# but don't export as it would create confusion with Euler integral function Dbeta(s::Number) β = 4.0^(-s) * ( zeta(s,0.25) - zeta(s,0.75) ) end diff --git a/test/li_test.jl b/test/li_test.jl index ee975d89..3e55e4ed 100644 --- a/test/li_test.jl +++ b/test/li_test.jl @@ -1,144 +1,144 @@ +using SpecialFunctions using Base.Test -include("li.jl") +const SF = SpecialFunctions + + +# golden ratio is used in some tests ϕ = (sqrt(5)+1)/2.0 -@testset "Check different input formats" begin - @test Li(-1.0, 0.0) ≈ 0.0 - @test Li(-1, 0.0) ≈ 0.0 - @test Li(-1.0, Complex(0.0)) ≈ 0.0 + 0.0im # don't seem to need imaginary bit here - @test Li(Complex(-1.0), 0.0) ≈ 0.0 + 0.0im - @test Li(Complex(-1.0), 0.0) ≈ 0.0 - @test Li(1, 0) ≈ 0.0 - x = collect(0.0:0.1:0.9) - @test all([Li.(1, x)[i] ≈ Li(1, x[i]) for i=1:length(x)]) - @test Li(Complex(-1.0), Complex(0.3)) ≈ Li(-1.0, 0.3) -end +@testset "Polylogarithm Li" begin + @testset "Check different input formats" begin + @test SF.Li(-1.0, 0.0) ≈ 0.0 + @test SF.Li(-1, 0.0) ≈ 0.0 + @test SF.Li(-1.0, Complex(0.0)) ≈ 0.0 + 0.0im # don't seem to need imaginary bit here + @test SF.Li(Complex(-1.0), 0.0) ≈ 0.0 + 0.0im + @test SF.Li(Complex(-1.0), 0.0) ≈ 0.0 + @test SF.Li(1, 0) ≈ 0.0 + x = collect(0.0:0.1:0.9) + @test all([SF.Li.(1, x)[i] ≈ SF.Li(1, x[i]) for i=1:length(x)]) + @test SF.Li(Complex(-1.0), Complex(0.3)) ≈ SF.Li(-1.0, 0.3) + @test SF.polylog(Complex(-1.0), Complex(0.3)) ≈ SF.Li(-1.0, 0.3) + end -@testset "Test s = n (a real integer)" begin - # simple case - @test Li(1, 0.5) ≈ log(2) + @testset "Test s = n (a real integer)" begin + # simple case + @test SF.Li(1, 0.5) ≈ log(2) - @testset " dilogarithm for real z" begin - @test Li(2,-1.0) ≈ -pi^2/12.0 - @test Li(2, 0.0) ≈ 0.0 - @test Li(2, 0.5) ≈ pi^2/12 - 0.5*log(2)^2 - @test Li(2, 1.0) ≈ pi^2/6.0 - @test Li(2, 2.0) ≈ pi^2/4 - im*pi*log(2) - @test Li(2, -ϕ) ≈ -pi^2/10 - log(ϕ)^2 - @test Li(2, -1/ϕ) ≈ -pi^2/15 + log(ϕ)^2/2 - @test Li(2, 1/ϕ^2) ≈ pi^2/15 - log(ϕ)^2 - @test Li(2, 1/ϕ) ≈ pi^2/10 - log(ϕ)^2 - # wiki has this one, but no ref: - @test Li(2, ϕ) ≈ 11*pi^2/15 + log(Complex(-1/ϕ))^2/2 - @test Li(2, ϕ^2) ≈ -11*pi^2/15 - log(Complex(-ϕ))^2 + @testset " dilogarithm for real z" begin + @test SF.Li(2,-1.0) ≈ -pi^2/12.0 + @test SF.Li(2, 0.0) ≈ 0.0 + @test SF.Li(2, 0.5) ≈ pi^2/12 - 0.5*log(2)^2 + @test SF.Li(2, 1.0) ≈ pi^2/6.0 + @test SF.Li(2, 2.0) ≈ pi^2/4 - im*pi*log(2) + @test SF.Li(2, -ϕ) ≈ -pi^2/10 - log(ϕ)^2 + @test SF.Li(2, -1/ϕ) ≈ -pi^2/15 + log(ϕ)^2/2 + @test SF.Li(2, 1/ϕ^2) ≈ pi^2/15 - log(ϕ)^2 + @test SF.Li(2, 1/ϕ) ≈ pi^2/10 - log(ϕ)^2 + @test SF.Li(2, ϕ) ≈ 11*pi^2/15 + log(Complex(-1/ϕ))^2/2 # wiki has this one, but no ref + @test SF.Li(2, ϕ^2) ≈ -11*pi^2/15 - log(Complex(-ϕ))^2 - # identities - Z = [3.0 + 0.4im, -3.0 + 0.4im, 3.0 - 0.4im, -3.0 + -0.4im] - for i=1:length(Z) - z = Z[i] - @test Li(2, z) + Li(2, 1/z) ≈ -pi^2/6.0 - log(Complex(-z))^2/2.0 + # identities + Z = [3.0 + 0.4im, -3.0 + 0.4im, 3.0 - 0.4im, -3.0 + -0.4im] + for i=1:length(Z) + z = Z[i] + @test SF.Li(2, z) + SF.Li(2, 1/z) ≈ -pi^2/6.0 - log(Complex(-z))^2/2.0 + end end - end - @testset " trilogarithm for real z" begin - @test Li(3,-1.0) ≈ -3*zeta(3)/4 - @test Li(3, 0.0) ≈ 0.0 - @test Li(3, 0.5) ≈ log(2)^3/6.0 - pi^2*log(2)/12.0 + (7.0/8.0)*zeta(3) - @test Li(3, 1.0) ≈ zeta(3) - @test Li(3, ϕ^(-2)) ≈ 4*zeta(3)/5 + 2*log(ϕ)^3/3 - 2*pi^2*log(ϕ)/15 - - # identities - z = 3.0 + 0.4im # test for random complex z - z = 1.5 # test for random complex z - # @test Li(3, z) + Li(3, 1-z) + Li(3, 1 - 1/z) ≈ zeta(3) + log(z)^3/6 + pi^2*log(z)/6 - 0.5*log(z)^2*log(complex(1-z)) - # @test Li(3, z) - Li(3, -z) ≈ 0.25 * Li(3, z^2) - end - - @testset " general case for real z" begin - # X = collect(-0.95:0.1:0.95) - X = collect(-3.0:0.1:3.0) - for i=1:length(X) - x = X[i] - # println(x) - @test Li(1, x) ≈ -log(Complex(1-x)) - @test Li(0, x) ≈ x ./ (1-x) - @test Li(-1, x) ≈ x ./ (1-x).^2 - @test Li(-2, x) ≈ x .* (1+x) ./ (1-x).^3 - @test Li(-3, x) ≈ x .* (1+4*x+x.^2) ./ (1-x).^4 - @test Li(-4, x) ≈ x .* (1+x) .* (1+10*x+x.^2) ./ (1-x).^5 + @testset " trilogarithm for real z" begin + @test SF.Li(3,-1.0) ≈ -3*SF.zeta(3)/4 + @test SF.Li(3, 0.0) ≈ 0.0 + @test SF.Li(3, 0.5) ≈ log(2)^3/6.0 - pi^2*log(2)/12.0 + (7.0/8.0)*SF.zeta(3) + @test SF.Li(3, 1.0) ≈ SF.zeta(3) + @test SF.Li(3, ϕ^(-2)) ≈ 4*SF.zeta(3)/5 + 2*log(ϕ)^3/3 - 2*pi^2*log(ϕ)/15 end - end - - @testset " general case for complex z" begin - X = collect(-3.0:0.5:3.0) - Y = [-1.3, -0.4, 0.4, 1.5] - for i=1:length(X) - for j=1:length(Y) - z = Complex(X[i], Y[j]) - # println(z) - @test Li(1, z) ≈ -log(Complex(1-z)) - @test Li(0, z) ≈ z ./ (1-z) - @test Li(-1, z) ≈ z ./ (1-z).^2 - @test Li(-2, z) ≈ z .* (1+z) ./ (1-z).^3 - @test Li(-3, z) ≈ z .* (1+4*z+z.^2) ./ (1-z).^4 - @test Li(-4, z) ≈ z .* (1+z) .* (1+10*z+z.^2) ./ (1-z).^5 + + @testset " general case for real z" begin + # X = collect(-0.95:0.1:0.95) + X = collect(-3.0:0.1:3.0) + for i=1:length(X) + x = X[i] + # println(x) + @test SF.Li(1, x) ≈ -log(Complex(1-x)) + @test SF.Li(0, x) ≈ x ./ (1-x) + @test SF.Li(-1, x) ≈ x ./ (1-x).^2 + @test SF.Li(-2, x) ≈ x .* (1+x) ./ (1-x).^3 + @test SF.Li(-3, x) ≈ x .* (1+4*x+x.^2) ./ (1-x).^4 + @test SF.Li(-4, x) ≈ x .* (1+x) .* (1+10*x+x.^2) ./ (1-x).^5 end end - end -end + + @testset " general case for complex z" begin + X = collect(-3.0:0.5:3.0) + Y = [-1.3, -0.4, 0.4, 1.5] + for i=1:length(X) + for j=1:length(Y) + z = Complex(X[i], Y[j]) + # println(z) + @test SF.Li(1, z) ≈ -log(Complex(1-z)) + @test SF.Li(0, z) ≈ z ./ (1-z) + @test SF.Li(-1, z) ≈ z ./ (1-z).^2 + @test SF.Li(-2, z) ≈ z .* (1+z) ./ (1-z).^3 + @test SF.Li(-3, z) ≈ z .* (1+4*z+z.^2) ./ (1-z).^4 + @test SF.Li(-4, z) ≈ z .* (1+z) .* (1+10*z+z.^2) ./ (1-z).^5 + end + end + end + end -@testset "Particular values |z| == 1" begin - S_r = [2.1 2.5 3.0] - S_i = [-1.3, -1.0, -0.5, 0.0, 0.5, 1.0, 1.3] - for i=1:length(S_r) - for j=1:length(S_i) - s = Complex(S_r[i], S_i[j]) - @test Li(s, -1.0) ≈ -eta(s) - @test Li(s, 1.0) ≈ zeta(s) - @test Li(s, im) ≈ - 2.0.^(-s).*eta.(s) + im*Dbeta.(s) + @testset "Particular values |z| == 1" begin + S_r = [2.1 2.5 3.0] + S_i = [-1.3, -1.0, -0.5, 0.0, 0.5, 1.0, 1.3] + for i=1:length(S_r) + for j=1:length(S_i) + s = Complex(S_r[i], S_i[j]) + @test SF.Li(s, -1.0) ≈ -SF.eta(s) + @test SF.Li(s, 1.0) ≈ SF.zeta(s) + @test SF.Li(s, im) ≈ - 2.0.^(-s).*SF.eta.(s) + im*SF.Dbeta.(s) + end end end -end -@testset "Additional Identities" begin - z = 0.5 - for n=1:5 - @test Li(-n,z) + (-1)^n * Li(-n, 1/z) ≈ 0.0 - end + @testset "Additional Identities" begin + z = 0.5 + for n=1:5 + @test SF.Li(-n,z) + (-1)^n * SF.Li(-n, 1/z) ≈ 0.0 + end - # for real s, and real z<1, Li should be real - S = [-1, 0.1, 2] - Z = [-2, -1.0, 0.1, 0.95] - for i=1:length(S) - for j=1:length(Z) - s = S[i] - z = Z[j] - # println("s = $s; z = $z") - @test abs( imag( Li(s, z) ) ) < 1.0e-14 + # for real s, and real z<1, SF.Li should be real + S = [-1, 0.1, 2] + Z = [-2, -1.0, 0.1, 0.95] + for i=1:length(S) + for j=1:length(Z) + s = S[i] + z = Z[j] + # println("s = $s; z = $z") + @test abs( imag( SF.Li(s, z) ) ) < 1.0e-14 + end end - end - # for real s, and real z>=1, the imaginary part is give - S = [-1.5, 0.1, 2] - Z = [1.05, 3.0] - for i=1:length(S) - for j=1:length(Z) - s = S[i] - z = Z[j] - # println("s = $s; z = $z") - μ = log(z) - @test imag( Li(s, z) ) ≈ -pi*μ^(s-1)/gamma(s) + # for real s, and real z>=1, the imaginary part is give + S = [-1.5, 0.1, 2] + Z = [1.05, 3.0] + for i=1:length(S) + for j=1:length(Z) + s = S[i] + z = Z[j] + # println("s = $s; z = $z") + μ = log(z) + @test imag( SF.Li(s, z) ) ≈ -pi*μ^(s-1)/gamma(s) + end end - end - S_r = [2.1 2.5 3.0] - S_i = [-1.3, -1.0, -0.5, 0.0, 0.5, 1.0, 1.3] - z = 3.0 - 0.1im - for i=1:length(S_r) - for j=1:length(S_i) - s = Complex(S_r[i], S_i[j]) - @test Li(s,z) + Li(s,-z) ≈ complex(2)^(1-s) * Li(s, z^2) + S_r = [2.1 2.5 3.0] + S_i = [-1.3, -1.0, -0.5, 0.0, 0.5, 1.0, 1.3] + z = 3.0 - 0.1im + for i=1:length(S_r) + for j=1:length(S_i) + s = Complex(S_r[i], S_i[j]) + @test SF.Li(s,z) + SF.Li(s,-z) ≈ complex(2)^(1-s) * SF.Li(s, z^2) + end end end end From b69a9f6a8e499f5fcbf03f0403296d70459820fe Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Thu, 1 Jun 2017 16:57:32 +0930 Subject: [PATCH 03/14] corrected subscripts in plots of L_n(x) --- examples/li_ex.jl | 22 +++++++++++++++------- examples/li_ex.pdf | Bin 9125 -> 12254 bytes 2 files changed, 15 insertions(+), 7 deletions(-) diff --git a/examples/li_ex.jl b/examples/li_ex.jl index be68a1e1..00238ca6 100644 --- a/examples/li_ex.jl +++ b/examples/li_ex.jl @@ -2,27 +2,35 @@ -using Plots; plotlyjs() +using Plots +plotlyjs() + default(show = true) # shouldn't need display commands theme(:solarized_light) using SpecialFunctions const SF = SpecialFunctions +using LaTeXStrings + # replicate (approximately) the plot from http://mathworld.wolfram.com/Polylogarithm.html x = -2.0 : 0.01 : 1.0 plt = plot( x, real(SF.Li.(-2,x)), - label="Li_{-2}(x)", legendfont=font(20, "Courier"), legend=:bottomright, + label="Li$(Char(8331))$(Char(8322))(x)", + legendfont=font(20, "Courier"), legend=:bottomright, xaxis = ("x", font(20, "Courier")), yaxis = ("", font(20, "Courier"), (-1, 1), -1.0:0.25:1.0), size = (1200, 800) - ) - -for i=-1:2 - plot!( x, real(SF.Li.(i,x)), label="Li_{$i}(x)") + ); + +for i=-1:-1 + plot!( x, real(SF.Li.(i,x)), label="Li$(Char(8331))$(Char(8320 - i))(x)"); end +for i=0:2 + plot!( x, real(SF.Li.(i,x)), label="Li$(Char(8320 + i))(x)"); +end +gui() sleep(1) savefig("li_ex.pdf") - diff --git a/examples/li_ex.pdf b/examples/li_ex.pdf index 05966231af2384c950c2165fbb16d396e9c42192..ee38d1cfba7adcfe0c9efc447dd9b1a0559a1b60 100644 GIT binary patch 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z?s>YK+c{df|I^{pwDYr~ Date: Fri, 2 Jun 2017 10:49:26 +0930 Subject: [PATCH 04/14] edits to bernoulli to (i) optimise speed, (ii) get rid on non-compliant warnings, and (iii) make the docs a little more precise --- examples/li_ex.jl | 2 -- src/bernoulli.jl | 44 +++++++++++++++++++++++--------------------- 2 files changed, 23 insertions(+), 23 deletions(-) diff --git a/examples/li_ex.jl b/examples/li_ex.jl index 00238ca6..f88b9333 100644 --- a/examples/li_ex.jl +++ b/examples/li_ex.jl @@ -1,7 +1,5 @@ # do plots of the Li function, - - using Plots plotlyjs() diff --git a/src/bernoulli.jl b/src/bernoulli.jl index ab46c3f0..7d877c1d 100644 --- a/src/bernoulli.jl +++ b/src/bernoulli.jl @@ -1,17 +1,10 @@ # # Bernoulli numbers (as rationals up to B_{34}) and Bernoulli polynomials -# The alg. for the latter could use some optimisation, as -# (1) the choice of domains for the different approaches was a bit arbitrary -# (2) the errors start to creep up for larger n, looks roughly like 1 order per increase in n of 5 -# """ bernoulli(n) - created: Tue May 23 2017 - email: matthew.roughan@adelaide.edu.au - - Calculates (first-kind or NIST) Bernoulli numbers, B_n (or at least the first 35 of them) + Calculates the first 34 Bernoulli numbers B_n (of the first-kind or NIST type) e.g., see + http://mathworld.wolfram.com/BernoulliNumber.html @@ -21,7 +14,10 @@ N.B. Bernoulli numbers of second kind only seem to differ in that B_1 = + 1/2 (instead of -1/2) ## Arguments -* `n::Integer`: the index into the series, n=0,1,2,3,... +* `n::Integer`: the index into the series, n=0,1,2,3,...,34 (for larger n use ``bernoulli(n,0.0)`` ) + + We only provide the 1st 34 as beyond this, we can't return Int64 rationals, so best to compute + the real approximation using ``bernoulli(n,0.0)`` ## Examples ```jldoctest @@ -33,11 +29,10 @@ function bernoulli(n::Int) # this just does a lookup -- seemed like it would be easier to code and faster # for the size of numbers I am working with if n<0 - warn("n should be > 0") throw(DomainError) end if n>34 - warn("If n > 34, then the numerator needs Int128 at least, and worse, so this code is not the code you want. Try using bernoulli(n, 0.0) to get a floating point approximation to the result.") + # warn("If n > 34, then the numerator needs Int128 at least, and worse, so this code is not the code you want. Try using bernoulli(n, 0.0) to get a floating point approximation to the result.") throw(DomainError) end @@ -67,8 +62,10 @@ end (c) M Roughan, 2017 Calculates Bernoulli polynomials from the Hurwitz-zeta function using - ``ζ(-n,x) = -B_{n+1}(x)/(n+1), for Re(x)>0 `` - Its probably not the fastest approach, but we get it almost for free. + +```ζ(-n,x) = -B_{n+1}(x)/(n+1), for Re(x)>0,``` + + which is faster than direct calculation of the polynomial except for n<=4. e.g., see @@ -87,7 +84,6 @@ julia> bernoulli(6, 1.2) """ function bernoulli(n::Int, x::Real) if n<0 - warn("n should be > 0") throw(DomainError) end if n == 0 @@ -100,15 +96,21 @@ function bernoulli(n::Int, x::Real) return x^3 - 1.5*x^2 + 0.5*x elseif n == 4 return x^4 - 2.0*x^3 + x^2 - 1/30.0 - elseif n == 5 - return x^5 - 2.5*x^4 +(5.0/3.0)*x^3 - x/6.0 + # elseif n == 5 + # return x^5 - 2.5*x^4 +(5.0/3.0)*x^3 - x/6.0 # this isn't faster than just doing the zeta directly end - if n <= 34 - # direct summation for reasonably small values of coefficients - k = 0:n - return sum( binomial.