Added complex sqrt and some complex set functions (#195)

* Added complex sqrt and some complex set functions

Added complex sqrt and some complex set functions

Added complex sqrt and some complex set functions

Fixed test

Fixed complex sqrt test

Added more tests

Fixed complex in

added tests back in

* Updated LICENSE.md to include sqrt code attribution

LICENSE.md

@@ -224,3 +224,30 @@ distributed under the License is distributed on an "AS IS" BASIS,
 WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 See the License for the specific language governing permissions and
 limitations under the License.
+
+The code for ssqs and sqrt{::Complex{<:Interval}} is modified from Base
+module in Julia 0.6.4, licensed also under the MIT license:
+
+> Copyright (c) 2009-2016: Jeff Bezanson, Stefan Karpinski, Viral B. Shah,
+> and other contributors:
+>
+> https://github.com/JuliaLang/julia/contributors
+>
+> Permission is hereby granted, free of charge, to any person obtaining
+> a copy of this software and associated documentation files (the
+> "Software"), to deal in the Software without restriction, including
+> without limitation the rights to use, copy, modify, merge, publish,
+> distribute, sublicense, and/or sell copies of the Software, and to
+> permit persons to whom the Software is furnished to do so, subject to
+> the following conditions:
+>
+> The above copyright notice and this permission notice shall be
+> included in all copies or substantial portions of the Software.
+>
+> THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+> EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+> MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+> NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
+> LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
+> OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
+> WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

src/intervals/complex.jl

@@ -0,0 +1,77 @@
+function ssqs(x::T, y::T,RND::RoundingMode) where T<:AbstractFloat
+    k::Int = 0
+    ρ = +(*(x,x,RND),*(y,y,RND),RND)
+    if !isfinite(ρ) && (isinf(x) || isinf(y))
+        ρ = convert(T, Inf)
+    elseif isinf(ρ) || (ρ==0 && (x!=0 || y!=0)) || ρ<nextfloat(zero(T))/(2*eps(T)^2)
+        m::T = max(abs(x), abs(y))
+        k = m==0 ? m : exponent(m)
+        xk, yk = ldexp(x,-k), ldexp(y,-k)
+        ρ = +(*(xk,xk,RND),*(yk,yk,RND),RND)
+    end
+    ρ, k
+end
+
+# function ssqs(x::Interval{T}, y::Interval{T}) where T<:AbstractFloat
+#     x = abs(x); y = abs(y)
+#     ρl,kl = ssqs(inf(x),inf(y),RoundDown)
+#     ρu,ku = ssqs(sup(x),sup(y),RoundUp)
+#     ρl, kl, ρu, ku
+# end
+
+function sqrt_realpart(x::T, y::T,RND::RoundingMode) where T<:AbstractFloat
+    # @assert x ≥ 0
+    ρ, k = ssqs(x,y,RND)
+    if isfinite(x) ρ = +(ldexp(x,-k),sqrt(ρ,RND),RND) end
+
+    if isodd(k)
+        k = div(k-1,2)
+    else
+        k = div(k,2)-1
+        ρ *= 2
+    end
+    ρ = ldexp(sqrt(ρ,RND),k) #sqrt((abs(z)+abs(x))/2) without over/underflow
+end
+
+function sqrt(z::Complex{Interval{T}}) where T<:AbstractFloat
+    x, y = reim(z)
+
+    (inf(x) < 0 && inf(y) < 0 && sup(y) >= 0) && error("Interval lies across branch cut")
+    x == y == 0 && return Complex(zero(x),y)
+
+    (inf(x) < 0 && sup(x) > 0) && return sqrt((inf(x)..0) + im*y) ∪ sqrt((0..sup(x)) + im*y)
+
+    absx = abs(x)
+
+    absy= abs(y)
+    ρl = sqrt_realpart(inf(absx),inf(absy),RoundDown)
+    ρu = sqrt_realpart(sup(absx),sup(absy),RoundUp)
+
+    ηu = sup(y)
+    if isfinite(ηu)
+        if ηu >= 0
+            ηu_re = sqrt_realpart(inf(absx),ηu,RoundDown)
+        else
+            ηu_re = sqrt_realpart(sup(absx),ηu,RoundUp)
+        end
+        ηu = ηu_re ≠ 0 ?  /(ηu,2ηu_re,RoundUp) : zero(T)
+    end
+
+    ηl = inf(y)
+    if ηl >= 0
+        ηl_re = sqrt_realpart(sup(absx),ηl,RoundUp)
+    else
+        ηl_re = sqrt_realpart(inf(absx),ηl,RoundDown)
+    end
+    ηl = ηl_re ≠ 0 ? /(ηl,2ηl_re,RoundDown) : zero(T)
+
+    ξ = ρl..ρu
+    η = ηl..ηu
+
+    if mid(x)<0
+        ξ = flipsign(η,mid(η))
+        η = flipsign((ρl..ρu),mid(η))
+    end
+
+    Complex(ξ,η)
+end

src/intervals/intervals.jl

@@ -122,7 +122,7 @@ include("arithmetic.jl")
 include("functions.jl")
 include("trigonometric.jl")
 include("hyperbolic.jl")
-
+include("complex.jl")
 
