add isapprox for arrays, with ≈ and ≉ synonyms

Author: stevengj

NEWS.md

@@ -317,7 +317,7 @@ Library improvements
 
     * The `MathConst` type has been renamed `Irrational` ([#11922]).
 
-    * `isapprox` now has simpler and more sensible default tolerances ([#12393]).
+    * `isapprox` now has simpler and more sensible default tolerances ([#12393]), supports arrays, and has synonyms `≈` ([U+2248](http://www.fileformat.info/info/unicode/char/2248/index.htm), LaTeX `\approx`) and `≉` ([U+2249](http://www.fileformat.info/info/unicode/char/2249/index.htm), LaTeX `\napprox`) for `isapprox` and `!isapprox`, respectively ([#12472]).
 
   * Numbers
 
@@ -1563,11 +1563,13 @@ Too numerous to mention.
 [#11891]: https://github.com/JuliaLang/julia/issues/11891
 [#11922]: https://github.com/JuliaLang/julia/issues/11922
 [#11985]: https://github.com/JuliaLang/julia/issues/11985
+[#12025]: https://github.com/JuliaLang/julia/issues/12025
 [#12031]: https://github.com/JuliaLang/julia/issues/12031
 [#12034]: https://github.com/JuliaLang/julia/issues/12034
 [#12087]: https://github.com/JuliaLang/julia/issues/12087
 [#12137]: https://github.com/JuliaLang/julia/issues/12137
 [#12162]: https://github.com/JuliaLang/julia/issues/12162
 [#12393]: https://github.com/JuliaLang/julia/issues/12393
 [#12458]: https://github.com/JuliaLang/julia/issues/12458
+[#12472]: https://github.com/JuliaLang/julia/issues/12472
 [#12491]: https://github.com/JuliaLang/julia/issues/12491

base/docs/helpdb.jl

@@ -2935,11 +2935,18 @@ cis
 doc"""
 ```rst
 ::
-           isapprox(x::Number, y::Number; rtol::Real=sqrt(eps), atol::Real=0)
+           isapprox(x, y; rtol::Real=sqrt(eps), atol::Real=0)
 
-Inexact equality comparison: ``true`` if ``abs(x-y) <= atol + rtol*max(abs(x), abs(y))``.  The default ``atol`` is zero and the default ``rtol`` depends on the types of ``x`` and ``y``.
+Inexact equality comparison: ``true`` if ``norm(x-y) <= atol + rtol*max(norm(x), norm(y))``.  The default ``atol`` is zero and the default ``rtol`` depends on the types of ``x`` and ``y``.
 
 For real or complex floating-point values, ``rtol`` defaults to ``sqrt(eps(typeof(real(x-y))))``.  This corresponds to requiring equality of about half of the significand digits.   For other types, ``rtol`` defaults to zero.
+
+``x`` and ``y`` may also be arrays of numbers, in which case ``norm``
+defaults to ``vecnorm`` but may be changed by passing a
+``norm::Function`` keyword argument.  (For numbers, ``norm`` is the
+same thing as ``abs``.)
+
+The binary operator ``≈`` is equivalent to ``isapprox`` with the default arguments, and ``x ≉ y`` is equivalent to ``!isapprox(x,y)``.
 ```
 """
 isapprox

base/exports.jl

@@ -455,6 +455,8 @@ export
     zero,
     √,
     ∛,
+    ≈,
+    ≉,
 
 # specfun
     airy,

base/floatfuncs.jl

@@ -166,10 +166,14 @@ for f in (:round, :ceil, :floor, :trunc)
 end
 
 # isapprox: approximate equality of numbers
-isapprox(x::Number, y::Number; rtol::Real=rtoldefault(x,y), atol::Real=0) =
+function isapprox(x::Number, y::Number; rtol::Real=rtoldefault(x,y), atol::Real=0)
     x == y || (isfinite(x) && isfinite(y) && abs(x-y) <= atol + rtol*max(abs(x), abs(y)))
+end
+
+const ≈ = isapprox
+≉(x,y) = !(x ≈ y)
 