(n,k) .* bernoulli.(k) .* x.^(n-k) ) - elseif x > 0 + # direct summation is slower than the zeta function below, even for small n + # if n <= 34 + # # direct summation for reasonably small values of coefficients + # total = 0.0 + # for k=0:n + # total += binomial.(n,k) .* bernoulli.(k) .* x.^(n-k) + # end + # return total + # else + + if x > 0 return -n*zeta(1-n, x) else # comments in the gamma.jl note that identity with zeta(s,z) only works for Re(z)>0 From bcad7921383a6009f5a877836c4b2ea2f17877bc Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Fri, 2 Jun 2017 11:23:09 +0930 Subject: [PATCH 05/14] improve speed of harmonic by replaceing vectorized code with loops --- src/bernoulli.jl | 4 ++-- src/harmonic.jl | 14 +++++++++++--- 2 files changed, 13 insertions(+), 5 deletions(-) diff --git a/src/bernoulli.jl b/src/bernoulli.jl index 7d877c1d..1cabff0f 100644 --- a/src/bernoulli.jl +++ b/src/bernoulli.jl @@ -96,8 +96,8 @@ function bernoulli(n::Int, x::Real) return x^3 - 1.5*x^2 + 0.5*x elseif n == 4 return x^4 - 2.0*x^3 + x^2 - 1/30.0 - # elseif n == 5 - # return x^5 - 2.5*x^4 +(5.0/3.0)*x^3 - x/6.0 # this isn't faster than just doing the zeta directly + elseif n == 5 + return x^5 - 2.5*x^4 +(5.0/3.0)*x^3 - x/6.0 end # direct summation is slower than the zeta function below, even for small n diff --git a/src/harmonic.jl b/src/harmonic.jl index eda23aff..c72588c8 100644 --- a/src/harmonic.jl +++ b/src/harmonic.jl @@ -17,7 +17,11 @@ function harmonic(n::Integer) # γ = euler_mascheroni_const = 0.577215664901532860606512090082402431042 # http://oeis.org/A001620 if n <=10 # get exact values for small n - return sum( 1.0./(1:n) ) + total = 0.0 + for k=1:n + total += 1.0 / k + end + return total else # numerical approximation for larger n return γ + digamma(n+1) @@ -35,7 +39,7 @@ end * `n::Integer`: index 1 of the Harmonic number to calculate * `r::Real`: index 2 of the Harmonic number to calculate -It should be possible to extend this to complex r, but hey. +It should be possible to extend this to complex r, but that requires more testing. ## Examples ```jldoctest @@ -44,5 +48,9 @@ julia> harmonic(2,1) ``` """ function harmonic(n::Integer, r::Real) - sum( 1.0 ./ (1:n).^r ) + total = 0.0 + for k=1:n + total += 1.0 / k^r + end + return total end From 3e75940b6330469f8479f61974eefc66ccf37c5c Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Fri, 2 Jun 2017 11:53:47 +0930 Subject: [PATCH 06/14] corrected throws, and added some tests --- src/bernoulli.jl | 10 +++------- src/harmonic.jl | 8 +++++++- src/li.jl | 6 ++---- test/bernoulli_test.jl | 18 ++++++++++++++---- test/harmonic_test.jl | 5 +++++ 5 files changed, 31 insertions(+), 16 deletions(-) diff --git a/src/bernoulli.jl b/src/bernoulli.jl index 1cabff0f..ed1e236f 100644 --- a/src/bernoulli.jl +++ b/src/bernoulli.jl @@ -29,11 +29,11 @@ function bernoulli(n::Int) # this just does a lookup -- seemed like it would be easier to code and faster # for the size of numbers I am working with if n<0 - throw(DomainError) + throw(DomainError()) end if n>34 # warn("If n > 34, then the numerator needs Int128 at least, and worse, so this code is not the code you want. Try using bernoulli(n, 0.0) to get a floating point approximation to the result.") - throw(DomainError) + throw(DomainError()) end # Denominator of Bernoulli number B_n @@ -57,10 +57,6 @@ end """ bernoulli(n, x) - created: Tue May 23 2017 - email: matthew.roughan@adelaide.edu.au - (c) M Roughan, 2017 - Calculates Bernoulli polynomials from the Hurwitz-zeta function using ```ζ(-n,x) = -B_{n+1}(x)/(n+1), for Re(x)>0,``` @@ -84,7 +80,7 @@ julia> bernoulli(6, 1.2) """ function bernoulli(n::Int, x::Real) if n<0 - throw(DomainError) + throw(DomainError()) end if n == 0 return 1 # zeta formula doesn't hold for n=0, so return explicit value diff --git a/src/harmonic.jl b/src/harmonic.jl index c72588c8..213c3569 100644 --- a/src/harmonic.jl +++ b/src/harmonic.jl @@ -15,7 +15,10 @@ julia> harmonic(2) """ function harmonic(n::Integer) # γ = euler_mascheroni_const = 0.577215664901532860606512090082402431042 # http://oeis.org/A001620 - if n <=10 + if n <= 0 + throw(DomainError()) + end + if n <= 10 # get exact values for small n total = 0.0 for k=1:n @@ -49,6 +52,9 @@ julia> harmonic(2,1) """ function harmonic(n::Integer, r::Real) total = 0.0 + if n <= 0 + throw(DomainError()) + end for k=1:n total += 1.0 / k^r end diff --git a/src/li.jl b/src/li.jl index c5ef356e..2d882294 100644 --- a/src/li.jl +++ b/src/li.jl @@ -136,8 +136,7 @@ end function Li_direct(s::Number, z::Number, accuracy=1.0e-18) # calculate using direct definition if abs(z) > 1 || ( abs(z) ≈ 1 && real(s) <= 2) - warn("Should have |z| < 1 or (|z|==1 and Re(s)>2)") - throw(DomainError) + throw(DomainError()) end if abs(z) > 1/2 warn("Slow convergence for |z| > 1/2") @@ -157,8 +156,7 @@ function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) μ = log(complex(z)) # println("μ = $μ") if abs(μ) > 2*pi - warn("Should have |μ| < 2*pi, but |μ| = $(abs(μ))") - throw(DomainError) + throw(DomainError()) end L = Int(ceil(-log10(accuracy)*log2(10))) # summation limit from Crandall, which is conservative if isinteger(s) diff --git a/test/bernoulli_test.jl b/test/bernoulli_test.jl index 3be85b1e..72819162 100644 --- a/test/bernoulli_test.jl +++ b/test/bernoulli_test.jl @@ -3,6 +3,16 @@ using Base.Test const SF = SpecialFunctions +# test functions, see runtests.jl +relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x) +relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x))) +≅(a,b) = relerrc(a,b) ≤ 1e-13 + +# test functions, similar to runtests.jl, but can't do relative error when some values are zero +abserr(z, x) = z == x ? 0.0 : abs(z - x) +abserrc(z, x) = max(abserr(real(z),real(x)), abserr(imag(z),imag(x))) +≒(a,b) = abserrc(a,b) ≤ 1e-13 + @testset "Bernoulli polynomials and numbers" begin @testset " consistency" begin for i=0:34 @@ -39,14 +49,14 @@ const SF = SpecialFunctions for n=1:9 # println(" n = $n, x = $x") # start to get errors of order 1.0e-14 for n=5, 1.0e-13 around n=10, ... - @test abs( SF.bernoulli(n, 1-x) - (-1.0)^n * SF.bernoulli(n, x) ) < 1.0e-13 - @test abs( SF.bernoulli(n, x+1) - SF.bernoulli(n, x) - n*x^(n-1) ) < 1.0e-13 - @test abs( (-1)^n * SF.bernoulli(n, -x) - SF.bernoulli(n, x) - n*x^(n-1) ) < 1.0e-13 + @test SF.bernoulli(n, 1-x) ≒ (-1.0)^n * SF.bernoulli(n, x) + @test SF.bernoulli(n, x+1) ≒ SF.bernoulli(n, x) + n*x^(n-1) + @test (-1)^n * SF.bernoulli(n, -x) ≒ SF.bernoulli(n, x) + n*x^(n-1) # Raabe (1851) m = 6.0 k = 0:m-1 - @test abs( sum( SF.bernoulli.(n, x + k/m ) )/m - m^(-n)*SF.bernoulli(n, m*x ) ) < 1.0e-13 + @test sum( SF.bernoulli.(n, x + k/m ) )/m ≒ m^(-n)*SF.bernoulli(n, m*x ) end end diff --git a/test/harmonic_test.jl b/test/harmonic_test.jl index 3a196a85..a13257e1 100644 --- a/test/harmonic_test.jl +++ b/test/harmonic_test.jl @@ -34,4 +34,9 @@ const SF = SpecialFunctions end end + @testset " errors" begin + @test_throws DomainError SF.harmonic(-1) + @test_throws DomainError SF.harmonic(0) + @test_throws DomainError SF.harmonic(0, 1.2) + end end From 0ba12ab6cb89b1681437dc3141ee702c216dc7f2 Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Fri, 2 Jun 2017 11:54:52 +0930 Subject: [PATCH 07/14] corrected throws, and added some tests --- test/bernoulli_test.jl | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/test/bernoulli_test.jl b/test/bernoulli_test.jl index 72819162..4b72027b 100644 --- a/test/bernoulli_test.jl +++ b/test/bernoulli_test.jl @@ -14,6 +14,11 @@ abserrc(z, x) = max(abserr(real(z),real(x)), abserr(imag(z),imag(x))) ≒(a,b) = abserrc(a,b) ≤ 1e-13 @testset "Bernoulli polynomials and numbers" begin + @testset " errors" begin + @test_throws DomainError SF.bernoulli(-1) + @test_throws DomainError SF.bernoulli(-1, 0.0) + end + @testset " consistency" begin for i=0:34 @test Float64(SF.bernoulli(i)) ≈ SF.bernoulli(i, 0.0) From 01f5dc59d6ab4a62733270ad024551ff0df59b2b Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Fri, 2 Jun 2017 12:14:21 +0930 Subject: [PATCH 08/14] minor changes to Li to add in some special cases --- src/harmonic.jl | 2 +- src/li.jl | 15 ++++++++++----- 2 files changed, 11 insertions(+), 6 deletions(-) diff --git a/src/harmonic.jl b/src/harmonic.jl index 213c3569..75818477 100644 --- a/src/harmonic.jl +++ b/src/harmonic.jl @@ -51,10 +51,10 @@ julia> harmonic(2,1) ``` """ function harmonic(n::Integer, r::Real) - total = 0.0 if n <= 0 throw(DomainError()) end + total = 0.0 for k=1:n total += 1.0 / k^r end diff --git a/src/li.jl b/src/li.jl index 2d882294..77fa1dc6 100644 --- a/src/li.jl +++ b/src/li.jl @@ -142,7 +142,8 @@ function Li_direct(s::Number, z::Number, accuracy=1.0e-18) warn("Slow convergence for |z| > 1/2") end total = 0.0 - L = ceil(-log10(accuracy)*log2(10)) # summation limit from Crandall, which is conservative + L = ceil(-log10(accuracy)*log2(10)) # summation limit from Crandall, which is conservative, but based on real s + tmp = z; for n=1:L a = z^n / n^s total += a @@ -158,10 +159,10 @@ function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) if abs(μ) > 2*pi throw(DomainError()) end - L = Int(ceil(-log10(accuracy)*log2(10))) # summation limit from Crandall, which is conservative + L = Int(ceil(-log10(accuracy)*log2(10))) # summation limit from Crandall, which is conservative, but based on real s if isinteger(s) n = Int(round(s)) - if n>0 + if n>1 # Crandall's 1.4 for s integer total = μ^(n-1)*(harmonic(n-1) - log(-μ))/gamma(n) # println(" μ=$μ, total = $total") @@ -177,9 +178,13 @@ function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) total -= A end # println(" μ=$μ, total = $total") + elseif n==1 + total = -log(Complex(1-z)) elseif n==0 - total = z ./ (1-z) - elseif n<0 + total = z / (1-z) + elseif n==-1 + total = z / (1-z)^2 + elseif n<-1 # Crandall's 1.5 for s integer total = factorial(-n) * (-μ)^(n-1) for k=0:L From c236756ba3ce9a1b9d78c5448999c33f7a67c5bb Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Fri, 2 Jun 2017 12:30:18 +0930 Subject: [PATCH 09/14] replaced dumb loop calculations of power series with something a bit nicer --- src/li.jl | 23 ++++++++++++++++------- 1 file changed, 16 insertions(+), 7 deletions(-) diff --git a/src/li.jl b/src/li.jl index 77fa1dc6..6616cd9e 100644 --- a/src/li.jl +++ b/src/li.jl @@ -143,10 +143,10 @@ function Li_direct(s::Number, z::Number, accuracy=1.0e-18) end total = 0.0 L = ceil(-log10(accuracy)*log2(10)) # summation limit from Crandall, which is conservative, but based on real s - tmp = z; + a = z; for n=1:L - a = z^n / n^s total += a + a *= z * ( n/(n+1.0) )^s # println(" total = $total") end return total @@ -166,11 +166,14 @@ function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) # Crandall's 1.4 for s integer total = μ^(n-1)*(harmonic(n-1) - log(-μ))/gamma(n) # println(" μ=$μ, total = $total") + tmp = 1 for m=0:L if n - m != 1 - total += μ^m * zeta(n - m) / gamma(m+1) + # total += μ^m * zeta(n - m) / gamma(m+1) + total += tmp * zeta(n - m) end - # println(" m=$m, total = $total") + # println(" m=$m, total = $total, tmp=$tmp, ctmp=$(μ^m /gamma(m+1))") + tmp *= μ/(m+1) end # println(" μ=$μ, total = $total") A = 2*pi*im*log(complex(z))^(s-1)/gamma(n) @@ -187,17 +190,23 @@ function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) elseif n<-1 # Crandall's 1.5 for s integer total = factorial(-n) * (-μ)^(n-1) + tmp = 1 for k=0:L - total -= μ^k * bernoulli(k-n+1, 0.0) / ( gamma(k+1)*(k-n+1) ) - end + # total -= μ^k * bernoulli(k-n+1, 0.0) / ( gamma(k+1)*(k-n+1) ) + total -= tmp * bernoulli(k-n+1, 0.0) / (k-n+1) + tmp *= μ/(k+1) + end else error("Should not get this case") end else # equivalent of Crandalls 1.4 for s non-integer total = gamma(1-s) * (-μ)^(s-1) + tmp = 1 for k=0:L - total += μ^k * zeta(s-k)/factorial(Float64(k)) + # total += μ^k * zeta(s-k)/factorial(Float64(k)) + total += tmp * zeta(s-k) + tmp *= μ/(k+1) end A = 2*pi*im*log(complex(z))^(s-1)/(gamma(s)) if isreal(z) && real(z)>=1 From 91229206d02e95f05ef4d94b6ec290251355e978 Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Fri, 2 Jun 2017 12:32:49 +0930 Subject: [PATCH 10/14] removed indent --- examples/bernoulli_ex.jl | 22 ---------------------- examples/bernoulli_ex.pdf | Bin 8484 -> 0 bytes examples/li_ex.jl | 34 ---------------------------------- examples/li_ex.pdf | Bin 12254 -> 0 bytes examples/li_ex2.jl | 35 ----------------------------------- examples/li_ex2.jl~ | 38 -------------------------------------- examples/li_ex2.pdf | Bin 190879 -> 0 bytes src/gamma.jl | 2 +- 8 files changed, 1 insertion(+), 130 deletions(-) delete mode 100644 examples/bernoulli_ex.jl delete mode 100644 examples/bernoulli_ex.pdf delete mode 100644 examples/li_ex.jl delete mode 100644 examples/li_ex.pdf delete mode 100644 examples/li_ex2.jl delete mode 100644 examples/li_ex2.jl~ delete mode 100644 examples/li_ex2.pdf diff --git a/examples/bernoulli_ex.jl b/examples/bernoulli_ex.jl deleted file mode 100644 index 92eb161d..00000000 --- a/examples/bernoulli_ex.jl +++ /dev/null @@ -1,22 +0,0 @@ -# Example of Bernoulli functions -# Replicate (approximately) the plot from http://mathworld.wolfram.com/BernoulliPolynomial.html - -using Plots; plotlyjs() -default(show = true) # shouldn't need display commands - -using SpecialFunctions -const SF = SpecialFunctions - -x = -0.4:0.01:1.0 -plt = plot( x, SF.bernoulli.