Define specialised abs2 and abs for complex intervals (#250)

* Define abs2 and abs for complex intervals

* Add test

* Add test for norm

* Modify tests according to review

* Update test/interval_tests/complex.jl

Co-Authored-By: dpsanders <dpsanders@gmail.com>

Author: dpsanders

src/IntervalArithmetic.jl

@@ -13,7 +13,7 @@ using SetRounding
 using Markdown
 
 using LinearAlgebra
-import LinearAlgebra: ×, dot
+import LinearAlgebra: ×, dot, norm
 export ×, dot
 
 
@@ -33,6 +33,7 @@ import Base:
     expm1, log1p,
     precision,
     isfinite, isnan, isinf, iszero,
+    abs, abs2, 
     show,
     isinteger, setdiff,
     parse, hash

src/intervals/complex.jl

@@ -103,3 +103,17 @@ function log(z::Complex{T}) where T<:Interval
 
     return log(ρ) + im * θ
 end
+
+
+function abs2(z::Complex{T}) where T<:Interval
+    return real(z)^2 + imag(z)^2
+end
+
+function abs(z::Complex{T}) where T<:Interval
+    return sqrt(abs2(z))
+end
+#
+# # \left( |x|^p \right)^{1/p}.
+# function norm(z::Complex{T}, p=2) where T<:Interval
+#     return (abs(z)^(p))^(1 / p)
+# end 

test/interval_tests/complex.jl

@@ -1,6 +1,8 @@
 using IntervalArithmetic
 using Test
 
+using LinearAlgebra
+
 @testset "Complex interval operations" begin
     a = @interval 1im
     b = @interval 4im + 3
@@ -44,3 +46,13 @@ end
     a = -3.1
     @test x ⊆ (x^a)^(1/a)
 end
+
+@testset "abs2 and abs" begin
+    x = (0..3) + (0..4)*im
+    @test abs2(x) == 0..25
+    @test abs(x) == norm(x) == 0..5
+
+    y = (-1..1) + (-2..2)*im
+    @test abs(y).lo == 0.0
+    @test abs2(y).lo == 0.0
+end