diff --git a/src/SpecialFunctions.jl b/src/SpecialFunctions.jl index 7dc0c529..0d15d0e1 100644 --- a/src/SpecialFunctions.jl +++ b/src/SpecialFunctions.jl @@ -55,7 +55,10 @@ if VERSION >= v"0.6.0-dev.2767" hankelh1x, hankelh2, hankelh2x, - zeta + zeta, + polylog, + bernoulli, + harmonic end end @@ -70,4 +73,8 @@ include("erf.jl") include("gamma.jl") include("deprecated.jl") +include("harmonic.jl") +include("bernoulli.jl") +include("polylog.jl") + end # module diff --git a/src/bernoulli.jl b/src/bernoulli.jl new file mode 100644 index 00000000..ed1e236f --- /dev/null +++ b/src/bernoulli.jl @@ -0,0 +1,116 @@ +# +# Bernoulli numbers (as rationals up to B_{34}) and Bernoulli polynomials + +""" + bernoulli(n) + + Calculates the first 34 Bernoulli numbers B_n (of the first-kind or NIST type) + e.g., see + + + http://mathworld.wolfram.com/BernoulliNumber.html + + https://en.wikipedia.org/wiki/Bernoulli_number + + http://dlmf.nist.gov/24 + + 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,...,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 +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 + 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) + + Calculates Bernoulli polynomials from the Hurwitz-zeta function using + +```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 + + + 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 + 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 + + # 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 + # 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/gamma.jl b/src/gamma.jl index 116a1fce..d7275baf 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/harmonic.jl b/src/harmonic.jl new file mode 100644 index 00000000..75818477 --- /dev/null +++ b/src/harmonic.jl @@ -0,0 +1,62 @@ +""" + 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 <= 0 + throw(DomainError()) + end + if n <= 10 + # get exact values for small 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) + 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 that requires more testing. + +## Examples +```jldoctest +julia> harmonic(2,1) +1.5 +``` +""" +function harmonic(n::Integer, r::Real) + if n <= 0 + throw(DomainError()) + end + total = 0.0 + for k=1:n + total += 1.0 / k^r + end + return total +end diff --git a/src/polylog.jl b/src/polylog.jl new file mode 100644 index 00000000..d4ebb31b --- /dev/null +++ b/src/polylog.jl @@ -0,0 +1,243 @@ +""" + 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_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) + 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 = $total") + elseif n==1 + total = -log(complex(1-z)) + elseif n==0 + total = z / (1-z) + elseif n==-1 + total = z / (1-z)^2 + elseif n<-1 + # Crandall's 1.5 for s integer + total = factorial(-n) * (-μ)^(n-1) + tmp = 1 + for k=0:L + # total -= μ^k * bernoulli(k-n+1, 0.0) / ( gamma(k+1)*(k-n+1) ) + total -= tmp * bernoulli(k-n+1, 0.0) / (k-n+1) + tmp *= μ/(k+1) + if abs(tmp)/abs(total) < 1.0e-30 + break + end + end + else + error("Should not get this case") + end + + # should have a case in here for when s is close to a real, positive integer + # seed Wood 9.4 + # elseif abs(s - round(real(s))) < 0.01 # not sure where the right cut off is + + + # could also use Wood 14.1 (square formula) to extend the range over which we sum directly - maybe??? + + else + # equivalent of Crandalls 1.4 for s non-integer, Wood (9.3) + total = gamma(1-s) * (-μ)^(s-1) + # println(" μ=$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/harmonic_test.jl b/test/harmonic_test.jl new file mode 100644 index 00000000..a13257e1 --- /dev/null +++ b/test/harmonic_test.jl @@ -0,0 +1,42 @@ +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 + + @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 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..b826fc8a --- /dev/null +++ b/test/polylog_test2.jl @@ -0,0 +1,81 @@ +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) + 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) + 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)))), |error3|=$(ceil(log10(abs(diff3))))") + 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 +-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/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