(1,x), - label="B_1(x)", legendfont=font(20, "Courier"), legend=:bottomright, - xaxis = ("x", font(20, "Courier")), - yaxis = ("", font(20, "Courier"), (-0.15, 0.15), -0.15:0.05:1.5), - size = (1200, 800) - ) - -for i=2:5 - plot!( x, SF.bernoulli.(i,x), label="B_$i(x)") -end - -savefig("bernoulli_ex.pdf") diff --git a/examples/bernoulli_ex.pdf b/examples/bernoulli_ex.pdf deleted file mode 100644 index ae0d37dc84981f08aa0df2f64394e2408e02584c..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 8484 zcmbVy1z1#D)HV&y5Ynj(3JQ`#4lUiC0s;dJFto$arHC{LqJnga(j^^|(jg@sBHb+^ z-++3r_ulV${_pwEnRCvp_pH6wUOVPl>)peoE-lLr=HLS`eb^{C1@M5lK#pcM01*)o zr!vgJ3TX}EMM0VX5D3I63%5hU5U8ge6bX}tnLAp*0AgYQXCwj!wFkH-zIvlpc%y=# z^=Oa9@(tUxG|78ELrj?`7!Nb~`en4Rur+D%l7TG`dFdOLqK3MSPQO>FUSBXa?g>sO 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zUpel-uN~lOb+xYIpbszxBH5w)7S|L1dw`a1x)+K*R%fZP!fIO1xhLCBB_;xaSK IsmtU3KS;LI#sB~S diff --git a/examples/li_ex2.jl b/examples/li_ex2.jl deleted file mode 100644 index 106860e4..00000000 --- a/examples/li_ex2.jl +++ /dev/null @@ -1,35 +0,0 @@ -# do plots of the Li function, -# this one takes a while - -using Plots; plotlyjs() -using PlotUtils # already from Plots, but as a reminder -using PlotThemes # already from Plots, but as a reminder -default(show = true) # shouldn't need display commands - -using SpecialFunctions -const SF = SpecialFunctions - -# one of the figures from https://en.wikipedia.org/wiki/Polylogarithm -# presuming this is a phase plot (there is not scale) -# step = 0.02 $ took about 200 seconds -# step = 0.01 # took about 11 minutes -step = 0.0025 -xs = -2.0 : step : 2.0 -ys = -2.0 : step : 2.0 -Z = [Complex(x, y) for x in xs, y in ys]' -tic() -L = SF.Li.(-2,Z) -theTime = toc() -a = angle(L)/pi -i = abs(imag(L)) .< 1.0e-8 -a[i] = (-sign(real(L[i]))+1.0)/2.0 -j = abs(L) .< 1.0e-8 -a[j] = NaN -heatmap(xs, ys, a, - xaxis = ("real", font(20, "Courier")), - yaxis = ("imag", font(20, "Courier")), - size = (800, 800), - color = :Spectral) - -sleep(1) -savefig("li_ex2.pdf") diff --git a/examples/li_ex2.jl~ b/examples/li_ex2.jl~ deleted file mode 100644 index 8f7bab99..00000000 --- a/examples/li_ex2.jl~ +++ /dev/null @@ -1,38 +0,0 @@ -# do plots of the Li function, - - - -using Plots; plotlyjs() -default(show = true) # shouldn't need display commands - -using SpecialFunctions -const SF = SpecialFunctions - -# replicate (approximately) the plot from http://mathworld.wolfram.com/Polylogarithm.html -x = -2.0 : 0.01 : 1.0 -plt = plot( x, real(SF.Li.(-2,x)), - label="Li_{-2}(x)", legendfont=font(20, "Courier"), legend=:bottomright, - xaxis = ("x", font(20, "Courier")), - yaxis = ("", font(20, "Courier"), (-1, 1), -1.0:0.25:1.0), - size = (1200, 800) - ) - -for i=-1:2 - plot!( x, real(SF.Li.(i,x)), label="Li_{$i}(x)") -end - -sleep(1) -savefig("li_ex.pdf") - - -# 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z?xwsf1x)EQSAQ2US@xe-Td_zmYsoo@3|eP2H+T&2Xi%#?u0T5FhyWUeAqs%H;Nu5{ z$KgO$d;|r@00NPS2PYW%puCr3XfzfefRczahDJlK6=w_!;6PYb#c+5KN%j~(z+xfz zIAcTt8XOX5OdBA8)5#tK&joaaIb(n}fk;^0hc z9cMW#dTm`S_+El)Syx*NzY2SRKp^7RlmjtX z(+3f=1{VSm3uS9__CdsHuf~)JBE7m_2vRnz7<4k|q|#xPY(YIvfvzj)d0|eYF~GEL lSUN%b&{GKo-!JeT=*J+_8Ss?|5YZR_5ZJL}-yuVRe*;s+$YcNj diff --git a/src/gamma.jl b/src/gamma.jl index 157b2205..d7275baf 100644 --- a/src/gamma.jl +++ b/src/gamma.jl @@ -321,7 +321,7 @@ end """ hurwitz_zeta(s, z) - An alias for zeta(s, z) +An alias for zeta(s, z) """ hurwitz_zeta(s::ComplexOrReal{Float64}, z::ComplexOrReal{Float64}) = zeta(s, z) From 21f6d02775e4fd163fe760f51885145d28883038 Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Fri, 2 Jun 2017 16:47:49 +0930 Subject: [PATCH 11/14] adding tests, and debugging --- src/li.jl | 29 +- test/bernoulli_test.jl | 70 --- test/li_test.jl | 18 +- test/li_test2.jl | 79 +++ test/li_testdata.dat | 1002 ++++++++++++++++++++++++++++++++++++ test/li_testdata.py | 26 + test/li_testdata_alpha.dat | 42 ++ 7 files changed, 1188 insertions(+), 78 deletions(-) delete mode 100644 test/bernoulli_test.jl create mode 100644 test/li_test2.jl create mode 100644 test/li_testdata.dat create mode 100755 test/li_testdata.py create mode 100644 test/li_testdata_alpha.dat diff --git a/src/li.jl b/src/li.jl index 6616cd9e..936696d5 100644 --- a/src/li.jl +++ b/src/li.jl @@ -97,9 +97,11 @@ function Li(s::Number, z::Number, accuracy::Real=1.0e-18) return -(-1.0)^s .*Li_direct(s, 1/z, accuracy) elseif abs(z) >= 1/T # use inversion formula to calculate in terms of Li_s(1/z) - G = (2*pi*im).^s .* zeta( 1-s, 0.5 - log(-1/complex(z))/(2*pi*im) ) ./ gamma(s) - F = complex(-1.0)^s .* Li_direct(s, 1/z, accuracy) - A = 2*pi*im*log(complex(z))^(s-1)/(gamma(s)) + z = convert(Complex{Float64}, z) + G = (2*pi*im)^s * zeta( 1-s, 0.5 + log(-z)/(2*pi*im) ) / gamma(s) + F = complex(-1.0)^s * Li_direct(s, 1/z, accuracy) + + A = 2*pi*im*log(z)^(s-1)/(gamma(s)) if ( isreal(z) && real(z)>=1 ) Θ = 1 else @@ -148,18 +150,21 @@ function Li_direct(s::Number, z::Number, accuracy=1.0e-18) total += a a *= z * ( n/(n+1.0) )^s # println(" total = $total") + if abs(a)/abs(total) < 1.0e-30 + break + end end return total end function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) # calculate using power series around μ = log(z) = 0 - μ = log(complex(z)) + μ = log(convert(Complex{Float64}, z)) # println("μ = $μ") if abs(μ) > 2*pi throw(DomainError()) end - L = Int(ceil(-log10(accuracy)*log2(10))) # summation limit from Crandall, which is conservative, but based on real s + L = Int(ceil(-log10(accuracy)*log2(10))) # revisit this limit if isinteger(s) n = Int(round(s)) if n>1 @@ -174,6 +179,9 @@ function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) end # println(" m=$m, total = $total, tmp=$tmp, ctmp=$(μ^m /gamma(m+1))") tmp *= μ/(m+1) + if abs(tmp)/abs(total) < 1.0e-30 + break + end end # println(" μ=$μ, total = $total") A = 2*pi*im*log(complex(z))^(s-1)/gamma(n) @@ -182,7 +190,7 @@ function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) end # println(" μ=$μ, total = $total") elseif n==1 - total = -log(Complex(1-z)) + total = -log(complex(1-z)) elseif n==0 total = z / (1-z) elseif n==-1 @@ -195,6 +203,9 @@ function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) # total -= μ^k * bernoulli(k-n+1, 0.0) / ( gamma(k+1)*(k-n+1) ) total -= tmp * bernoulli(k-n+1, 0.0) / (k-n+1) tmp *= μ/(k+1) + if abs(tmp)/abs(total) < 1.0e-30 + break + end end else error("Should not get this case") @@ -202,12 +213,18 @@ function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) else # equivalent of Crandalls 1.4 for s non-integer total = gamma(1-s) * (-μ)^(s-1) + # println(" μ=$μ, total = $total") tmp = 1 for k=0:L # total += μ^k * zeta(s-k)/factorial(Float64(k)) total += tmp * zeta(s-k) + # println(" tmp=$(tmp* zeta(s-k)), total = $total") tmp *= μ/(k+1) + if abs(tmp)/abs(total) < 1.0e-30 + break + end end + A = 2*pi*im*log(complex(z))^(s-1)/(gamma(s)) if isreal(z) && real(z)>=1 total -= A diff --git a/test/bernoulli_test.jl b/test/bernoulli_test.jl deleted file mode 100644 index 4b72027b..00000000 --- a/test/bernoulli_test.jl +++ /dev/null @@ -1,70 +0,0 @@ -using SpecialFunctions -using Base.Test - -const SF = SpecialFunctions - -# test functions, see runtests.jl -relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x) -relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x))) -≅(a,b) = relerrc(a,b) ≤ 1e-13 - -# test functions, similar to runtests.jl, but can't do relative error when some values are zero -abserr(z, x) = z == x ? 0.0 : abs(z - x) -abserrc(z, x) = max(abserr(real(z),real(x)), abserr(imag(z),imag(x))) -≒(a,b) = abserrc(a,b) ≤ 1e-13 - -@testset "Bernoulli polynomials and numbers" begin - @testset " errors" begin - @test_throws DomainError SF.bernoulli(-1) - @test_throws DomainError SF.bernoulli(-1, 0.0) - end - - @testset " consistency" begin - for i=0:34 - @test Float64(SF.bernoulli(i)) ≈ SF.bernoulli(i, 0.0) - end - end - - @testset " numbers explicit cases" begin - for i=2:34 - # first two values of zeta don't work - @test Float64(SF.bernoulli(i)) ≈ -i*SF.zeta( 1 - i ) - end - end - - @testset " polynomials explicit cases" begin - X = collect(-2.0 : 0.1 : 3.0) - for i=1:length(X) - x = X[i] - # println(" x = $x") - @test SF.bernoulli(0, x) ≈ 1.0 - @test SF.bernoulli(1, x) ≈ x - 0.5 - @test SF.bernoulli(2, x) ≈ x^2 - x + 1.0/6.0 - @test SF.bernoulli(3, x) ≈ x^3 - 1.5*x^2 + 0.5*x - @test SF.bernoulli(4, x) ≈ x^4 - 2.0*x^3 + x^2 - 1/30.0 - @test SF.bernoulli(5, x) ≈ x^5 - 2.5*x^4 + (5.0/3.0)*x^3 - x/6.0 - @test SF.bernoulli(6, x) ≈ x^6 - 3.0*x^5 + (5.0/2.0)*x^4 - 0.5*x^2 + 1/42.0 - end - end - - @testset " symmetries" begin - X = collect(0.0: 0.1 : 1.0) - for i=1:length(X) - x = X[i] - for n=1:9 - # println(" n = $n, x = $x") - # start to get errors of order 1.0e-14 for n=5, 1.0e-13 around n=10, ... - @test SF.bernoulli(n, 1-x) ≒ (-1.0)^n * SF.bernoulli(n, x) - @test SF.bernoulli(n, x+1) ≒ SF.bernoulli(n, x) + n*x^(n-1) - @test (-1)^n * SF.bernoulli(n, -x) ≒ SF.bernoulli(n, x) + n*x^(n-1) - - # Raabe (1851) - m = 6.0 - k = 0:m-1 - @test sum( SF.bernoulli.(n, x + k/m ) )/m ≒ m^(-n)*SF.bernoulli(n, m*x ) - - end - end - end -end - diff --git a/test/li_test.jl b/test/li_test.jl index 3e55e4ed..b7295248 100644 --- a/test/li_test.jl +++ b/test/li_test.jl @@ -3,6 +3,15 @@ using Base.Test const SF = SpecialFunctions +# test functions, see runtests.jl +relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x) +relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x))) +≅(a,b) = relerrc(a,b) ≤ 1e-13 + +# test functions, similar to runtests.jl, but can't do relative error when some values are zero +abserr(z, x) = z == x ? 0.0 : abs(z - x) +abserrc(z, x) = max(abserr(real(z),real(x)), abserr(imag(z),imag(x))) +≒(a,b) = abserrc(a,b) ≤ 1e-13 # golden ratio is used in some tests ϕ = (sqrt(5)+1)/2.0 @@ -22,9 +31,11 @@ const SF = SpecialFunctions end @testset "Test s = n (a real integer)" begin - # simple case + # simple cases @test SF.Li(1, 0.5) ≈ log(2) - + @test !isfinite( SF.Li(1, 1.0) ) + @test SF.Li(1, 2) ≒ -pi*im # comes from -log(complex(1-2)) + @testset " dilogarithm for real z" begin @test SF.Li(2,-1.0) ≈ -pi^2/12.0 @test SF.Li(2, 0.0) ≈ 0.0 @@ -98,6 +109,9 @@ const SF = SpecialFunctions @test SF.Li(s, im) ≈ - 2.0.^(-s).*SF.eta.(s) + im*SF.Dbeta.(s) end end + + # the singularity + @test !isfinite( SF.Li(0.5, 1.0) ) end @testset "Additional Identities" begin diff --git a/test/li_test2.jl b/test/li_test2.jl new file mode 100644 index 00000000..55c54dd8 --- /dev/null +++ b/test/li_test2.jl @@ -0,0 +1,79 @@ +using SpecialFunctions +using Base.Test + +const SF = SpecialFunctions + +# test functions, see runtests.jl +relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x) +relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x))) +≅(a,b) = relerrc(a,b) ≤ 1e-9 + +# test functions, similar to runtests.jl, but can't do relative error when some values are zero +abserr(z, x) = z == x ? 0.0 : abs(z - x) +abserrc(z, x) = max(abserr(real(z),real(x)), abserr(imag(z),imag(x))) +≒(a,b) = abserrc(a,b) ≤ 1e-10 + +S = Array(Complex, (0,)) +Z = Array(Complex, (0,)) +errors = Array(Complex, (0,)) +rel_errors = Array(Complex, (0,)) +L = 0 +L2 = 0 +diff = 0 +rel_diff = 0 + +file = "li_testdata_alpha.dat" +f = open(file, "r") +for line in eachline(f) + if ismatch(r"#", line) + else + v = split(line, ',') + + s = parse(Float64, v[1]) + im*parse(Float64, v[2]) + z = parse(Float64, v[3]) + im*parse(Float64, v[4]) + L = parse(Float64, v[5]) + im*parse(Float64, v[6]) + L2 = SF.Li(s,z,1.0e-18) + diff = L - L2 + rel_diff = complex(real(diff)/real(L), imag(diff)/imag(L)) + + push!(S, s) + push!(Z, z) + push!(errors, diff) + push!