add separate assess method; add docs on AlefeldPotra methods (#459)

* add separate assess method; add docs on AlefeldPotra methods

* version bump

Project.toml

@@ -1,6 +1,6 @@
 name = "Roots"
 uuid = "f2b01f46-fcfa-551c-844a-d8ac1e96c665"
-version = "2.2.3"
+version = "2.2.4"
 
 [deps]
 Accessors = "7d9f7c33-5ae7-4f3b-8dc6-eff91059b697"

src/Bracketing/alefeld_potra_shi.jl

@@ -16,6 +16,9 @@ An abstract type for Alefeld-Potra-Shi type bracketing problems, as discussed in
 The `update_step` method calls a `calculateΔ` method that can be customized to turn an algorithm based on interpolation into a bracketed algorithm. See [`Roots.BracketedHalley`](@ref) for an example.
 
 This implementation deviates slightly from the printed algorithm, as it may use an initial call to `_middle` rather than a secant step, depending on the signs of ``a`` and ``b``.
+
+!!! note
+    These algorithms do not check the size of `f` for convergence, so the `atol` or `rtol` are not utilized.
 """
 =#
 abstract type AbstractAlefeldPotraShi <: AbstractBracketingMethod end

src/convergence.jl

@@ -278,6 +278,17 @@ function assess_convergence(M::Any, state::AbstractUnivariateZeroState, options)
     return (:not_converged, false)
 end
 
+function assess_convergence(M::AbstractBisection, state::AbstractUnivariateZeroState, options::O) where {O <: Union{ExactOptions, XExactOptions}}
+    # return convergence_flag, boolean
+    # no check if f == ∞
+    is_exact_zero_f(M, state, options) && return (:exact_zero, true)
+    isnan_f(M, state) && return (:nan, true)
+    is_approx_zero_f(M, state, options) && return (:f_converged, true)
+    iszero_Δx(M, state, options) && return (:x_converged, true)
+    return (:not_converged, false)
+end
+
+
 # speeds up exact f values by just a bit (2% or so) over the above, so guess this is worth it.
 function assess_convergence(
     M::AbstractBracketingMethod,

test/test_find_zero.jl

@@ -583,3 +583,13 @@ end
         @test ForwardDiff.gradient(H, [1.0, 2])[1] ≈ -0.4416107917053284
     end
 end
+
+@testset "bracketing_atol" begin
+    ## issue $457
+    f(x) = x^2 - 4
+    @test find_zero(f, (0,Inf)) ≈ 2 # 2.0 correct
+    @test find_zero(f, (0,Inf), atol=1) ≈ 1.9997558593749998
+    @test find_zero(f, (0,Inf),atol=1e-5) ≈ 1.9999998807907102
+    @test find_zero(f, (0,8), atol=1) ≈ 1.99609375
+    @test find_zero(f, (0,8), atol=1e-3) ≈ 2.0000152587890625
+end