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add sine and cosine integrals #32
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const pidiv2 = π/2 | ||
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import Base.Math: @horner | ||
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There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. changed to |
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# This program computes the sine integral ∫_0^x sin(t)/t dt using the rational approximations of A.J. MacLeod, Numer. Algor., 12:259--272, 1996. | ||
function sinint(x::Float64) | ||
t = x*x | ||
absx = abs(x) | ||
if absx ≤ 6.0 | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Seems like you could omit the |
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return x * @horner(t, 1.00000000000000000000E0, | ||
-0.44663998931312457298E-1, | ||
0.11209146443112369449E-2, | ||
-0.13276124407928422367E-4, | ||
0.85118014179823463879E-7, | ||
-0.29989314303147656479E-9, | ||
0.55401971660186204711E-12, | ||
-0.42406353433133212926E-15) / | ||
@horner(t, 1.00000000000000000000E0, | ||
0.10891556624243098264E-1, | ||
0.59334456769186835896E-4, | ||
0.21231112954641805908E-6, | ||
0.54747121846510390750E-9, | ||
0.10378561511331814674E-11, | ||
0.13754880327250272679E-14, | ||
0.10223981202236205703E-17) | ||
elseif absx ≤ 12.0 | ||
invt = inv(t) | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. why not just There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. i think these are equivalent There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. better then to just use There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. i don't know why it's defined if it can't be used in a completely equivalent case, but i don't care enough to discuss it further: it's changed. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. It can, i.e these are all equivalent and get compiled to the same code There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. @MikaelSlevinsky, it seems like that definition would fail (spurious underflow) for There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more.
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Still fails for There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Yes, sorry about the dispatch loop. Anyways, surely there are multiple corner cases, but it's a reasonable request that julia> function realcomplextest(x::AbstractArray, f::Function)
for y in x
test = f(y) - f(y+0im)
if test != 0
println(y," ",test)
end
end
end
realcomplextest (generic function with 1 method)
julia> realcomplextest(linspace(0,50,10001), sin)
julia> realcomplextest(linspace(0,50,10001), cos)
julia> realcomplextest(linspace(0,50,10001), exp)
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Sure, that would be a nice property to have. |
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return sign(x)*pidiv2 - cos(x) * @horner(invt, 0.99999999962173909991E0, | ||
0.36451060338631902917E3, | ||
0.44218548041288440874E5, | ||
0.22467569405961151887E7, | ||
0.49315316723035561922E8, | ||
0.43186795279670283193E9, | ||
0.11847992519956804350E10, | ||
0.45573267593795103181E9) / | ||
(x * @horner(invt, 1.00000000000000000000E0, | ||
0.36651060273229347594E3, | ||
0.44927569814970692777E5, | ||
0.23285354882204041700E7, | ||
0.53117852017228262911E8, | ||
0.50335310667241870372E9, | ||
0.16575285015623175410E10, | ||
0.11746532837038341076E10)) - | ||
sin(x)*invt * @horner(invt, 0.99999999920484901956E0, | ||
0.51385504875307321394E3, | ||
0.92293483452013810811E5, | ||
0.74071341863359841727E7, | ||
0.28142356162841356551E9, | ||
0.49280890357734623984E10, | ||
0.35524762685554302472E11, | ||
0.79194271662085049376E11, | ||
