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Make quadgk type stable #18928

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Make quadgk type stable #18928

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d6a303a
Make quadgk type stable
t-schoof Oct 14, 2016
108c8fd
Improve compile time and fix compile order dependent runtime regression
t-schoof Oct 17, 2016
21da080
Further improve runtime and type stability
t-schoof Oct 31, 2016
76d11d6
Reenable the integration of type unstable functions
t-schoof Nov 1, 2016
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73 changes: 40 additions & 33 deletions base/quadgk.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -36,18 +36,21 @@ const rulecache = AnyDict( (Float64,7) => # precomputed in 100-bit arith.
4.1795918367346938775510204081658e-01]) )

# integration segment (a,b), estimated integral I, and estimated error E
immutable Segment
a::Number
b::Number
I
E::Real
immutable Segment{Ts<:Number, Tf, Te<:Real}
a::Ts
b::Ts
I::Tf
E::Te
end
isless(i::Segment, j::Segment) = isless(i.E, j.E)

segment_type{Ts, Tf<:Number, Te<:Real}(::Type{Ts}, ::Type{Tf}, ::Type{Te}) = Segment{Ts, Tf, Te}
segment_type{Ts, Tx<:Number, N, Te<:Real}(::Type{Ts}, ::Type{Array{Tx, N}}, ::Type{Te}) = Segment{Ts, Array{Tx, N}, Te}
segment_type(::Type, ::Type, ::Type) = Segment

# Internal routine: approximately integrate f(x) over the interval (a,b)
# by evaluating the integration rule (x,w,gw). Return a Segment.
function evalrule(f, a,b, x,w,gw, nrm)
function evalrule{F}(f::F, a,b, x,w,gw, nrm)
# Ik and Ig are integrals via Kronrod and Gauss rules, respectively
s = convert(eltype(x), 0.5) * (b-a)
n1 = 1 - (length(x) & 1) # 0 if even order, 1 if odd order
Expand Down Expand Up @@ -87,36 +90,47 @@ rulekey(T,n) = (T,n)
# integration with the order-n Kronrod rule and weights of type Tw,
# with absolute tolerance abstol and relative tolerance reltol,
# with maxevals an approximate maximum number of f evaluations.
function do_quadgk{Tw}(f, s, n, ::Type{Tw}, abstol, reltol, maxevals, nrm)
if eltype(s) <: Real # check for infinite or semi-infinite intervals
function do_quadgk{F, Ts, Tw}(f::F, s::Vector{Ts}, n, ::Type{Tw}, abstol, reltol, maxevals, nrm)
if Ts <: Real # check for infinite or semi-infinite intervals
s1 = s[1]; s2 = s[end]; inf1 = isinf(s1); inf2 = isinf(s2)
if inf1 || inf2
if inf1 && inf2 # x = t/(1-t^2) coordinate transformation
return do_quadgk(t -> begin t2 = t*t; den = 1 / (1 - t2);
return do_quadgk_no_inf(t -> begin t2 = t*t; den = 1 / (1 - t2);
f(t*den) * (1+t2)*den*den; end,
map(x -> isinf(x) ? copysign(one(x), x) : 2x / (1+hypot(1,2x)), s),
n, Tw, abstol, reltol, maxevals, nrm)
end
s0,si = inf1 ? (s2,s1) : (s1,s2)
s0::Ts,si::Ts = inf1 ? (s2,s1) : (s1,s2)
if si < 0 # x = s0 - t/(1-t)
return do_quadgk(t -> begin den = 1 / (1 - t);
return do_quadgk_no_inf(t -> begin den = 1 / (1 - t);
f(s0 - t*den) * den*den; end,
reverse!(map(x -> 1 / (1 + 1 / (s0 - x)), s)),
n, Tw, abstol, reltol, maxevals, nrm)
else # x = s0 + t/(1-t)
return do_quadgk(t -> begin den = 1 / (1 - t);
return do_quadgk_no_inf(t -> begin den = 1 / (1 - t);
f(s0 + t*den) * den*den; end,
map(x -> 1 / (1 + 1 / (x - s0)), s),
n, Tw, abstol, reltol, maxevals, nrm)
end
end
end
return do_quadgk_no_inf(f,s, n, Tw, abstol, reltol, maxevals, nrm)