(rel_errors, rel_diff) + + # println(" L = $L, L2 =$L2, diff = $diff, \t\t rel diff = $rel_diff") + println(" s=$s, z=$z, |error|=$(ceil(log10(abs(diff))))") + end +end + +# test particular values from Wolfram alpha +# @testset "Li tests from Wolfram alpha" begin + +# @test SF.Li(-5.000001, 0.2) ≅ 6.904305364987799724230921116283822405784472717903797757356 +# @test SF.Li(-2.000001, 0.2) ≅ 0.468750244180354668389455477288861311399345621141103816518 +# @test SF.Li(-1.000001, 0.2) ≅ 0.312500094184052926057717828023167764362038234067425674984 +# @test SF.Li( 0.000000, 0.2) ≅ 0.25 +# @test SF.Li( 0.000001, 0.2) ≅ 0.249999960605813559100141040899469061220130457763090931203 +# @test SF.Li( 1.000001, 0.2) ≅ 0.223143533840628659834227863703211050928449673084530824304 +# @test SF.Li( 2.000001, 0.2) ≅ 0.211003767368667938482167666994614717744311348708200162427 +# @test SF.Li( 3.000001, 0.2) ≅ 0.205324191902667822690683746355210013921766645655171501890 +# @test SF.Li( 5.000001, 0.2) ≅ 0.201284594885756384926736867628954299128217148725340685679 + +# @test SF.Li(-5.000001, 0.99) ≅ 1.16439554606503726735945485177408748716681200000005e14 +# @test SF.Li(-2.000001, 0.99) ≅ 1.97011088076187708022430311310006082286580942497213e6 +# @test SF.Li(-1.000001, 0.99) ≅ 9900.04972775403451817385172007212640686495346816117 +# @test SF.Li( 0.000000, 0.99) ≅ 99 +# @test SF.Li( 0.000001, 0.99) ≅ 98.9995988050883517986218058007486858307489619231956 +# @test SF.Li( 1.000001, 0.99) ≅ 4.605161353580737753746071585981515640203854412492030454580 # error ~ 1.0e11 +# @test SF.Li( 2.000001, 0.99) ≅ 1.588624649826159899085393338960947692670814871114246941712 # error ~ 1.0e11 +# @test SF.Li( 3.000001, 0.99) ≅ 1.185832744102043627496839259642313182039620569301974825714 +# @test SF.Li( 5.000001, 0.99) ≅ 1.026110449210262375841139988837721699091019919425544651696 + +# @test SF.Li( 1.000001, 1.20) ≅ 1.60944129470676213105637136369425072009730867713452098010 - 3.14158912002728351076393334268714060723117930694360052182im +# @test SF.Li( 1.000001, 1.50) ≅ 0.69315092620532843831758654220946253504678190003195418383 - 3.1415916309839162783984857611820275451978840493033368692im +# @test SF.Li( 1.000001, 1.70) ≅ 0.35667861585275033749153552050383961466011713941159572502 - 3.1415924761565638528178063690254522808240730779853094090im +# @test SF.Li( 1.000001, 1.999999) ≒ +# @test SF.Li( 1.000001, 2.00) ≅ +# @test SF.Li( 1.000001, 2.10) ≅ -0.0953067628118213777754205736836756047381793027303437193 - 3.14159352922832327133509242559303053858610422117044987im + +# end diff --git a/test/li_testdata.dat b/test/li_testdata.dat new file mode 100644 index 00000000..5e43d45f --- /dev/null +++ b/test/li_testdata.dat @@ -0,0 +1,1002 @@ +# test data from li_testdata.py +# using mpmath library +# s(real, imag), z(real, imag), Li(real, imag) +2.57636950631092576103,2.89889121739954180867,0.13267161787652381744,-1.52908730194326358820,0.22742653414510977106,-1.28004045152782230943 +-2.18434643020828289650,0.06266903366343373749,-2.04420634002174583799,-2.87365889020505971274,0.12933937360075600798,-0.04815800746168075186 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+2.40007489210605529095,2.87660762026954452963,5.69370195484752983361,-0.50901867863753191834,3.18184282108034643954,-0.65118188288410305820 +-0.28088304419904919218,0.87630768879533871285,0.18980544059841819582,-2.14549658946724441932,-0.21347130074107478759,-0.43572201313373565279 diff --git a/test/li_testdata.py b/test/li_testdata.py new file mode 100755 index 00000000..00a36ec3 --- /dev/null +++ b/test/li_testdata.py @@ -0,0 +1,26 @@ +#!/usr/bin/python +# python code using mpmath to create a test data sets +from mpmath import * +mp.dps = 30 +mp.pretty = True + +import random +random.seed(1) + +N = 1000 +mu = 0.0 +sigma = 2.0 +fid = open('li_testdata.dat', 'w') +fid.write('# test data from li_testdata.py\n') +fid.write('# using mpmath library\n') +fid.write('# s(real, imag), z(real, imag), Li(real, imag)\n') +for i in range(1,N): + s = random.gauss(mu, sigma) + j*random.gauss(mu, sigma) + z = random.gauss(mu, sigma) + j*random.gauss(mu, sigma) + L = polylog(s, z) + fid.write('%.20f,%.20f,%.20f,%.20f,%.20f,%.20f\n' % (s.real, s.imag, z.real, z.imag, L.real, L.imag)) + + +fid.close() + + diff --git a/test/li_testdata_alpha.dat b/test/li_testdata_alpha.dat new file mode 100644 index 00000000..a4ea2d97 --- /dev/null +++ b/test/li_testdata_alpha.dat @@ -0,0 +1,42 @@ +# test data from Wolfram alpha +# +# s(real, imag), z(real, imag), Li(real, imag) +# +# start with values with near integer s, because these might be problematic as they are near special cases +# in particular for the series sum around z=1, which we use when 0.5 < |z| < 2.0 +-5.000001, 0.0, 0.2, 0.0, 6.904305364987799724230921116283822405784472717903797757356, 0.0 +-2.000001, 0.0, 0.2, 0.0, 0.468750244180354668389455477288861311399345621141103816518, 0.0 +-1.000001, 0.0, 0.2, 0.0, 0.312500094184052926057717828023167764362038234067425674984, 0.0 + 0.000000, 0.0, 0.2, 0.0, 0.25, 0.0 + 0.000001, 0.0, 0.2, 0.0, 0.249999960605813559100141040899469061220130457763090931203, 0.0 + 1.000001, 0.0, 0.2, 0.0, 0.223143533840628659834227863703211050928449673084530824304, 0.0 + 2.000001, 0.0, 0.2, 0.0, 0.211003767368667938482167666994614717744311348708200162427, 0.0 + 3.000001, 0.0, 0.2, 0.0, 0.205324191902667822690683746355210013921766645655171501890, 0.0 + 5.000001, 0.0, 0.2, 0.0, 0.201284594885756384926736867628954299128217148725340685679, 0.0 +# + 1.000001, 0.0, 0.5, 0.0, 0.693147007663231669849516514839299401059360031508002626249, 0.0 +# +-5.000001, 0.0, 0.99, 0.0, 1.16439554606503726735945485177408748716681200000005e14, 0.0 +-2.000001, 0.0, 0.99, 0.0, 1.97011088076187708022430311310006082286580942497213e6, 0.0 +-1.000001, 0.0, 0.99, 0.0, 9900.04972775403451817385172007212640686495346816117, 0.0 + 0.000000, 0.0, 0.99, 0.0, 99, 0.0 + 0.000001, 0.0, 0.99, 0.0, 98.9995988050883517986218058007486858307489619231956, 0.0 + 1.000001, 0.0, 0.99, 0.0, 4.605161353580737753746071585981515640203854412492030454580, 0.0 + 2.000001, 0.0, 0.99, 0.0, 1.588624649826159899085393338960947692670814871114246941712, 0.0 + 3.000001, 0.0, 0.99, 0.0, 1.185832744102043627496839259642313182039620569301974825714, 0.0 + 5.000001, 0.0, 0.99, 0.0, 1.026110449210262375841139988837721699091019919425544651696, 0.0 +# + 1.000001, 0.0, 1.20, 0.0, 1.60944129470676213105637136369425072009730867713452098010, -3.14158912002728351076393334268714060723117930694360052182 + 1.000001, 0.0, 1.50, 0.0, 0.69315092620532843831758654220946253504678190003195418383, -3.1415916309839162783984857611820275451978840493033368692 + 1.000001, 0.0, 1.70, 0.0, 0.35667861585275033749153552050383961466011713941159572502, -3.1415924761565638528178063690254522808240730779853094090 + 1.000001, 0.0, 1.999999, 0.0, 4.48462808108367607858388644733186363064154361287440e-6,-3.14159331552720658388549501733501317621202043844284 + 1.000001, 0.0, 2.00, 0.0, 3.48462690882559763270126765927498392238379428297186e-6, -3.14159331552947276581779433082131101718301247144061 + 1.000001, 0.0, 2.10, 0.0, -0.0953067628118213777754205736836756047381793027303437193, -3.14159352922832327133509242559303053858610422117044987 +# +# +# difficult cases are for s close to a positive integer, and using expansion around z=1 + 1.000001, 0.0, 0.99, 0.0, 4.605161353580737753746071585981515640203854412492030454580, 0.0 + 1.000001, 0.0, 1.20, 0.0, 1.60944129470676213105637136369425072009730867713452098010, -3.14158912002728351076393334268714060723117930694360052182 + 1.000001, 0.0, 1.50, 0.0, 0.69315092620532843831758654220946253504678190003195418383, -3.1415916309839162783984857611820275451978840493033368692 + 1.000001, 0.0, 1.70, 0.0, 0.35667861585275033749153552050383961466011713941159572502, -3.1415924761565638528178063690254522808240730779853094090 + 1.000001, 0.0, 1.999999, 0.0, 4.48462808108367607858388644733186363064154361287440e-6,-3.14159331552720658388549501733501317621202043844284 From 9fd9675e808141e466718efe7e43c8e9d935afeb Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Fri, 2 Jun 2017 16:54:14 +0930 Subject: [PATCH 12/14] changed Li to polylog --- src/SpecialFunctions.jl | 3 +- src/li.jl | 234 --------- test/li_test.jl | 158 ------ test/li_test2.jl | 79 --- test/li_testdata.dat | 1002 ------------------------------------ test/li_testdata.py | 26 - test/li_testdata_alpha.dat | 42 -- 7 files changed, 1 insertion(+), 1543 deletions(-) delete mode 100644 src/li.jl delete mode 100644 test/li_test.jl delete mode 100644 test/li_test2.jl delete mode 100644 test/li_testdata.dat delete mode 100755 test/li_testdata.py delete mode 100644 test/li_testdata_alpha.dat diff --git a/src/SpecialFunctions.jl b/src/SpecialFunctions.jl index 9f8863c5..0d15d0e1 100644 --- a/src/SpecialFunctions.jl +++ b/src/SpecialFunctions.jl @@ -56,7 +56,6 @@ if VERSION >= v"0.6.0-dev.2767" hankelh2, hankelh2x, zeta, - Li, polylog, bernoulli, harmonic @@ -76,6 +75,6 @@ include("deprecated.jl") include("harmonic.jl") include("bernoulli.jl") -include("li.jl") +include("polylog.jl") end # module diff --git a/src/li.jl b/src/li.jl deleted file mode 100644 index 936696d5..00000000 --- a/src/li.jl +++ /dev/null @@ -1,234 +0,0 @@ -""" - Li(s, z) - - Calculates the Polylogarithm function ``Li_s(z)`` defined by - -```math - L_s = \sum_{n=1}^{\infty} \frac{z^n}{n^s} -``` - - For ideas going into this see - - + Crandall, "Note on fast polylogarithm computation", 2006, - which focusses on the case where s=n (integer and real) - http://www.wolfgang-ehrhardt.de/Polylog.pdf - - + Vepstas, "AN EFFICIENT ALGORITHM FOR ACCELERATING THE CONVERGENCE - OF OSCILLATORY SERIES, USEFUL FOR COMPUTING THE - POLYLOGARITHM AND HURWITZ ZETA FUNCTIONS", 2007 - which treats the general case, but presumes arbitrary precision arithmetic - https://arxiv.org/abs/math/0702243 - - + Wood, "The computation of Plylogarithms", 1992 - which focusses on s=n, integer and real, and which has formatting issues making - it hard to read correctly. - https://www.cs.kent.ac.uk/pubs/1992/110/ - - + Maximon, "The dilogarithm function for complex argument", 2003 - which provides useful test cases for s=2 - - + Zagier, "The dilogarithm function in geometry and number theory", 1989 - similar to Maximon - - Of these the only one that actually specifies a full algorithm is - Crandall, and he also treats special cases more carefully, so this - is the one that I have paid most attention to. However, extending it - for s on the whole complex plane requires some additions, and many - of these are actually most nicely documented on the wikipedia page - - + https://en.wikipedia.org/wiki/Polylogarithm - - With further details at - - + http://mathworld.wolfram.com/Polylogarithm.html - + http://dlmf.nist.gov/25.12#ii - + http://mathworld.wolfram.com/Trilogarithm.html - + http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/ - - The wiki page points out some errors in earlier works, but not all - parts on the page have references, and not all statements seem to - come from any of the listed references? - - The code draws heavily on existing functions, in particular the - Hurwitz-zeta function, which is aliased to zeta(s,q) in Julia. - - Accuracy has been tested using many of the identities known for Li - and relations to known functions as special cases, and by comparison - to `polylog(s, z)` in the `mpmath` arbitrary-precision package in Python. - - http://mpmath.org/doc/current/functions/zeta.html - - The latter shows deviations of the order of - - + 10^{Im(s) - 20} for Im(s) < 0 - + 10^{Im(s) - 20} for Im(s) > 0 - - It isn't clear whether we can do better than this with - double-precision arithmetic. - -## Arguments -* `s::Complex`: the 'fractional' coefficient -* `z::Complex`: the point at which to calculate it -* `accuracy::Real=1.0e-18`: nominal accuracy of calculation, but mainly useful for testing - -## Examples -```jldoctest -julia> Li(-1.0, 0.0) -(0.0,1) -``` -""" -function Li(s::Number, z::Number, accuracy::Real=1.0e-18) - T = 0.5 # threshold at which we change algorithms - if z ≈ 1.0 - if real(s) > 1 - return zeta(s) - else - return Inf - end - elseif z ≈ -1.0 - return -eta(s) - elseif s ≈ 0.0 - return z ./ (1-z) - elseif abs(z) <= T - return Li_direct(s, z, accuracy) - elseif abs(z) >= 1/T && isinteger(s) && real(s) < 0 - # use inversion formula to calculate in terms of Li_n(1/z) - # but note for negative integer s, it collapses to something small - return -(-1.0)^s .