 # default tolerance arguments
 rtoldefault{T<:AbstractFloat}(::Type{T}) = sqrt(eps(T))
 rtoldefault{T<:Real}(::Type{T}) = 0
-rtoldefault{T<:Number,S<:Number}(x::T, y::S) = rtoldefault(promote_type(real(T),real(S)))
+rtoldefault{T<:Number,S<:Number}(x::Union(T,Type{T}), y::Union(S,Type{S})) = rtoldefault(promote_type(real(T),real(S)))

base/linalg.jl

@@ -5,7 +5,8 @@ module LinAlg
 importall Base
 importall ..Base.Operators
 import Base: USE_BLAS64, size, copy, copy_transpose!, power_by_squaring,
-             print_matrix, transpose!, unsafe_getindex, unsafe_setindex!
+             print_matrix, transpose!, unsafe_getindex, unsafe_setindex!,
+             isapprox
 
 export
 # Modules

base/linalg/generic.jl

@@ -535,3 +535,8 @@ det(x::Number) = x
 logdet(A::AbstractMatrix) = logdet(lufact(A))
 logabsdet(A::AbstractMatrix) = logabsdet(lufact(A))
 
+# isapprox: approximate equality of arrays [like isapprox(Number,Number)]
+function isapprox{T<:Number,S<:Number}(x::AbstractArray{T}, y::AbstractArray{S}; rtol::Real=Base.rtoldefault(T,S), atol::Real=0, norm::Function=vecnorm)
+    d = norm(x - y)
+    return isfinite(d) ? d <= atol + rtol*max(norm(x), norm(y)) : x == y
+end

test/floatapprox.jl

@@ -1,33 +1,40 @@
 # This file is a part of Julia. License is MIT: http://julialang.org/license
 
 # Floating point numbers - basic tests
-@test isapprox(4.00000000000001, 4.0)
-@test isapprox(5.0,4.999999999999993)
-@test !isapprox(4.000000002, 4.00300002)
+@test 4.00000000000001 ≈ 4.0
+@test 5.0 ≈ 4.999999999999993
+@test 4.000000002 ≉ 4.00300002
 
 # Other tolerance levels
 @test isapprox(4.32, 4.3; rtol=0.1, atol=0.01)
 @test isapprox(1.001, 1.002; rtol=0.001, atol=0.0001)
 @test !isapprox(4.5, 4.9; rtol=0.001, atol=0.001)
 
 # Complex numbers
-@test isapprox(1.0 + 1.0im, 1.0 + 1.00000000000001im)
-@test isapprox(0.9999999999999 + 1.0im, 1.0 + 1.000000000000001im)
+@test 1.0 + 1.0im ≈ 1.0 + 1.00000000000001im
+@test 0.9999999999999 + 1.0im ≈ 1.0 + 1.000000000000001im
 @test isapprox(0.9999 + 1.0im, 1.0 + 1.1im; rtol = 0.0001, atol=1.1)
 
 # Complex <-> reals
-@test isapprox(1.0 + 0im, 1.0000000000001)
+@test 1.0 + 0im ≈ 1.0000000000001
 @test isapprox(0.9999999999999, 1.0 + 0im)
 @test !isapprox(1.0+1im, 1.000000000000001)
 
 # Comparing NaNs
-@test !isapprox(4.0,NaN)
-@test !isapprox(NaN,4.0)
-@test !isapprox(complex(2.3,NaN), complex(NaN,2.3))
-@test !isapprox(NaN, NaN)
-@test !isapprox(complex(NaN,NaN), complex(NaN,NaN))
-@test !isapprox(complex(NaN,2.3), complex(NaN,2.3))
-@test !isapprox(complex(2.3,NaN), complex(2.3,NaN))
+@test 4.0 ≉ NaN
+@test NaN ≉ 4.0
+@test complex(2.3,NaN) ≉ complex(NaN,2.3)
+@test NaN ≉ NaN
+@test complex(NaN,NaN) ≉ complex(NaN,NaN)
+@test complex(NaN,2.3) ≉ complex(NaN,2.3)
+@test complex(2.3,NaN) ≉ complex(2.3,NaN)
+
+# Comparing Infs
+@test Inf ≈ Inf
+@test Inf ≉ 1
+@test Inf ≉ -Inf
+@test complex(0.0,Inf) ≈ complex(0.0,Inf)
+@test complex(0.0,Inf) ≉ complex(0.0,-Inf)
 
 # Tests for integers and rationals
 @test isapprox(4,4)
@@ -46,3 +53,9 @@
 
 # issue #12375:
 @test !isapprox(1e17, 1)
+
+# Tests for arrays:
+@test [1,2,3] ≈ [1,2,3+1e-9]
+@test [0,1] ≈ [1e-9, 1]
+@test [0,Inf] ≈ [0,Inf]
+@test [0,Inf] ≉ [0,-Inf]