Remove third-party root finding package in favor of bisection (implemented by AI)

Project.toml

@@ -3,4 +3,3 @@ DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
 Interpolations = "a98d9a8b-a2ab-59e6-89dd-64a1c18fca59"
 Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
 PolyLog = "85e3b03c-9856-11eb-0374-4dc1f8670e7f"
-Roots = "f2b01f46-fcfa-551c-844a-d8ac1e96c665"

bisection.jl

@@ -0,0 +1,62 @@
+"Bisection method: Finds a zero in the interval [a, b]"
+function bisection(f::Function, a::Float64, b::Float64; tol::Float64 = 1e-8, max_iter::Int = 1000)::Float64
+    fa::Float64 = f(a)
+    fb::Float64 = f(b)
+    
+    # Ensure the endpoints bracket a root.
+    if fa * fb > 0.0
+        error("No sign change detected in the interval: f(a)*f(b) > 0")
+    end
+
+    c::Float64 = a
+    for _ in 1:max_iter
+        c = (a + b) / 2.0
+        fc::Float64 = f(c)
+        
+        # Check for convergence using the function value or the interval width.
+        if abs(fc) < tol || abs(b - a) < tol
+            return c
+        elseif fa * fc < 0.0
+            b = c
+            fb = fc
+        else
+            a = c
+            fa = fc
+        end
+    end
+    return c  # Return the current approximation if max iterations reached.
+end
+
+"Function to find all zeros in [x_start, x_end] by scanning for sign changes."
+function find_all_zeros(f::Function, x_start::Float64, x_end::Float64; partitions::Int = 1000, tol::Float64 = 1e-8)::Vector{Float64}
+    zeros_list::Vector{Float64} = Float64[]
+    Δ::Float64 = (x_end - x_start) / partitions
+    x_prev::Float64 = x_start
+    f_prev::Float64 = f(x_prev)
+    
+    for i in 1:partitions
+        x_curr::Float64 = x_start + i * Δ
+        f_curr::Float64 = f(x_curr)
+        
+        # Check for a sign change or an exact zero.
+        if f_prev * f_curr < 0.0 || f_prev == 0.0
+            try
+                root::Float64 = bisection(f, x_prev, x_curr; tol=tol)
+                # Add the root if it's new (avoid duplicates from neighboring intervals)
+                if isempty(zeros_list) || abs(root - zeros_list[end]) > tol
+                    push!(zeros_list, root)
+                end
+            catch err
+                # Skip this interval if bisection fails (should not occur if a sign change exists)
+            end
+        elseif f_curr == 0.0
+            if isempty(zeros_list) || abs(x_curr - zeros_list[end]) > tol
+                push!(zeros_list, x_curr)
+            end
+        end
+        
+        x_prev = x_curr
+        f_prev = f_curr
+    end
+    return zeros_list
+end

nucleons.jl

@@ -1,4 +1,5 @@
-using DifferentialEquations, Roots
+using DifferentialEquations
+include("bisection.jl")
 
 const ħc = 197.33 # MeVfm
 const M_n = 939.0 # MeV/c2
@@ -87,18 +88,18 @@ function boundaryValueFunc(κ, p, s::system; dtype=Float64, algo=Tsit5())
     return func
 end
 
-"Find all bound energies between E_min (=850) and E_max (=938) where
+"Find all bound energies between E_min (=850.0) and E_max (=938.0) where
     tol_digits determines the precision for root finding and the threshold for identifying duplicates,
     the other parameters are the same from dirac!(...)."
-function findEs(κ, p, s::system, E_min=850, E_max=938, tol_digits=5)
+function findEs(κ, p, s::system, E_min=850.0, E_max=938.0, tol_digits=5)
     func = boundaryValueFunc(κ, p, s)
-    Es = find_zeros(func, (E_min, E_max); xatol=1/10^tol_digits)
+    Es = find_all_zeros(func, E_min, E_max; partitions=20, tol=1/10^tol_digits)
     return unique(E -> round(E; digits=tol_digits), Es)
 end
 
 "Find all orbitals and return two lists containing κ values and corresponding energies for a single species where
     the other parameters are defined above"
-function findAllOrbitals(p, s::system, E_min=850, E_max=938)
+function findAllOrbitals(p, s::system, E_min=850.0, E_max=938.0)
     κs = Int[]
     Es = Float64[]
     # start from κ=-1 and go both up and down
@@ -160,7 +161,7 @@ end
 
 "Solve the Dirac equation and calculate scalar and vector densities of a nucleon species where
     the other parameters are defined above"
-function solveNucleonDensity(p, s::system, E_min=850, E_max=938)
+function solveNucleonDensity(p, s::system, E_min=850.0, E_max=938.0)
     κs, Es = findAllOrbitals(p, s, E_min, E_max)
     occs = fillNucleons(Z_or_N(s, p), κs, Es)
     return calculateNucleonDensity(κs, Es, occs, p, s)

test/Pb208_nucleon_basic.jl

@@ -21,8 +21,8 @@ s.W0 = Vs
 s.B0 = Rs
 s.A0 = As
 
-E_min = 850
-E_max = 938
+E_min = 850.0
+E_max = 938.0
 
 boundEs = findEs(κ, p, s, E_min, E_max)