*Li_direct(s, 1/z, accuracy) - elseif abs(z) >= 1/T - # use inversion formula to calculate in terms of Li_s(1/z) - z = convert(Complex{Float64}, z) - G = (2*pi*im)^s * zeta( 1-s, 0.5 + log(-z)/(2*pi*im) ) / gamma(s) - F = complex(-1.0)^s * Li_direct(s, 1/z, accuracy) - - A = 2*pi*im*log(z)^(s-1)/(gamma(s)) - if ( isreal(z) && real(z)>=1 ) - Θ = 1 - else - Θ = 0 - end - # println("G = $G, F=$F, Θ=$Θ, A=$A") - return ( G - F - Θ*A ) - else - # power series around mu=0, for z = e^mu - Li_series_mu(s, z, accuracy) - end -end - - -""" - Polylog(s, z) - - This is just an alias of Li(s, z) - -""" -polylog(s::Number, z::Number ) = Li(s, z) - - -#################################################################### -#### these are component functions and aren't exported at this point - -# Dirichlet beta function, for testing results -# https://en.wikipedia.org/wiki/Dirichlet_beta_function -# but don't export as it would create confusion with Euler integral -function Dbeta(s::Number) - β = 4.0^(-s) * ( zeta(s,0.25) - zeta(s,0.75) ) -end - -function Li_direct(s::Number, z::Number, accuracy=1.0e-18) - # calculate using direct definition - if abs(z) > 1 || ( abs(z) ≈ 1 && real(s) <= 2) - throw(DomainError()) - end - if abs(z) > 1/2 - warn("Slow convergence for |z| > 1/2") - end - total = 0.0 - L = ceil(-log10(accuracy)*log2(10)) # summation limit from Crandall, which is conservative, but based on real s - a = z; - for n=1:L - total += a - a *= z * ( n/(n+1.0) )^s - # println(" total = $total") - if abs(a)/abs(total) < 1.0e-30 - break - end - end - return total -end - -function Li_series_mu(s::Number, z::Number, accuracy=1.0e-18) - # calculate using power series around μ = log(z) = 0 - μ = log(convert(Complex{Float64}, z)) - # println("μ = $μ") - if abs(μ) > 2*pi - throw(DomainError()) - end - L = Int(ceil(-log10(accuracy)*log2(10))) # revisit this limit - if isinteger(s) - n = Int(round(s)) - if n>1 - # Crandall's 1.4 for s integer - total = μ^(n-1)*(harmonic(n-1) - log(-μ))/gamma(n) - # println(" μ=$μ, total = $total") - tmp = 1 - for m=0:L - if n - m != 1 - # total += μ^m * zeta(n - m) / gamma(m+1) - total += tmp * zeta(n - m) - end - # println(" m=$m, total = $total, tmp=$tmp, ctmp=$(μ^m /gamma(m+1))") - tmp *= μ/(m+1) - if abs(tmp)/abs(total) < 1.0e-30 - break - end - end - # println(" μ=$μ, total = $total") - A = 2*pi*im*log(complex(z))^(s-1)/gamma(n) - if isreal(z) && real(z)>=1 - total -= A - end - # println(" μ=$μ, total = $total") - elseif n==1 - total = -log(complex(1-z)) - elseif n==0 - total = z / (1-z) - elseif n==-1 - total = z / (1-z)^2 - elseif n<-1 - # Crandall's 1.5 for s integer - total = factorial(-n) * (-μ)^(n-1) - tmp = 1 - for k=0:L - # total -= μ^k * bernoulli(k-n+1, 0.0) / ( gamma(k+1)*(k-n+1) ) - total -= tmp * bernoulli(k-n+1, 0.0) / (k-n+1) - tmp *= μ/(k+1) - if abs(tmp)/abs(total) < 1.0e-30 - break - end - end - else - error("Should not get this case") - end - else - # equivalent of Crandalls 1.4 for s non-integer - total = gamma(1-s) * (-μ)^(s-1) - # println(" μ=$μ, total = $total") - tmp = 1 - for k=0:L - # total += μ^k * zeta(s-k)/factorial(Float64(k)) - total += tmp * zeta(s-k) - # println(" tmp=$(tmp* zeta(s-k)), total = $total") - tmp *= μ/(k+1) - if abs(tmp)/abs(total) < 1.0e-30 - break - end - end - - A = 2*pi*im*log(complex(z))^(s-1)/(gamma(s)) - if isreal(z) && real(z)>=1 - total -= A - end - end - return total -end diff --git a/test/li_test.jl b/test/li_test.jl deleted file mode 100644 index b7295248..00000000 --- a/test/li_test.jl +++ /dev/null @@ -1,158 +0,0 @@ -using SpecialFunctions -using Base.Test - -const SF = SpecialFunctions - -# test functions, see runtests.jl -relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x) -relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x))) -≅(a,b) = relerrc(a,b) ≤ 1e-13 - -# test functions, similar to runtests.jl, but can't do relative error when some values are zero -abserr(z, x) = z == x ? 0.0 : abs(z - x) -abserrc(z, x) = max(abserr(real(z),real(x)), abserr(imag(z),imag(x))) -≒(a,b) = abserrc(a,b) ≤ 1e-13 - -# golden ratio is used in some tests -ϕ = (sqrt(5)+1)/2.0 - -@testset "Polylogarithm Li" begin - @testset "Check different input formats" begin - @test SF.Li(-1.0, 0.0) ≈ 0.0 - @test SF.Li(-1, 0.0) ≈ 0.0 - @test SF.Li(-1.0, Complex(0.0)) ≈ 0.0 + 0.0im # don't seem to need imaginary bit here - @test SF.Li(Complex(-1.0), 0.0) ≈ 0.0 + 0.0im - @test SF.Li(Complex(-1.0), 0.0) ≈ 0.0 - @test SF.Li(1, 0) ≈ 0.0 - x = collect(0.0:0.1:0.9) - @test all([SF.Li.(1, x)[i] ≈ SF.Li(1, x[i]) for i=1:length(x)]) - @test SF.Li(Complex(-1.0), Complex(0.3)) ≈ SF.Li(-1.0, 0.3) - @test SF.polylog(Complex(-1.0), Complex(0.3)) ≈ SF.Li(-1.0, 0.3) - end - - @testset "Test s = n (a real integer)" begin - # simple cases - @test SF.Li(1, 0.5) ≈ log(2) - @test !isfinite( SF.Li(1, 1.0) ) - @test SF.Li(1, 2) ≒ -pi*im # comes from -log(complex(1-2)) - - @testset " dilogarithm for real z" begin - @test SF.Li(2,-1.0) ≈ -pi^2/12.0 - @test SF.Li(2, 0.0) ≈ 0.0 - @test SF.Li(2, 0.5) ≈ pi^2/12 - 0.5*log(2)^2 - @test SF.Li(2, 1.0) ≈ pi^2/6.0 - @test SF.Li(2, 2.0) ≈ pi^2/4 - im*pi*log(2) - @test SF.Li(2, -ϕ) ≈ -pi^2/10 - log(ϕ)^2 - @test SF.Li(2, -1/ϕ) ≈ -pi^2/15 + log(ϕ)^2/2 - @test SF.Li(2, 1/ϕ^2) ≈ pi^2/15 - log(ϕ)^2 - @test SF.Li(2, 1/ϕ) ≈ pi^2/10 - log(ϕ)^2 - @test SF.Li(2, ϕ) ≈ 11*pi^2/15 + log(Complex(-1/ϕ))^2/2 # wiki has this one, but no ref - @test SF.Li(2, ϕ^2) ≈ -11*pi^2/15 - log(Complex(-ϕ))^2 - - # identities - Z = [3.0 + 0.4im, -3.0 + 0.4im, 3.0 - 0.4im, -3.0 + -0.4im] - for i=1:length(Z) - z = Z[i] - @test SF.Li(2, z) + SF.Li(2, 1/z) ≈ -pi^2/6.0 - log(Complex(-z))^2/2.0 - end - end - - @testset " trilogarithm for real z" begin - @test SF.Li(3,-1.0) ≈ -3*SF.zeta(3)/4 - @test SF.Li(3, 0.0) ≈ 0.0 - @test SF.Li(3, 0.5) ≈ log(2)^3/6.0 - pi^2*log(2)/12.0 + (7.0/8.0)*SF.zeta(3) - @test SF.Li(3, 1.0) ≈ SF.zeta(3) - @test SF.Li(3, ϕ^(-2)) ≈ 4*SF.zeta(3)/5 + 2*log(ϕ)^3/3 - 2*pi^2*log(ϕ)/15 - end - - @testset " general case for real z" begin - # X = collect(-0.95:0.1:0.95) - X = collect(-3.0:0.1:3.0) - for i=1:length(X) - x = X[i] - # println(x) - @test SF.Li(1, x) ≈ -log(Complex(1-x)) - @test SF.Li(0, x) ≈ x ./ (1-x) - @test SF.Li(-1, x) ≈ x ./ (1-x).^2 - @test SF.Li(-2, x) ≈ x .* (1+x) ./ (1-x).^3 - @test SF.Li(-3, x) ≈ x .* (1+4*x+x.^2) ./ (1-x).^4 - @test SF.Li(-4, x) ≈ x .* (1+x) .* (1+10*x+x.^2) ./ (1-x).^5 - end - end - - @testset " general case for complex z" begin - X = collect(-3.0:0.5:3.0) - Y = [-1.3, -0.4, 0.4, 1.5] - for i=1:length(X) - for j=1:length(Y) - z = Complex(X[i], Y[j]) - # println(z) - @test SF.Li(1, z) ≈ -log(Complex(1-z)) - @test SF.Li(0, z) ≈ z ./ (1-z) - @test SF.Li(-1, z) ≈ z ./ (1-z).^2 - @test SF.Li(-2, z) ≈ z .* (1+z) ./ (1-z).^3 - @test SF.Li(-3, z) ≈ z .* (1+4*z+z.^2) ./ (1-z).^4 - @test SF.Li(-4, z) ≈ z .* (1+z) .* (1+10*z+z.^2) ./ (1-z).^5 - end - end - end - end - - @testset "Particular values |z| == 1" begin - S_r = [2.1 2.5 3.0] - S_i = [-1.3, -1.0, -0.5, 0.0, 0.5, 1.0, 1.3] - for i=1:length(S_r) - for j=1:length(S_i) - s = Complex(S_r[i], S_i[j]) - @test SF.Li(s, -1.0) ≈ -SF.eta(s) - @test SF.Li(s, 1.0) ≈ SF.zeta(s) - @test SF.Li(s, im) ≈ - 2.0.^(-s).*SF.eta.(s) + im*SF.Dbeta.(s) - end - end - - # the singularity - @test !isfinite( SF.Li(0.5, 1.0) ) - end - - @testset "Additional Identities" begin - z = 0.5 - for n=1:5 - @test SF.Li(-n,z) + (-1)^n * SF.Li(-n, 1/z) ≈ 0.0 - end - - # for real s, and real z<1, SF.Li should be real - S = [-1, 0.1, 2] - Z = [-2, -1.0, 0.1, 0.95] - for i=1:length(S) - for j=1:length(Z) - s = S[i] - z = Z[j] - # println("s = $s; z = $z") - @test abs( imag( SF.Li(s, z) ) ) < 1.0e-14 - end - end - - # for real s, and real z>=1, the imaginary part is give - S = [-1.5, 0.1, 2] - Z = [1.05, 3.0] - for i=1:length(S) - for j=1:length(Z) - s = S[i] - z = Z[j] - # println("s = $s; z = $z") - μ = log(z) - @test imag( SF.Li(s, z) ) ≈ -pi*μ^(s-1)/gamma(s) - end - end - - S_r = [2.1 2.5 3.0] - S_i = [-1.3, -1.0, -0.5, 0.0, 0.5, 1.0, 1.3] - z = 3.0 - 0.1im - for i=1:length(S_r) - for j=1:length(S_i) - s = Complex(S_r[i], S_i[j]) - @test SF.Li(s,z) + SF.Li(s,-z) ≈ complex(2)^(1-s) * SF.Li(s, z^2) - end - end - end -end diff --git a/test/li_test2.jl b/test/li_test2.jl deleted file mode 100644 index 55c54dd8..00000000 --- a/test/li_test2.jl +++ /dev/null @@ -1,79 +0,0 @@ -using SpecialFunctions -using Base.Test - -const SF = SpecialFunctions - -# test functions, see runtests.jl -relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x) -relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x))) -≅(a,b) = relerrc(a,b) ≤ 1e-9 - -# test functions, similar to runtests.jl, but can't do relative error when some values are zero -abserr(z, x) = z == x ? 0.0 : abs(z - x) -abserrc(z, x) = max(abserr(real(z),real(x)), abserr(imag(z),imag(x))) -≒(a,b) = abserrc(a,b) ≤ 1e-10 - -S = Array(Complex, (0,)) -Z = Array(Complex, (0,)) -errors = Array(Complex, (0,)) -rel_errors = Array(Complex, (0,)) -L = 0 -L2 = 0 -diff = 0 -rel_diff = 0 - -file = "li_testdata_alpha.dat" -f = open(file, "r") -for line in eachline(f) - if ismatch(r"#", line) - else - v = split(line, ',') - - s = parse(Float64, v[1]) + im*parse(Float64, v[2]) - z = parse(Float64, v[3]) + im*parse(Float64, v[4]) - L = parse(Float64, v[5]) + im*parse(Float64, v[6]) - L2 = SF.Li(s,z,1.0e-18) - diff = L - L2 - rel_diff = complex(real(diff)/real(L), imag(diff)/imag(L)) - - push!(S, s) - push!(Z, z) - push!(errors, diff) - push!(rel_errors, rel_diff) - - # println(" L = $L, L2 =$L2, diff = $diff, \t\t rel diff = $rel_diff") - println(" s=$s, z=$z, |error|=$(ceil(log10(abs(diff))))") - end -end - -# test particular values from Wolfram alpha -# @testset "Li tests from Wolfram alpha" begin - -# @test SF.Li(-5.000001, 0.2) ≅ 6.904305364987799724230921116283822405784472717903797757356 -# @test SF.Li(-2.000001, 0.2) ≅ 0.468750244180354668389455477288861311399345621141103816518 -# @test SF.Li(-1.000001, 0.2) ≅ 0.312500094184052926057717828023167764362038234067425674984 -# @test SF.Li( 0.000000, 0.2) ≅ 0.25 -# @test SF.Li( 0.000001, 0.2) ≅ 0.249999960605813559100141040899469061220130457763090931203 -# @test SF.Li( 1.000001, 0.2) ≅ 0.223143533840628659834227863703211050928449673084530824304 -# @test SF.Li( 2.000001, 0.2) ≅ 0.211003767368667938482167666994614717744311348708200162427 -# @test SF.Li( 3.000001, 0.2) ≅ 0.205324191902667822690683746355210013921766645655171501890 -# @test SF.Li( 5.000001, 0.2) ≅ 0.201284594885756384926736867628954299128217148725340685679 - -# @test SF.Li(-5.000001, 0.99) ≅ 1.16439554606503726735945485177408748716681200000005e14 -# @test SF.Li(-2.000001, 0.99) ≅ 1.97011088076187708022430311310006082286580942497213e6 -# @test SF.Li(-1.000001, 0.99) ≅ 9900.04972775403451817385172007212640686495346816117 -# @test SF.Li( 0.000000, 0.99) ≅ 99 -# @test SF.Li( 0.000001, 0.99) ≅ 98.9995988050883517986218058007486858307489619231956 -# @test SF.Li( 1.000001, 0.99) ≅ 4.605161353580737753746071585981515640203854412492030454580 # error ~ 1.0e11 -# @test SF.Li( 2.000001, 0.99) ≅ 1.588624649826159899085393338960947692670814871114246941712 # error ~ 1.0e11 -# @test SF.Li( 3.000001, 0.99) ≅ 1.185832744102043627496839259642313182039620569301974825714 -# @test SF.Li( 5.000001, 0.99) ≅ 1.026110449210262375841139988837721699091019919425544651696 - -# @test SF.Li( 1.000001, 1.20) ≅ 1.60944129470676213105637136369425072009730867713452098010 - 3.14158912002728351076393334268714060723117930694360052182im -# @test SF.Li( 1.000001, 1.50) ≅ 0.69315092620532843831758654220946253504678190003195418383 - 3.1415916309839162783984857611820275451978840493033368692im -# @test SF.Li( 1.000001, 1.70) ≅ 0.35667861585275033749153552050383961466011713941159572502 - 3.1415924761565638528178063690254522808240730779853094090im -# @test SF.Li( 1.000001, 1.999999) ≒ -# @test SF.Li( 1.000001, 2.00) ≅ -# @test SF.Li( 1.000001, 2.10) ≅ -0.0953067628118213777754205736836756047381793027303437193 - 3.14159352922832327133509242559303053858610422117044987im - -# end diff --git a/test/li_testdata.dat b/test/li_testdata.dat deleted file mode 100644 index 5e43d45f..00000000 --- a/test/li_testdata.dat +++ /dev/null @@ -1,1002 +0,0 @@ -# test data from li_testdata.py -# using mpmath library -# s(real, imag), z(real, imag), Li(real, imag) -2.57636950631092576103,2.89889121739954180867,0.13267161787652381744,-1.52908730194326358820,0.22742653414510977106,-1.28004045152782230943 --2.18434643020828289650,0.06266903366343373749,-2.04420634002174583799,-2.87365889020505971274,0.12933937360075600798,-0.04815800746168075186 -0.39862395296750768869,0.26674920931720969008,1.09293660067646314360,-1.82794188747062524847,-0.66760355167941398680,-0.96417825863038475465 -0.01001056725314488846,-0.12948352074537577971,-3.01165800252148363469,1.07599435732206760719,-0.75279632600315304902,0.14841623345829757508 -0.64142220199769650613,4.77822408648137209042,0.40593835461992844316,-0.28940461632984104678,0.40023759633717770345,-0.17532047368394246267 -2.46551435005784735921,0.39758249638685100313,1.81806205230641815263,-0.73108853246503358037,1.51148893026643205140,-1.57786846239759315225 -0.43634362672116489001,2.04857735697403597896,1.39249404490592310069,0.25694449726571277903,41.71385136831121798195,1.52618389902109252709 --2.16461604919208916442,0.89044354472209708007,0.15372696343850464351,1.44093519371850731225,1.38691892283118933804,-2.40697690029367006304 -0.43246587560014609908,2.17637082227104627918,-0.10312053084785084700,0.40392812015884255183,-0.21797177269881662376,0.49922954476336078855 -1.33355287725129900700,-2.17376922434895902825,-0.80332052014803778661,-1.00005713790473693692,-1.34253239608775731817,-0.74045035995397889828 -3.96123143904649221980,-0.18572405131216915275,1.30444044761514410524,1.23875012434743791978,1.19979011846723793155,1.44894225295413314747 --0.56174686037378518400,-3.10167365426089425284,1.92968794135228649544,-0.81439353932718372420,-34.37985149865603062835,-310.72209452809499907744 -1.43591331348979966975,-2.61052965021992910977,-0.87596600393965950992,2.51364268921072264718,-0.00240060623286146232,1.77862481057026045050 -2.86200797610503654411,-2.60491724227671461733,-2.66561495817788651763,-0.08852887520366388430,-2.58492881130488783015,0.59636078882324561334 -1.45648264663743942648,0.32100935875865949098,0.60710941565396747688,-1.97767284994922598607,-0.34651118436440153658,-1.50914700734366702406 -1.17358342762760714528,2.23370468227367124925,-0.87134504056796591165,-2.86697612732693185578,0.02191048737842231364,-1.66959799807934339277 --1.51764164736749784801,1.52331601167396524232,-3.46739438367286778586,-0.18375357561120775340,0.55793543949284429839,-0.43103843044477591073 --1.98201657029098687879,-0.26226888395543751509,-0.48904317349607717702,0.03171356654463203251,-0.07680467931235855639,0.03648050325174385239 -3.00241479260680144137,0.84142088721576069332,2.66742549073045198682,-0.28282542860690790265,2.44525567768755536235,-1.35612886097764895332 --0.95917321565393154348,0.75762116453197803079,-5.67158157336007473504,-0.07977762938413437976,0.01167117801068017109,-0.38069096777907401874 -0.32033957815045399631,-2.47041748802708260868,0.92873044514510194603,-1.11848960865392044184,3.75112911588832753651,-7.63900216285321853604 --4.91820038055391428600,-0.42663678073589322848,-1.95769148859355879289,-1.04119174520029278419,-0.14675605138638325720,0.50950796408382825753 --0.30456883741073870198,2.50195069846764628352,0.20629635788318034795,-0.05697124996734893171,0.15734020053052566457,-0.09088719830683018863 -0.77800883233477435841,-3.62418473164922927410,2.48024774124713598411,-2.15417384059296823651,28.21635549594314085198,-63.10598120235564323366 -0.87819014227001401807,-2.25356099093727380023,-1.95297072922188053035,-0.79257575054300066775,-1.95625305800968041225,0.64972884357600035887 -3.79149692523689152424,1.39532886893695096120,-1.20839298809733430851,-0.56862040039246619649,-1.10566489480446561444,-0.57429793392406625063 --2.30279746843796928601,-0.06832002103614588651,-1.14614485950032252148,1.44377580526245252734,0.17041272226224379871,-0.01448773674151459037 --2.71413512039784432517,-0.66928640711966458365,-1.68401719373821956438,-1.43741691800371840237,0.37073273095053277393,-0.22865458346746941887 -1.42232310660463934227,0.25261150892671274670,1.17039821256232734470,2.37814969807393072898,-0.79837107862665712688,2.20501357618023607898 -2.29913946302378713327,-2.74358544283092165017,1.07393138030001566996,-3.52253691384292366706,-0.51719208756770840285,-8.15901682867007238542 --0.12772004181612056639,3.83823658666309031062,-0.38731719070881998057,-0.73838387719184028768,0.05594991857385048922,-0.81520000185779839530 -0.34066167886494569395,0.03585684501928015910,0.05318747093745866683,-1.51457170861579704102,-0.66688998930687237721,-0.68255534951796159593 -2.16349339764403714170,1.77811958301283312700,-0.42445876611790778199,0.62947009262530928542,-0.54654624434945053846,0.59936265306917058115 -1.31689436365501877368,2.06434124067226187904,0.78550870672114081472,1.39020321916685141161,0.79812585532765567553,3.48361853783185226874 --0.52642133366774257475,-2.13875148392894276483,-0.99066145233585711782,2.03831805674157084951,0.30279632813687445969,0.71107113756092732437 -1.95544529470547767680,0.29253802331978351337,-1.13479649924998171606,0.61535653417777003149,-0.97592426794115616495,0.36984512894420268347 -3.32686518099973405072,2.70890051511893092950,-1.36711658057389162657,-0.08701193265562583157,-1.36062528114901626353,-0.24448372270119145266 --2.90341785323817624942,-2.27113454100386347179,0.37650341788782715202,0.04953097506133657957,-1.93219787742304327516,-0.79661411257734271452 -1.92951174131699221270,2.53525277606702204025,1.66982203179625887657,2.63955414184140302325,1.48231666425040664592,8.55645877554493416994 --1.09417348158409288850,-2.25770066689586412778,1.00106554569333172644,5.35737321005578959188,0.67151614766503708864,0.00555369548256008389 -0.71357545727120230694,-2.30349485744346393901,0.48478404021509668986,2.85122693781474767505,0.52992898391528886481,1.17578897325882958391 --2.06870467472840546819,1.60677618978963088203,-1.22233163169028724582,2.54567165255331273599,2.77320321306179184973,-0.84055132497828688454 -1.57088055424708072927,0.60812636413520937406,4.00012107057584032077,-0.81781945489823182349,0.82609696499190876651,-2.65426613070186334653 --1.37218437889669275798,3.70965888736754711985,-1.75283999732885553691,4.39778582258539962879,14.73997055296313085648,-40.52708585201452962110 --0.08026000995000538707,-2.07334338980241605910,-0.00386723229507326656,0.26099126485103318895,-0.00165821839426662861,0.20260818782841760743 -0.40229017080170370591,-0.38392075078513954178,2.16248402262439398669,-4.63977051154448094650,-2.09955306080444881189,-0.45924952940128005441 --1.10920562748263318298,-0.52434290851183762427,3.63934226643014202551,-3.98542305242313066316,0.87518086485723589618,0.61458120835071861077 --0.67945090682829800688,-2.28621496118836198264,-1.32960792172058095417,1.28090642650475650122,0.21084028977952817829,0.82469453395591452161 -0.82183038346911618799,2.88014128717291706394,-1.19909242827884132687,0.53733355064605836660,-1.65606137084607185628,-0.12515458373575635842 -2.34678460682534151260,1.80679248259090829976,-0.67200563670512425407,2.25618821573730743424,-1.69887764400868146453,2.20013318994669315742 --1.84760168994507711915,3.60673636903974292878,0.30874356954873793946,-0.22527031173367617622,0.19665059952279206890,0.01394525259271747405 -0.54169341994591413414,1.69807464434043620116,3.48298001167853765736,-0.28403380902366798244,0.70480791644954676745,-0.70489972650889953787 --0.73513632646878190879,1.17307020994419142212,-1.74314065957547525976,-3.39234193828196595177,0.05354516216166078230,-0.42656358074546274484 -1.67185665023741614199,-0.75882800689206086897,2.25361619810877611414,-2.05339898515298635573,-0.42030236918342528396,-4.17777410728974984977 --5.79250188704576096654,0.56596537668784030473,0.30986639234384538710,3.20084370422189978811,-9.01808159029483746849,21.40867965811729689563 -1.05109981599858293144,0.61877161739142749752,1.17326369946115360854,-0.73440127914186070779,0.30608418261238529157,-1.23199148980728900860 -0.15503892540664135358,-2.70378743245048758226,1.03830135106726406669,-1.61065967260576936582,-2.32527697533546673370,-14.42644115772991142421 --0.89166365640015055050,1.39948687288938811157,1.83106048366111973102,-2.01434681797823911253,0.15663103759285479377,-0.19921504652669783098 -4.00769088178234156317,-1.18327680795679590631,1.67003186602721598319,1.90260102833674604739,1.32200806643155255315,1.96475227538506858060 -0.44812911112920333911,0.34434643721426638763,3.59440709625736554145,1.78034237272737616564,-2.59742273049532235163,1.19169058685757933702 -0.89090077303996106917,-3.64855457256029103164,-1.49418737198113382725,2.32537442164026897373,0.26947428173695237019,2.41019426223886590321 -0.38809994889293314424,-1.91008105365219638117,-1.28487820822796483533,-0.61234214269290132560,-1.25710611266099969718,0.34287948237106180560 -1.37213120767885876639,0.77402645903904321312,1.99474445929363630725,-1.63404461577669435357,0.28974984892751637355,-1.68830146106600520461 -1.97204834410828477864,-1.00376018200548089432,-0.59783929373324551371,3.46624347789627229588,-0.69070186587592108385,2.02858790516247999847 -0.14819963768736743592,-0.27876363294676420601,-0.42055445565994536983,-0.76969459443142485000,-0.53120047207626241992,-0.32050330785615155360 -3.11879465770158903482,2.75315909254188140309,1.43385313198680197644,0.36837694677857063530,1.03672061607615750845,-0.15848446521068598747 -2.08510280406691395072,-0.15542344245059303010,0.90524147842666657926,0.80373769394874872152,0.67858170303970966053,1.15906288036870197544 -0.17789838462358123494,3.29684324165760234138,3.51049897417332523730,2.64730057192176904124,-101.03639955076174317128,50.41481221106550236755 --3.82629491006355726412,3.67187307194324974446,1.40475528546299499588,-0.90113826679406827669,-0.31701618420331001413,0.29311336084789407774 --0.04878828407038606624,2.27665646417371014820,2.34971960704918281237,1.71008603771938472171,-31.09371873503737759847,10.80575843829999449497 -0.28171775859002384790,0.07161669167101274158,1.66411334282586720335,-0.18283482369229134279,-1.70559038337728008372,-1.54906094105174951814 --1.79613006557046039902,-1.24673645918604036886,-0.27929619983787040693,0.66503307719534898723,0.02352640445273061764,0.09940968976709667337 -4.52740098115719380445,-2.73961037542987684290,0.95441890249159744286,-0.18333017882651711505,0.94909112108665605323,-0.13874288339050097618 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-2.40007489210605529095,2.87660762026954452963,5.69370195484752983361,-0.50901867863753191834,3.18184282108034643954,-0.65118188288410305820 --0.28088304419904919218,0.87630768879533871285,0.18980544059841819582,-2.14549658946724441932,-0.21347130074107478759,-0.43572201313373565279 diff --git a/test/li_testdata.py b/test/li_testdata.py deleted file mode 100755 index 00a36ec3..00000000 --- a/test/li_testdata.py +++ /dev/null @@ -1,26 +0,0 @@ -#!/usr/bin/python -# python code using mpmath to create a test data sets -from mpmath import * -mp.dps = 30 -mp.pretty = True - -import random -random.seed(1) - -N = 1000 -mu = 0.0 -sigma = 2.0 -fid = open('li_testdata.dat', 'w') -fid.write('# test data from li_testdata.py\n') -fid.write('# using mpmath library\n') -fid.write('# s(real, imag), z(real, imag), Li(real, imag)\n') -for i in range(1,N): - s = random.gauss(mu, sigma) + j*random.gauss(mu, sigma) - z = random.gauss(mu, sigma) + j*random.gauss(mu, sigma) - L = polylog(s, z) - fid.write('%.20f,%.20f,%.20f,%.20f,%.20f,%.20f\n' % (s.real, s.imag, z.real, z.imag, L.real, L.imag)) - - -fid.close() - - diff --git a/test/li_testdata_alpha.dat b/test/li_testdata_alpha.dat deleted file mode 100644 index a4ea2d97..00000000 --- a/test/li_testdata_alpha.dat +++ /dev/null @@ -1,42 +0,0 @@ -# test data from Wolfram alpha -# -# s(real, imag), z(real, imag), Li(real, imag) -# -# start with values with near integer s, because these might be problematic as they are near special cases -# in particular for the series sum around z=1, which we use when 0.5 < |z| < 2.0 --5.000001, 0.0, 0.2, 0.0, 6.904305364987799724230921116283822405784472717903797757356, 0.0 --2.000001, 0.0, 0.2, 0.0, 0.468750244180354668389455477288861311399345621141103816518, 0.0 --1.000001, 0.0, 0.2, 0.0, 0.312500094184052926057717828023167764362038234067425674984, 0.0 - 0.000000, 0.0, 0.2, 0.0, 0.25, 0.0 - 0.000001, 0.0, 0.2, 0.0, 0.249999960605813559100141040899469061220130457763090931203, 0.0 - 1.000001, 0.0, 0.2, 0.0, 0.223143533840628659834227863703211050928449673084530824304, 0.0 - 2.000001, 0.0, 0.2, 0.0, 0.211003767368667938482167666994614717744311348708200162427, 0.0 - 3.000001, 0.0, 0.2, 0.0, 0.205324191902667822690683746355210013921766645655171501890, 0.0 - 5.000001, 0.0, 0.2, 0.0, 0.201284594885756384926736867628954299128217148725340685679, 0.0 -# - 1.000001, 0.0, 0.5, 0.0, 0.693147007663231669849516514839299401059360031508002626249, 0.0 -# --5.000001, 0.0, 0.99, 0.0, 1.16439554606503726735945485177408748716681200000005e14, 0.0 --2.000001, 0.0, 0.99, 0.0, 1.97011088076187708022430311310006082286580942497213e6, 0.0 --1.000001, 0.0, 0.99, 0.0, 9900.04972775403451817385172007212640686495346816117, 0.0 - 0.000000, 0.0, 0.99, 0.0, 99, 0.0 - 0.000001, 0.0, 0.99, 0.0, 98.9995988050883517986218058007486858307489619231956, 0.0 - 1.000001, 0.0, 0.99, 0.0, 4.605161353580737753746071585981515640203854412492030454580, 0.0 - 2.000001, 0.0, 0.99, 0.0, 1.588624649826159899085393338960947692670814871114246941712, 0.0 - 3.000001, 0.0, 0.99, 0.0, 1.185832744102043627496839259642313182039620569301974825714, 0.0 - 5.000001, 0.0, 0.99, 0.0, 1.026110449210262375841139988837721699091019919425544651696, 0.0 -# - 1.000001, 0.0, 1.20, 0.0, 1.60944129470676213105637136369425072009730867713452098010, -3.14158912002728351076393334268714060723117930694360052182 - 1.000001, 0.0, 1.50, 0.0, 