bessely_zero fails for abs(nu) < 0.1587 and n = 1

[simonp0420]

This works fine:

julia> bessely_zero(0.1587, 1)
1.1173051438870518

but this fails

julia> bessely_zero(0.1586, 1)
ERROR: DomainError with -0.0002719861885149225:
`x` must be nonnegative.
Stacktrace:
  [1] bessely(nu::Float64, x::Float64)
    @ SpecialFunctions C:\Users\peter\.julia\packages\SpecialFunctions\mf0qH\src\bessel.jl:612
  [2] #5
    @ C:\Users\peter\Documents\julia\packages\FunctionZeros.jl\src\FunctionZeros.jl:63 [inlined]
  [3] Callable_Function
    @ C:\Users\peter\.julia\packages\Roots\ohGnS\src\functions.jl:29 [inlined]
  [4] init_state
    @ C:\Users\peter\.julia\packages\Roots\ohGnS\src\DerivativeFree\derivative_free.jl:6 [inlined]
  [5] #init#49
    @ C:\Users\peter\.julia\packages\Roots\ohGnS\src\hybrid.jl:16 [inlined]
  [6] init (repeats 2 times)
    @ C:\Users\peter\.julia\packages\Roots\ohGnS\src\hybrid.jl:5 [inlined]
  [7] #init#85
    @ C:\Users\peter\.julia\packages\Roots\ohGnS\src\DerivativeFree\order0.jl:37 [inlined]
  [8] init
    @ C:\Users\peter\.julia\packages\Roots\ohGnS\src\DerivativeFree\order0.jl:27 [inlined]
  [9] solve(𝑭𝑿::Roots.ZeroProblem{…}, M::Roots.Order0, p::Nothing; verbose::Bool, kwargs::@Kwargs{…})
    @ Roots C:\Users\peter\.julia\packages\Roots\ohGnS\src\find_zero.jl:497
 [10] solve
    @ C:\Users\peter\.julia\packages\Roots\ohGnS\src\find_zero.jl:490 [inlined]
 [11] #find_zero#40
    @ C:\Users\peter\.julia\packages\Roots\ohGnS\src\find_zero.jl:220 [inlined]
 [12] find_zero (repeats 2 times)
    @ C:\Users\peter\.julia\packages\Roots\ohGnS\src\find_zero.jl:210 [inlined]
 [13] find_zero
    @ C:\Users\peter\.julia\packages\Roots\ohGnS\src\find_zero.jl:243 [inlined]
 [14] bessely_zero(nu::Float64, n::Int64)
    @ FunctionZeros C:\Users\peter\Documents\julia\packages\FunctionZeros.jl\src\FunctionZeros.jl:63
 [15] top-level scope
    @ REPL[55]:1
Some type information was truncated. Use `show(err)` to see complete types.

Similar failure for other positive and negative values of nu when abs(nu) < 0.1587, including nu = 0.

One possible solution:

function bessely_zero(nu, n; order=2)
    if isone(n) && abs(nu) < 0.1587 
        return Roots.find_zero((x) -> SpecialFunctions.bessely(nu, x),
                                           0.5 * (nu + besselj_zero(nu, n)); order)
    else
        return Roots.find_zero((x) -> SpecialFunctions.bessely(nu, x),
                                           bessel_zero_asymptotic(nu, n, 2); order)
    end
end

This produces

julia> using BenchmarkTools

julia> @btime bessely_zero(0.0, 1)
  1.050 μs (0 allocations: 0 bytes)
0.8935769662791675

julia> @btime bessely_zero(0.1586, 1)
  18.100 μs (24 allocations: 384 bytes)
1.117167411163268

[jlapeyre]

• Fixed in Derivative zeros and fast Float64 lookup tables #23