 # Syntax for intervals
 

src/intervals/set_operations.jl

@@ -13,6 +13,9 @@ function in(x::T, a::Interval) where T<:Real
     a.lo <= x <= a.hi
 end
 
+in(x::T, a::Complex{<:Interval}) where T<:Real = x ∈ real(a) && 0 ∈ imag(a)
+in(x::Complex{T}, a::Complex{<:Interval}) where T<:Real = real(x) ∈ real(a) && imag(x) ∈ imag(a)
+
 
 
 """
@@ -52,6 +55,10 @@ function isdisjoint(a::Interval, b::Interval)
     islessprime(b.hi, a.lo) || islessprime(a.hi, b.lo)
 end
 
+function isdisjoint(a::Complex{<:Interval}, b::Complex{<:Interval})
+    isdisjoint(real(a),real(b)) || isdisjoint(imag(a),imag(b))
+end
+
 
 # Intersection
 """
@@ -71,6 +78,14 @@ end
 intersect(a::Interval{T}, b::Interval{S}) where {T,S} =
     intersect(promote(a, b)...)
 
+function intersect(a::Complex{Interval{T}},b::Complex{Interval{T}}) where T
+    isdisjoint(a,b) && return complex(emptyinterval(T),emptyinterval(T)) # for type stability
+
+    complex(intersect(real(a),real(b)),intersect(imag(a),imag(b)))
+end
+intersect(a::Interval{Complex{T}}, b::Interval{Complex{S}}) where {T,S} =
+    intersect(promote(a, b)...)
+
 
 ## Hull
 """
@@ -83,6 +98,8 @@ all of `a` and `b`.
 hull(a::Interval, b::Interval) = Interval(min(a.lo, b.lo), max(a.hi, b.hi))
 #
 # hull{T,S}(a::Interval{T}, b::Interval{S}) = hull(promote(a, b)...)
+hull(a::Complex{<:Interval},b::Complex{<:Interval}) =
+    complex(hull(real(a),real(b)),hull(imag(a),imag(b)))
 
 """
     union(a, b)
@@ -94,6 +111,7 @@ to `hull(a,b)`.
 union(a::Interval, b::Interval) = hull(a, b)
 #
 # union(a::Interval, b::Interval) = union(promote(a, b)...)
+union(a::Complex{<:Interval},b::Complex{<:Interval}) = hull(a, b)
 
 
 """

test/interval_tests/complex.jl

@@ -7,17 +7,34 @@ end
 
 @testset "Complex interval operations" begin
     a = @interval 1im
-    @test typeof(a)== Complex{IntervalArithmetic.Interval{Float64}}
+    b = @interval 4im + 3
+    c = (@interval -1 4) + (@interval 0 2)*im
+
+    @test typeof(a) == Complex{IntervalArithmetic.Interval{Float64}}
     @test a ==  Interval(0) + Interval(1)*im
     @test a * a == Interval(-1)
     @test a + a == Interval(2)*im
     @test a - a == 0
     @test a / a == 1
     @test a^2 == -1
+
+    @test 3+2im ∈ c
+    @test a ∪ b == (@interval 0 3) + (@interval 1 4)*im
+    @test c ∩ (a ∪ b) == (@interval 0 3) + (@interval 1 2)*im
+    @test a ∩ b == ∅ + ∅*im
+    @test isdisjoint(a,b) == true
 end
 
 @testset "Complex functions" begin
     Z = (3 ± 1e-7) + (4 ± 1e-7)*im
     @test sin(Z) == complex(sin(real(Z))*cosh(imag(Z)),sinh(imag(Z))*cos(real(Z)))
     @test exp(-im * Interval(π)) == Interval(-1.0, -0.9999999999999999) - Interval(1.224646799147353e-16, 1.2246467991473532e-16)*im
+
+    sZ = sqrt(Z)
+    @test sZ == Interval(1.99999996999999951619,2.00000003000000070585) + Interval(0.99999996999999984926,1.00000003000000048381)*im
+    @test sqrt(-Z) == imag(sZ) - real(sZ)*im
+
+    @test sqrt((@interval -1 0) + (@interval 0)*im) .== (@interval 0 1)*im
+    @test sqrt((@interval -1 1) + (@interval 0)*im) .== (@interval 0 1) + (@interval 0 1)*im
+    @test sqrt((@interval -9//32 Inf)*im + (@interval 0)) .== (@interval 0 Inf) + (@interval -3//8 Inf)*im
 end