Simple doubling in N size for adaptive_simpsons_rule seems best

Author: fusion809

src/Simpson.jl

@@ -79,15 +79,7 @@ function adaptive_simpsons_rule(f::Function, a::Number, b::Number, ε::Float64=1
     criterion = 100*ε;
     while criterion >= 15*ε
         criterion = abs((simpsons_rule(f, N, a, (a+b)/2)+simpsons_rule(f, N, (a+b)/2, b)-simpsons_rule(f, N, a, b))/simpsons_rule(f, N, a, b));
-        if criterion >= 15*ε
-            # The following N adjustment comes from the error formula for Simpson's rule
-            N = convert(Int64, round((((16*criterion)/(15*ε))^(1/4))*N));
-            if ! iseven(N)
-                N = N+1;
-            end
-        else
-            N = 2*N;
-        end
+        N = 2*N;
     end
     return simpsons_rule(f, N, a, b)
 end

test/sinxx.jl

@@ -8,7 +8,9 @@ sol_8 = 1.562225466889056293352345138804502677227824980541083456384;
 printstyled("Integrating sin(x)/x from 0 to 100 and comparing it to the exact result.\n"; color = :red)
 @testset "sinxx" begin
     printstyled("Running: adaptive_simpsons_rule with ε=1e-7\n"; color = :magenta)
-    @time @test adaptive_simpsons_rule(sinxx, 0, 100, 1e-7) ≈ sol_8
+    # Using 1e-7 returns and error, despite it being accurate to an absolute and 
+    # relative tolerance of 1e-7
+    @time @test adaptive_simpsons_rule(sinxx, 0, 100, 1e-8) ≈ sol_8
     printstyled("Running: chebyshev_quadrature with k=1\n"; color = :magenta)
     @time @test chebyshev_quadrature(sinxx, 29645, 1, 0, 100) ≈ sol_8
     printstyled("Running: chebyshev_quadrature with k=2\n"; color = :magenta)