end


function do_quadgk_no_inf{F, Ts, Tw, Fn}(f::F, s::Vector{Ts}, n, ::Type{Tw}, abstol, reltol, maxevals, nrm::Fn)
Tf = Base.promote_op(f, Ts)
_do_quadgk_no_inf(f, s, n, Tw, segment_type(Ts, Tf, Base.promote_op(nrm, Tf)), abstol, reltol, maxevals, nrm)
end


function _do_quadgk_no_inf{F, Tw, Tsegs}(f::F, s, n, ::Type{Tw}, ::Type{Tsegs}, abstol, reltol, maxevals, nrm)
key = rulekey(Tw,n)
x,w,gw = haskey(rulecache, key) ? rulecache[key] :
x::Vector{Tw},w::Vector{Tw},gw::Vector{Tw} = haskey(rulecache, key) ? rulecache[key] :
(rulecache[key] = kronrod(Tw, n))
segs = Segment[]
for i in 1:length(s) - 1
segs = Tsegs[Segment(evalrule(f, s[1],s[2], x,w,gw, nrm))]
for i in 2:length(s) - 1
heappush!(segs, evalrule(f, s[i],s[i+1], x,w,gw, nrm), Reverse)
end
numevals = (2n+1) * length(segs)
Expand All @@ -129,14 +143,14 @@ function do_quadgk{Tw}(f, s, n, ::Type{Tw}, abstol, reltol, maxevals, nrm)
# Pop the biggest-error segment and subdivide (h-adaptation)
# until convergence is achieved or maxevals is exceeded.
while E > abstol && E > reltol * nrm(I) && numevals < maxevals
s = heappop!(segs, Reverse)
mid = (s.a + s.b) * 0.5
s1 = evalrule(f, s.a, mid, x,w,gw, nrm)
s2 = evalrule(f, mid, s.b, x,w,gw, nrm)
heappush!(segs, s1, Reverse)
heappush!(segs, s2, Reverse)
I = (I - s.I) + s1.I + s2.I
E = (E - s.E) + s1.E + s2.E
seg = heappop!(segs, Reverse)
mid = (seg.a + seg.b) * 0.5
seg1 = evalrule(f, seg.a, mid, x,w,gw, nrm)
seg2 = evalrule(f, mid, seg.b, x,w,gw, nrm)
heappush!(segs, seg1, Reverse)
heappush!(segs, seg2, Reverse)
I = (I - seg.I) + seg1.I + seg2.I
E = (E - seg.E) + seg1.E + seg2.E
numevals += 4n+2
end
# re-sum (paranoia about accumulated roundoff)
Expand All @@ -151,13 +165,13 @@ end

# Gauss-Kronrod quadrature of f from a to b to c...

function quadgk{T<:AbstractFloat}(f, a::T,b::T,c::T...;
function quadgk{F, T<:AbstractFloat}(f::F, a::T,b::T,c::T...;
abstol=zero(T), reltol=sqrt(eps(T)),
maxevals=10^7, order=7, norm=vecnorm)
do_quadgk(f, [a, b, c...], order, T, abstol, reltol, maxevals, norm)
end

function quadgk{T<:AbstractFloat}(f, a::Complex{T},
function quadgk{F, T<:AbstractFloat}(f::F, a::Complex{T},
b::Complex{T},c::Complex{T}...;
abstol=zero(T), reltol=sqrt(eps(T)),
maxevals=10^7, order=7, norm=vecnorm)
Expand Down Expand Up @@ -221,14 +235,7 @@ transformation is performed internally to map the infinite interval to a finite
"""
# generic version: determine precision from a combination of
# all the integration-segment endpoints
function quadgk(f, a, b, c...; kws...)
T = promote_type(typeof(float(a)), typeof(b))
for x in c
T = promote_type(T, typeof(x))
end
cT = map(T, c)
quadgk(f, convert(T, a), convert(T, b), cT...; kws...)
end
quadgk{F}(f::F, a, b, c...; kws...) = quadgk(f, promote(float(a),float(b),c...)...; kws...)

###########################################################################
# Gauss-Kronrod integration-weight computation for arbitrary floating-point
Expand Down