0.17942522624413898907E11) / | ||
@horner(invt, 1.00000000000000000000E0, | ||
0.51985504708814870209E3, | ||
0.95292615508125947321E5, | ||
0.79215459679762667578E7, | ||
0.31977567790733781460E9, | ||
0.62273134702439012114E10, | ||
0.54570971054996441467E11, | ||
0.18241750166645704670E12, | ||
0.15407148148861454434E12) | ||
elseif absx < Inf | ||
invt = inv(t) | ||
return sign(x) * pidiv2 - cos(x) / x * (1.0 - | ||
@horner(invt, 0.19999999999999978257E1, | ||
0.22206119380434958727E4, | ||
0.84749007623988236808E6, | ||
0.13959267954823943232E9, | ||
0.10197205463267975592E11, | ||
0.30229865264524075951E12, | ||
0.27504053804288471142E13, | ||
0.21818989704686874983E13) / | ||
@horner(invt, 1.00000000000000000000E0, | ||
0.11223059690217167788E4, | ||
0.43685270974851313242E6, | ||
0.74654702140658116258E8, | ||
0.58580034751805687471E10, | ||
0.20157980379272098841E12, | ||
0.26229141857684496445E13, | ||
0.87852907334918467516E13)*invt) - | ||
sin(x) * invt * (1.0 - @horner(invt, 0.59999999999999993089E1, | ||
0.96527746044997139158E4, | ||
0.56077626996568834185E7, | ||
0.15022667718927317198E10, | ||
0.19644271064733088465E12, | ||
0.12191368281163225043E14, | ||
0.31924389898645609533E15, | ||
0.25876053010027485934E16, | ||
0.12754978896268878403E16) / | ||
@horner(invt, 1.00000000000000000000E0, | ||
0.16287957674166143196E4, | ||
0.96636303195787870963E6, | ||
0.26839734750950667021E9, | ||
0.37388510548029219241E11, | ||
0.26028585666152144496E13, | ||
0.85134283716950697226E14, | ||
0.11304079361627952930E16, | ||
0.42519841479489798424E16)*invt) | ||
elseif isnan(x) | ||
return NaN | ||
else | ||
return sign(x) * pidiv2 | ||
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end | ||
end | ||
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# This program computes the cosine integral γ + log x + ∫_0^x (cos(t)-1)/t dt using the rational approximations of A.J. MacLeod, Numer. Algor., 12:259--272, 1996. | ||
function cosint(x::Float64) | ||
t, r0, r1 = x*x, 0.616505485620716233797110404100, 3.384180422551186426397851146402 | ||
r01, r02 = 0.6162109375, 0.29454812071623379711E-3 | ||
r11, r12 = 3.3837890625, 0.39136005118642639785E-3 | ||
if x < 0.0 | ||
return throw(DomainError()) | ||
elseif x ≤ 3.0 | ||
return log(x/r0) + ((x - r01) - r02) * (x + r0) * | ||
@horner(t, -0.24607411378767540707E0, | ||
0.72113492241301534559E-2, | ||
-0.11867127836204767056E-3, | ||
0.90542655466969866243E-6, | ||
-0.34322242412444409037E-8, | ||
0.51950683460656886834E-11) / | ||
@horner(t, 1.00000000000000000000E0, | ||
0.12670095552700637845E-1, | ||
0.78168450570724148921E-4, | ||
0.29959200177005821677E-6, | ||
0.73191677761328838216E-9, | ||
0.94351174530907529061E-12) | ||
elseif x ≤ 6.0 | ||
return log(x/r1) + ((x - r11) - r12) * (x + r1) * | ||
@horner(t, -0.15684781827145408780E0, | ||
0.66253165609605468916E-2, | ||
-0.12822297297864512864E-3, | ||
0.12360964097729408891E-5, | ||
-0.66450975112876224532E-8, | ||
0.20326936466803159446E-10, | ||
-0.33590883135343844613E-13, | ||
0.23686934961435015119E-16) / | ||
@horner(t, 1.00000000000000000000E0, | ||
0.96166044388828741188E-2, | ||
0.45257514591257035006E-4, | ||
0.13544922659627723233E-6, | ||
0.27715365686570002081E-9, | ||
0.37718676301688932926E-12, | ||
0.27706844497155995398E-15) | ||
elseif x ≤ 12.0 | ||
invt = inv(t) | ||
return sin(x) * @horner(invt, 0.99999999962173909991E0, | ||
0.36451060338631902917E3, | ||
0.44218548041288440874E5, | ||
0.22467569405961151887E7, | ||
0.49315316723035561922E8, | ||
0.43186795279670283193E9, | ||
0.11847992519956804350E10, | ||
0.45573267593795103181E9) / | ||
(x * @horner(invt, 1.00000000000000000000E0, | ||
0.36651060273229347594E3, | ||
0.44927569814970692777E5, | ||
0.23285354882204041700E7, | ||