0.69315092620532843831758654220946253504678190003195418383, -3.1415916309839162783984857611820275451978840493033368692 - 1.000001, 0.0, 1.70, 0.0, 0.35667861585275033749153552050383961466011713941159572502, -3.1415924761565638528178063690254522808240730779853094090 - 1.000001, 0.0, 1.999999, 0.0, 4.48462808108367607858388644733186363064154361287440e-6,-3.14159331552720658388549501733501317621202043844284 - 1.000001, 0.0, 2.00, 0.0, 3.48462690882559763270126765927498392238379428297186e-6, -3.14159331552947276581779433082131101718301247144061 - 1.000001, 0.0, 2.10, 0.0, -0.0953067628118213777754205736836756047381793027303437193, -3.14159352922832327133509242559303053858610422117044987 -# -# -# difficult cases are for s close to a positive integer, and using expansion around z=1 - 1.000001, 0.0, 0.99, 0.0, 4.605161353580737753746071585981515640203854412492030454580, 0.0 - 1.000001, 0.0, 1.20, 0.0, 1.60944129470676213105637136369425072009730867713452098010, -3.14158912002728351076393334268714060723117930694360052182 - 1.000001, 0.0, 1.50, 0.0, 0.69315092620532843831758654220946253504678190003195418383, -3.1415916309839162783984857611820275451978840493033368692 - 1.000001, 0.0, 1.70, 0.0, 0.35667861585275033749153552050383961466011713941159572502, -3.1415924761565638528178063690254522808240730779853094090 - 1.000001, 0.0, 1.999999, 0.0, 4.48462808108367607858388644733186363064154361287440e-6,-3.14159331552720658388549501733501317621202043844284 From 384786cf676432d6b021fbecc5959a88a1a133f2 Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Fri, 2 Jun 2017 16:56:15 +0930 Subject: [PATCH 13/14] changed Li to polylog --- src/polylog.jl | 225 +++++++ test/polylog_test.jl | 158 +++++ test/polylog_test2.jl | 79 +++ test/polylog_testdata.dat | 1002 +++++++++++++++++++++++++++++++ test/polylog_testdata.py | 26 + test/polylog_testdata_alpha.dat | 42 ++ 6 files changed, 1532 insertions(+) create mode 100644 src/polylog.jl create mode 100644 test/polylog_test.jl create mode 100644 test/polylog_test2.jl create mode 100644 test/polylog_testdata.dat create mode 100755 test/polylog_testdata.py create mode 100644 test/polylog_testdata_alpha.dat diff --git a/src/polylog.jl b/src/polylog.jl new file mode 100644 index 00000000..ada19413 --- /dev/null +++ b/src/polylog.jl @@ -0,0 +1,225 @@ +""" + polylog(s, z) + + Calculates the Polylogarithm function ``Li_s(z)`` defined by + +```math + L_s = \sum_{n=1}^{\infty} \frac{z^n}{n^s} +``` + + For ideas going into this see + + + Crandall, "Note on fast polylogarithm computation", 2006, + which focusses on the case where s=n (integer and real) + http://www.wolfgang-ehrhardt.de/Polylog.pdf + + + Vepstas, "AN EFFICIENT ALGORITHM FOR ACCELERATING THE CONVERGENCE + OF OSCILLATORY SERIES, USEFUL FOR COMPUTING THE + POLYLOGARITHM AND HURWITZ ZETA FUNCTIONS", 2007 + which treats the general case, but presumes arbitrary precision arithmetic + https://arxiv.org/abs/math/0702243 + + + Wood, "The computation of Plylogarithms", 1992 + which focusses on s=n, integer and real, and which has formatting issues making + it hard to read correctly. + https://www.cs.kent.ac.uk/pubs/1992/110/ + + + Maximon, "The dilogarithm function for complex argument", 2003 + which provides useful test cases for s=2 + + + Zagier, "The dilogarithm function in geometry and number theory", 1989 + similar to Maximon + + Of these the only one that actually specifies a full algorithm is + Crandall, and he also treats special cases more carefully, so this + is the one that I have paid most attention to. However, extending it + for s on the whole complex plane requires some additions, and many + of these are actually most nicely documented on the wikipedia page + + + https://en.wikipedia.org/wiki/Polylogarithm + + With further details at + + + http://mathworld.wolfram.com/Polylogarithm.html + + http://dlmf.nist.gov/25.12#ii + + http://mathworld.wolfram.com/Trilogarithm.html + + http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/ + + The wiki page points out some errors in earlier works, but not all + parts on the page have references, and not all statements seem to + come from any of the listed references? + + The code draws heavily on existing functions, in particular the + Hurwitz-zeta function, which is aliased to zeta(s,q) in Julia. + + Accuracy has been tested using many of the identities known for Li + and relations to known functions as special cases, and by comparison + to `polylog(s, z)` in the `mpmath` arbitrary-precision package in Python. + + http://mpmath.org/doc/current/functions/zeta.html + + The latter shows deviations of the order of + + + 10^{Im(s) - 20} for Im(s) < 0 + + 10^{Im(s) - 20} for Im(s) > 0 + + It isn't clear whether we can do better than this with + double-precision arithmetic. + +## Arguments +* `s::Complex`: the 'fractional' coefficient +* `z::Complex`: the point at which to calculate it +* `accuracy::Real=1.0e-18`: nominal accuracy of calculation, but mainly useful for testing + +## Examples +```jldoctest +julia> polylog(-1.0, 0.0) +(0.0,1) +``` +""" +function polylog(s::Number, z::Number, accuracy::Real=1.0e-18) + T = 0.5 # threshold at which we change algorithms + if z ≈ 1.0 + if real(s) > 1 + return zeta(s) + else + return Inf + end + elseif z ≈ -1.0 + return -eta(s) + elseif s ≈ 0.0 + return z ./ (1-z) + elseif abs(z) <= T + return polylog_direct(s, z, accuracy) + elseif abs(z) >= 1/T && isinteger(s) && real(s) < 0 + # use inversion formula to calculate in terms of Li_n(1/z) + # but note for negative integer s, it collapses to something small + return -(-1.0)^s .*polylog_direct(s, 1/z, accuracy) + elseif abs(z) >= 1/T + # use inversion formula to calculate in terms of Li_s(1/z) + z = convert(Complex{Float64}, z) + G = (2*pi*im)^s * zeta( 1-s, 0.5 + log(-z)/(2*pi*im) ) / gamma(s) + F = complex(-1.0)^s * polylog_direct(s, 1/z, accuracy) + + A = 2*pi*im*log(z)^(s-1)/(gamma(s)) + if ( isreal(z) && real(z)>=1 ) + Θ = 1 + else + Θ = 0 + end + # println("G = $G, F=$F, Θ=$Θ, A=$A") + return ( G - F - Θ*A ) + else + # power series around mu=0, for z = e^mu + polylog_series_mu(s, z, accuracy) + end +end + + +#################################################################### +#### these are component functions and aren't exported at this point + +# Dirichlet beta function, for testing results +# https://en.wikipedia.org/wiki/Dirichlet_beta_function +# but don't export as it would create confusion with Euler integral +function Dbeta(s::Number) + β = 4.0^(-s) * ( zeta(s,0.25) - zeta(s,0.75) ) +end + +function polylog_direct(s::Number, z::Number, accuracy=1.0e-18) + # calculate using direct definition + if abs(z) > 1 || ( abs(z) ≈ 1 && real(s) <= 2) + throw(DomainError()) + end + if abs(z) > 1/2 + warn("Slow convergence for |z| > 1/2") + end + total = 0.0 + L = ceil(-log10(accuracy)*log2(10)) # summation limit from Crandall, which is conservative, but based on real s + a = z; + for n=1:L + total += a + a *= z * ( n/(n+1.0) )^s + # println(" total = $total") + if abs(a)/abs(total) < 1.0e-30 + break + end + end + return total +end + +function polylog_series_mu(s::Number, z::Number, accuracy=1.0e-18) + # calculate using power series around μ = log(z) = 0 + μ = log(convert(Complex{Float64}, z)) + # println("μ = $μ") + if abs(μ) > 2*pi + throw(DomainError()) + end + L = Int(ceil(-log10(accuracy)*log2(10))) # revisit this limit + if isinteger(s) + n = Int(round(s)) + if n>1 + # Crandall's 1.4 for s integer + total = μ^(n-1)*(harmonic(n-1) - log(-μ))/gamma(n) + # println(" μ=$μ, total = $total") + tmp = 1 + for m=0:L + if n - m != 1 + # total += μ^m * zeta(n - m) / gamma(m+1) + total += tmp * zeta(n - m) + end + # println(" m=$m, total = $total, tmp=$tmp, ctmp=$(μ^m /gamma(m+1))") + tmp *= μ/(m+1) + if abs(tmp)/abs(total) < 1.0e-30 + break + end + end + # println(" μ=$μ, total = $total") + A = 2*pi*im*log(complex(z))^(s-1)/gamma(n) + if isreal(z) && real(z)>=1 + total -= A + end + # println(" μ=$μ, total = $total") + elseif n==1 + total = -log(complex(1-z)) + elseif n==0 + total = z / (1-z) + elseif n==-1 + total = z / (1-z)^2 + elseif n<-1 + # Crandall's 1.5 for s integer + total = factorial(-n) * (-μ)^(n-1) + tmp = 1 + for k=0:L + # total -= μ^k * bernoulli(k-n+1, 0.0) / ( gamma(k+1)*(k-n+1) ) + total -= tmp * bernoulli(k-n+1, 0.0) / (k-n+1) + tmp *= μ/(k+1) + if abs(tmp)/abs(total) < 1.0e-30 + break + end + end + else + error("Should not get this case") + end + else + # equivalent of Crandalls 1.4 for s non-integer + total = gamma(1-s) * (-μ)^(s-1) + # println(" μ=$μ, total = $total") + tmp = 1 + for k=0:L + # total += μ^k * zeta(s-k)/factorial(Float64(k)) + total += tmp * zeta(s-k) + # println(" tmp=$(tmp* zeta(s-k)), total = $total") + tmp *= μ/(k+1) + if abs(tmp)/abs(total) < 1.0e-30 + break + end + end + + A = 2*pi*im*log(complex(z))^(s-1)/(gamma(s)) + if isreal(z) && real(z)>=1 + total -= A + end + end + return total +end diff --git a/test/polylog_test.jl b/test/polylog_test.jl new file mode 100644 index 00000000..4bf74eeb --- /dev/null +++ b/test/polylog_test.jl @@ -0,0 +1,158 @@ +using SpecialFunctions +using Base.Test + +const SF = SpecialFunctions + +# test functions, see runtests.jl +relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x) +relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x))) +≅(a,b) = relerrc(a,b) ≤ 1e-13 + +# test functions, similar to runtests.jl, but can't do relative error when some values are zero +abserr(z, x) = z == x ? 0.0 : abs(z - x) +abserrc(z, x) = max(abserr(real(z),real(x)), abserr(imag(z),imag(x))) +≒(a,b) = abserrc(a,b) ≤ 1e-13 + +# golden ratio is used in some tests +ϕ = (sqrt(5)+1)/2.0 + +@testset "Polylogarithm polylog" begin + @testset "Check different input formats" begin + @test SF.polylog(-1.0, 0.0) ≈ 0.0 + @test SF.polylog(-1, 0.0) ≈ 0.0 + @test SF.polylog(-1.0, Complex(0.0)) ≈ 0.0 + 0.0im # don't seem to need imaginary bit here + @test SF.polylog(Complex(-1.0), 0.0) ≈ 0.0 + 0.0im + @test SF.polylog(Complex(-1.0), 0.0) ≈ 0.0 + @test SF.polylog(1, 0) ≈ 0.0 + x = collect(0.0:0.1:0.9) + @test all([SF.polylog.(1, x)[i] ≈ SF.polylog(1, x[i]) for i=1:length(x)]) + @test SF.polylog(Complex(-1.0), Complex(0.3)) ≈ SF.polylog(-1.0, 0.3) + @test SF.polylog(Complex(-1.0), Complex(0.3)) ≈ SF.polylog(-1.0, 0.3) + end + + @testset "Test s = n (a real integer)" begin + # simple cases + @test SF.polylog(1, 0.5) ≈ log(2) + @test !isfinite( SF.polylog(1, 1.0) ) + @test SF.polylog(1, 2) ≒ -pi*im # comes from -log(complex(1-2)) + + @testset " dilogarithm for real z" begin + @test SF.polylog(2,-1.0) ≈ -pi^2/12.0 + @test SF.polylog(2, 0.0) ≈ 0.0 + @test SF.polylog(2, 0.5) ≈ pi^2/12 - 0.5*log(2)^2 + @test SF.polylog(2, 1.0) ≈ pi^2/6.0 + @test SF.polylog(2, 2.0) ≈ pi^2/4 - im*pi*log(2) + @test SF.polylog(2, -ϕ) ≈ -pi^2/10 - log(ϕ)^2 + @test SF.polylog(2, -1/ϕ) ≈ -pi^2/15 + log(ϕ)^2/2 + @test SF.polylog(2, 1/ϕ^2) ≈ pi^2/15 - log(ϕ)^2 + @test SF.polylog(2, 1/ϕ) ≈ pi^2/10 - log(ϕ)^2 + @test SF.polylog(2, ϕ) ≈ 11*pi^2/15 + log(Complex(-1/ϕ))^2/2 # wiki has this one, but no ref + @test SF.polylog(2, ϕ^2) ≈ -11*pi^2/15 - log(Complex(-ϕ))^2 + + # identities + Z = [3.0 + 0.4im, -3.0 + 0.4im, 3.0 - 0.4im, -3.0 + -0.4im] + for i=1:length(Z) + z = Z[i] + @test SF.polylog(2, z) + SF.polylog(2, 1/z) ≈ -pi^2/6.0 - log(Complex(-z))^2/2.0 + end + end + + @testset " trilogarithm for real z" begin + @test SF.polylog(3,-1.0) ≈ -3*SF.zeta(3)/4 + @test SF.polylog(3, 0.0) ≈ 0.0 + @test SF.polylog(3, 0.5) ≈ log(2)^3/6.0 - pi^2*log(2)/12.0 + (7.0/8.0)*SF.zeta(3) + @test SF.polylog(3, 1.0) ≈ SF.zeta(3) + @test SF.polylog(3, ϕ^(-2)) ≈ 4*SF.zeta(3)/5 + 2*log(ϕ)^3/3 - 2*pi^2*log(ϕ)/15 + end + + @testset " general case for real z" begin + # X = collect(-0.95:0.1:0.95) + X = collect(-3.0:0.1:3.0) + for i=1:length(X) + x = X[i] + # println(x) + @test SF.polylog(1, x) ≈ -log(Complex(1-x)) + @test SF.polylog(0, x) ≈ x ./ (1-x) + @test SF.polylog(-1, x) ≈ x ./ (1-x).^2 + @test SF.polylog(-2, x) ≈ x .* (1+x) ./ (1-x).^3 + @test SF.polylog(-3, x) ≈ x .* (1+4*x+x.^2) ./ (1-x).^4 + @test SF.polylog(-4, x) ≈ x .* (1+x) .* (1+10*x+x.^2) ./ (1-x).^5 + end + end + + @testset " general case for complex z" begin + X = collect(-3.0:0.5:3.0) + Y = [-1.3, -0.4, 0.4, 1.5] + for i=1:length(X) + for j=1:length(Y) + z = Complex(X[i], Y[j]) + # println(z) + @test SF.polylog(1, z) ≈ -log(Complex(1-z)) + @test SF.polylog(0, z) ≈ z ./ (1-z) + @test SF.polylog(-1, z) ≈ z ./ (1-z).^2 + @test SF.polylog(-2, z) ≈ z .* (1+z) ./ (1-z).^3 + @test SF.polylog(-3, z) ≈ z .* (1+4*z+z.^2) ./ (1-z).^4 + @test SF.polylog(-4, z) ≈ z .* (1+z) .* (1+10*z+z.^2) ./ (1-z).^5 + end + end + end + end + + @testset "Particular values |z| == 1" begin + S_r = [2.1 2.5 3.0] + S_i = [-1.3, -1.0, -0.5, 0.0, 0.5, 1.0, 1.3] + for i=1:length(S_r) + for j=1:length(S_i) + s = Complex(S_r[i], S_i[j]) + @test SF.polylog(s, -1.0) ≈ -SF.eta(s) + @test SF.polylog(s, 1.0) ≈ SF.zeta(s) + @test SF.polylog(s, im) ≈ - 2.0.