0.53117852017228262911E8, | ||
0.50335310667241870372E9, | ||
0.16575285015623175410E10, | ||
0.11746532837038341076E10)) - | ||
cos(x) * invt * @horner(invt, 0.99999999920484901956E0, | ||
0.51385504875307321394E3, | ||
0.92293483452013810811E5, | ||
0.74071341863359841727E7, | ||
0.28142356162841356551E9, | ||
0.49280890357734623984E10, | ||
0.35524762685554302472E11, | ||
0.79194271662085049376E11, | ||
0.17942522624413898907E11) / | ||
@horner(invt, 1.00000000000000000000E0, | ||
0.51985504708814870209E3, | ||
0.95292615508125947321E5, | ||
0.79215459679762667578E7, | ||
0.31977567790733781460E9, | ||
0.62273134702439012114E10, | ||
0.54570971054996441467E11, | ||
0.18241750166645704670E12, | ||
0.15407148148861454434E12) | ||
elseif x < Inf | ||
invt = inv(t) | ||
return sin(x)/x * (1.0 - @horner(invt, 0.19999999999999978257E1, | ||
0.22206119380434958727E4, | ||
0.84749007623988236808E6, | ||
0.13959267954823943232E9, | ||
0.10197205463267975592E11, | ||
0.30229865264524075951E12, | ||
0.27504053804288471142E13, | ||
0.21818989704686874983E13) / | ||
@horner(invt, 1.00000000000000000000E0, | ||
0.11223059690217167788E4, | ||
0.43685270974851313242E6, | ||
0.74654702140658116258E8, | ||
0.58580034751805687471E10, | ||
0.20157980379272098841E12, | ||
0.26229141857684496445E13, | ||
0.87852907334918467516E13)*invt) - | ||
cos(x)*invt * (1.0 - @horner(invt, 0.59999999999999993089E1, | ||
0.96527746044997139158E4, | ||
0.56077626996568834185E7, | ||
0.15022667718927317198E10, | ||
0.19644271064733088465E12, | ||
0.12191368281163225043E14, | ||
0.31924389898645609533E15, | ||
0.25876053010027485934E16, | ||
0.12754978896268878403E16) / | ||
@horner(invt, 1.00000000000000000000E0, | ||
0.16287957674166143196E4, | ||
0.96636303195787870963E6, | ||
0.26839734750950667021E9, | ||
0.37388510548029219241E11, | ||
0.26028585666152144496E13, | ||
0.85134283716950697226E14, | ||
0.11304079361627952930E16, | ||
0.42519841479489798424E16)*invt) | ||
elseif isnan(x) | ||
return NaN | ||
else | ||
return 0.0 | ||
end | ||
end | ||
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for f in (:sinint, :cosint) | ||
@eval begin | ||
($f)(x::Float32) = Float32(($f)(Float64(x))) | ||
($f)(x::Float16) = Float16(($f)(Float64(x))) | ||
($f)(x::Real) = ($f)(float(x)) | ||
($f)(x::AbstractFloat) = error("not implemented for ", typeof(x)) | ||
end | ||
end | ||
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""" | ||
sinint(x) | ||
Compute the sine integral function of `x`, defined by ``\\operatorname{Si}(x) := \\int_0^x\\frac{\\sin t}{t} dt`` | ||
for real `x`. | ||
""" | ||
sinint | ||
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""" | ||
cosint(x) | ||
Compute the cosine integral function of `x`, defined by ``\\operatorname{Ci}(x) := \\gamma + \\log x + \\int_0^x \\frac{\\cos t - 1}{t} dt`` | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. What is gamma here? Also, should put newlines after the function signatures in docstrings. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. gamma is the Euler-Mascheroni constant, I just tried to replicate the style of current repository https://github.com/JuliaMath/SpecialFunctions.jl/blob/master/src/erf.jl#L40, but i now see that it's inconsistent. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Yeah, I'd add something like "where |
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for real `x > 0`. | ||
""" | ||
cosint |
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I believe this is unnecessary, i.e just using the constant
π/2
is sufficient and will be compiled to a constant in the function anyway... can check comparing@code_llvm
or similarThere was a problem hiding this comment.
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removed