^(-s).*SF.eta.(s) + im*SF.Dbeta.(s) + end + end + + # the singularity + @test !isfinite( SF.polylog(0.5, 1.0) ) + end + + @testset "Additional Identities" begin + z = 0.5 + for n=1:5 + @test SF.polylog(-n,z) + (-1)^n * SF.polylog(-n, 1/z) ≈ 0.0 + end + + # for real s, and real z<1, SF.polylog should be real + S = [-1, 0.1, 2] + Z = [-2, -1.0, 0.1, 0.95] + for i=1:length(S) + for j=1:length(Z) + s = S[i] + z = Z[j] + # println("s = $s; z = $z") + @test abs( imag( SF.polylog(s, z) ) ) < 1.0e-14 + end + end + + # for real s, and real z>=1, the imaginary part is give + S = [-1.5, 0.1, 2] + Z = [1.05, 3.0] + for i=1:length(S) + for j=1:length(Z) + s = S[i] + z = Z[j] + # println("s = $s; z = $z") + μ = log(z) + @test imag( SF.polylog(s, z) ) ≈ -pi*μ^(s-1)/gamma(s) + end + end + + S_r = [2.1 2.5 3.0] + S_i = [-1.3, -1.0, -0.5, 0.0, 0.5, 1.0, 1.3] + z = 3.0 - 0.1im + for i=1:length(S_r) + for j=1:length(S_i) + s = Complex(S_r[i], S_i[j]) + @test SF.polylog(s,z) + SF.polylog(s,-z) ≈ complex(2)^(1-s) * SF.polylog(s, z^2) + end + end + end +end diff --git a/test/polylog_test2.jl b/test/polylog_test2.jl new file mode 100644 index 00000000..8e8f6f0c --- /dev/null +++ b/test/polylog_test2.jl @@ -0,0 +1,79 @@ +using SpecialFunctions +using Base.Test + +const SF = SpecialFunctions + +# test functions, see runtests.jl +relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x) +relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x))) +≅(a,b) = relerrc(a,b) ≤ 1e-9 + +# test functions, similar to runtests.jl, but can't do relative error when some values are zero +abserr(z, x) = z == x ? 0.0 : abs(z - x) +abserrc(z, x) = max(abserr(real(z),real(x)), abserr(imag(z),imag(x))) +≒(a,b) = abserrc(a,b) ≤ 1e-10 + +S = Array(Complex, (0,)) +Z = Array(Complex, (0,)) +errors = Array(Complex, (0,)) +rel_errors = Array(Complex, (0,)) +L = 0 +L2 = 0 +diff = 0 +rel_diff = 0 + +file = "polylog_testdata_alpha.dat" +f = open(file, "r") +for line in eachline(f) + if ismatch(r"#", line) + else + v = split(line, ',') + + s = parse(Float64, v[1]) + im*parse(Float64, v[2]) + z = parse(Float64, v[3]) + im*parse(Float64, v[4]) + L = parse(Float64, v[5]) + im*parse(Float64, v[6]) + L2 = SF.polylog(s,z,1.0e-18) + diff = L - L2 + rel_diff = complex(real(diff)/real(L), imag(diff)/imag(L)) + + push!(S, s) + push!(Z, z) + push!(errors, diff) + push!(rel_errors, rel_diff) + + # println(" L = $L, L2 =$L2, diff = $diff, \t\t rel diff = $rel_diff") + println(" s=$s, z=$z, |error|=$(ceil(log10(abs(diff))))") + end +end + +# test particular values from Wolfram alpha +# @testset "polylog tests from Wolfram alpha" begin + +# @test SF.polylog(-5.000001, 0.2) ≅ 6.904305364987799724230921116283822405784472717903797757356 +# @test SF.polylog(-2.000001, 0.2) ≅ 0.468750244180354668389455477288861311399345621141103816518 +# @test SF.polylog(-1.000001, 0.2) ≅ 0.312500094184052926057717828023167764362038234067425674984 +# @test SF.polylog( 0.000000, 0.2) ≅ 0.25 +# @test SF.polylog( 0.000001, 0.2) ≅ 0.249999960605813559100141040899469061220130457763090931203 +# @test SF.polylog( 1.000001, 0.2) ≅ 0.223143533840628659834227863703211050928449673084530824304 +# @test SF.polylog( 2.000001, 0.2) ≅ 0.211003767368667938482167666994614717744311348708200162427 +# @test SF.polylog( 3.000001, 0.2) ≅ 0.205324191902667822690683746355210013921766645655171501890 +# @test SF.polylog( 5.000001, 0.2) ≅ 0.201284594885756384926736867628954299128217148725340685679 + +# @test SF.polylog(-5.000001, 0.99) ≅ 1.16439554606503726735945485177408748716681200000005e14 +# @test SF.polylog(-2.000001, 0.99) ≅ 1.97011088076187708022430311310006082286580942497213e6 +# @test SF.polylog(-1.000001, 0.99) ≅ 9900.04972775403451817385172007212640686495346816117 +# @test SF.polylog( 0.000000, 0.99) ≅ 99 +# @test SF.polylog( 0.000001, 0.99) ≅ 98.9995988050883517986218058007486858307489619231956 +# @test SF.polylog( 1.000001, 0.99) ≅ 4.605161353580737753746071585981515640203854412492030454580 # error ~ 1.0e11 +# @test SF.polylog( 2.000001, 0.99) ≅ 1.588624649826159899085393338960947692670814871114246941712 # error ~ 1.0e11 +# @test SF.polylog( 3.000001, 0.99) ≅ 1.185832744102043627496839259642313182039620569301974825714 +# @test SF.polylog( 5.000001, 0.99) ≅ 1.026110449210262375841139988837721699091019919425544651696 + +# @test SF.polylog( 1.000001, 1.20) ≅ 1.60944129470676213105637136369425072009730867713452098010 - 3.14158912002728351076393334268714060723117930694360052182im +# @test SF.polylog( 1.000001, 1.50) ≅ 0.69315092620532843831758654220946253504678190003195418383 - 3.1415916309839162783984857611820275451978840493033368692im +# @test SF.polylog( 1.000001, 1.70) ≅ 0.35667861585275033749153552050383961466011713941159572502 - 3.1415924761565638528178063690254522808240730779853094090im +# @test SF.polylog( 1.000001, 1.999999) ≒ +# @test SF.polylog( 1.000001, 2.00) ≅ +# @test SF.polylog( 1.000001, 2.10) ≅ -0.0953067628118213777754205736836756047381793027303437193 - 3.14159352922832327133509242559303053858610422117044987im + +# end diff --git a/test/polylog_testdata.dat b/test/polylog_testdata.dat new file mode 100644 index 00000000..5e43d45f --- /dev/null +++ b/test/polylog_testdata.dat @@ -0,0 +1,1002 @@ +# test data from li_testdata.py +# using mpmath library +# s(real, imag), z(real, imag), Li(real, imag) +2.57636950631092576103,2.89889121739954180867,0.13267161787652381744,-1.52908730194326358820,0.22742653414510977106,-1.28004045152782230943 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+2.40007489210605529095,2.87660762026954452963,5.69370195484752983361,-0.50901867863753191834,3.18184282108034643954,-0.65118188288410305820 +-0.28088304419904919218,0.87630768879533871285,0.18980544059841819582,-2.14549658946724441932,-0.21347130074107478759,-0.43572201313373565279 diff --git a/test/polylog_testdata.py b/test/polylog_testdata.py new file mode 100755 index 00000000..00a36ec3 --- /dev/null +++ b/test/polylog_testdata.py @@ -0,0 +1,26 @@ +#!/usr/bin/python +# python code using mpmath to create a test data sets +from mpmath import * +mp.dps = 30 +mp.pretty = True + +import random +random.seed(1) + +N = 1000 +mu = 0.0 +sigma = 2.0 +fid = open('li_testdata.dat', 'w') +fid.write('# test data from li_testdata.py\n') +fid.write('# using mpmath library\n') +fid.write('# s(real, imag), z(real, imag), Li(real, imag)\n') +for i in range(1,N): + s = random.gauss(mu, sigma) + j*random.gauss(mu, sigma) + z = random.gauss(mu, sigma) + j*random.gauss(mu, sigma) + L = polylog(s, z) + fid.write('%.20f,%.20f,%.20f,%.20f,%.20f,%.20f\n' % (s.real, s.imag, z.real, z.imag, L.real, L.imag)) + + +fid.close() + + diff --git a/test/polylog_testdata_alpha.dat b/test/polylog_testdata_alpha.dat new file mode 100644 index 00000000..a4ea2d97 --- /dev/null +++ b/test/polylog_testdata_alpha.dat @@ -0,0 +1,42 @@ +# test data from Wolfram alpha +# +# s(real, imag), z(real, imag), Li(real, imag) +# +# start with values with near integer s, because these might be problematic as they are near special cases +# in particular for the series sum around z=1, which we use when 0.5 < |z| < 2.0 +-5.000001, 0.0, 0.2, 0.0, 6.904305364987799724230921116283822405784472717903797757356, 0.0 +-2.000001, 0.0, 0.2, 0.0, 0.468750244180354668389455477288861311399345621141103816518, 0.0 +-1.000001, 0.0, 0.2, 0.0, 0.312500094184052926057717828023167764362038234067425674984, 0.0 + 0.000000, 0.0, 0.2, 0.0, 0.25, 0.0 + 0.000001, 0.0, 0.2, 0.0, 0.249999960605813559100141040899469061220130457763090931203, 0.0 + 1.000001, 0.0, 0.2, 0.0, 0.223143533840628659834227863703211050928449673084530824304, 0.0 + 2.000001, 0.0, 0.2, 0.0, 0.211003767368667938482167666994614717744311348708200162427, 0.0 + 3.000001, 0.0, 0.2, 0.0, 0.205324191902667822690683746355210013921766645655171501890, 0.0 + 5.000001, 0.0, 0.2, 0.0, 0.201284594885756384926736867628954299128217148725340685679, 0.0 +# + 1.000001, 0.0, 0.5, 0.0, 0.693147007663231669849516514839299401059360031508002626249, 0.0 +# +-5.000001, 0.0, 0.99, 0.0, 1.16439554606503726735945485177408748716681200000005e14, 0.0 +-2.000001, 0.0, 0.99, 0.0, 1.97011088076187708022430311310006082286580942497213e6, 0.0 +-1.000001, 0.0, 0.99, 0.0, 9900.04972775403451817385172007212640686495346816117, 0.0 + 0.000000, 0.0, 0.99, 0.0, 99, 0.0 + 0.000001, 0.0, 0.99, 0.0, 98.9995988050883517986218058007486858307489619231956, 0.0 + 1.000001, 0.0, 0.99, 0.0, 4.605161353580737753746071585981515640203854412492030454580, 0.0 + 2.000001, 0.0, 0.99, 0.0, 1.588624649826159899085393338960947692670814871114246941712, 0.0 + 3.000001, 0.0, 0.99, 0.0, 1.185832744102043627496839259642313182039620569301974825714, 0.0 + 5.000001, 0.0, 0.99, 0.0, 1.026110449210262375841139988837721699091019919425544651696, 0.0 +# + 1.000001, 0.0, 1.20, 0.0, 1.60944129470676213105637136369425072009730867713452098010, -3.14158912002728351076393334268714060723117930694360052182 + 1.000001, 0.0, 1.50, 0.0, 0.69315092620532843831758654220946253504678190003195418383, -3.1415916309839162783984857611820275451978840493033368692 + 1.000001, 0.0, 1.70, 0.0, 0.35667861585275033749153552050383961466011713941159572502, -3.1415924761565638528178063690254522808240730779853094090 + 1.000001, 0.0, 1.999999, 0.0, 4.48462808108367607858388644733186363064154361287440e-6,-3.14159331552720658388549501733501317621202043844284 + 1.000001, 0.0, 2.00, 0.0, 3.48462690882559763270126765927498392238379428297186e-6, -3.14159331552947276581779433082131101718301247144061 + 1.000001, 0.0, 2.10, 0.0, -0.0953067628118213777754205736836756047381793027303437193, -3.14159352922832327133509242559303053858610422117044987 +# +# +# difficult cases are for s close to a positive integer, and using expansion around z=1 + 1.000001, 0.0, 0.99, 0.0, 4.605161353580737753746071585981515640203854412492030454580, 0.0 + 1.000001, 0.0, 1.20, 0.0, 1.60944129470676213105637136369425072009730867713452098010, -3.14158912002728351076393334268714060723117930694360052182 + 1.000001, 0.0, 1.50, 0.0, 0.69315092620532843831758654220946253504678190003195418383, -3.1415916309839162783984857611820275451978840493033368692 + 1.000001, 0.0, 1.70, 0.0, 0.35667861585275033749153552050383961466011713941159572502, -3.1415924761565638528178063690254522808240730779853094090 + 1.000001, 0.0, 1.999999, 0.0, 4.48462808108367607858388644733186363064154361287440e-6,-3.14159331552720658388549501733501317621202043844284 From a4342dcd63c7ad2e333ce97319e9d57fc96a09af Mon Sep 17 00:00:00 2001 From: Matthew Roughan Date: Mon, 5 Jun 2017 13:02:45 +0930 Subject: [PATCH 14/14] added poly_zeta for testing --- src/polylog.jl | 20 +++++++++++++++++++- test/polylog_test2.jl | 6 ++++-- 2 files changed, 23 insertions(+), 3 deletions(-) diff --git a/src/polylog.jl b/src/polylog.jl index ada19413..d4ebb31b 100644 --- a/src/polylog.jl +++ b/src/polylog.jl @@ -126,6 +126,16 @@ function Dbeta(s::Number) β = 4.0^(-s) * ( zeta(s,0.25) - zeta(s,0.75) ) end +function polylog_zeta(s::Number, z::Number, accuracy=1.0e-18) + # compute using the Hurwitz-zeta function identity + twopi = 2π + x = im * (log(complex(-z)) / twopi) + ss = 1-s + ip = im^ss + return ( gamma(ss)/twopi^(ss) ) * (ip * zeta(ss, 0.5-x) + conj(ip) * zeta(ss, 0.5+x)) +end + + function polylog_direct(s::Number, z::Number, accuracy=1.0e-18) # calculate using direct definition if abs(z) > 1 || ( abs(z) ≈ 1 && real(s) <= 2) @@ -201,8 +211,16 @@ function polylog_series_mu(s::Number, z::Number, accuracy=1.0e-18) else error("Should not get this case") end + + # should have a case in here for when s is close to a real, positive integer + # seed Wood 9.4 + # elseif abs(s - round(real(s))) < 0.01 # not sure where the right cut off is + + + # could also use Wood 14.1 (square formula) to extend the range over which we sum directly - maybe??? + else - # equivalent of Crandalls 1.4 for s non-integer + # equivalent of Crandalls 1.4 for s non-integer, Wood (9.3) total = gamma(1-s) * (-μ)^(s-1) # println(" μ=$μ, total = $total") tmp = 1 diff --git a/test/polylog_test2.jl b/test/polylog_test2.jl index 8e8f6f0c..b826fc8a 100644 --- a/test/polylog_test2.jl +++ b/test/polylog_test2.jl @@ -33,7 +33,9 @@ for line in eachline(f) z = parse(Float64, v[3]) + im*parse(Float64, v[4]) L = parse(Float64, v[5]) + im*parse(Float64, v[6]) L2 = SF.polylog(s,z,1.0e-18) - diff = L - L2 + L3 = SF.polylog_zeta(s,z) + diff = L - L2 + diff3 = L - L3 rel_diff = complex(real(diff)/real(L), imag(diff)/imag(L)) push!(S, s) @@ -42,7 +44,7 @@ for line in eachline(f) push!(rel_errors, rel_diff) # println(" L = $L, L2 =$L2, diff = $diff, \t\t rel diff = $rel_diff") - println(" s=$s, z=$z, |error|=$(ceil(log10(abs(diff))))") + println(" s=$s, z=$z, |error|=$(ceil(log10(abs(diff)))), |error3|=$(ceil(log10(abs(diff3))))") end end