# Special function (airy function) with arbitrary precision

**URL:** <https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546>\
**Category:** New to Julia\
**Created:** [January 9, 2024, 4:10am UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546 "2024-01-09T04:10:31Z")\
**Posts on this page:** 10\
**Page:** 1

<div class="post-metadata">

**Author:** ![baizhan](https://avatars.discourse-cdn.com/v4/letter/b/ecae2f/32.png) [@baizhan](https://discourse.julialang.org/u/baizhan)\
**Post date:** [January 9, 2024, 4:10am UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546/1 "2024-01-09T04:10:31Z")

</div>

I am solving equations which requires to evaluate Airy function and its derivative at high precision.  
Although the package `SpecialFunctions` provides `airyai` and `airyaiprime` function,  
the `airyaiprime` does not accept `BigFloat` as parameter.

I have also tried `ArbNumerics`. But when I do

```julia
>>>using ArbNumerics

>>>airyai(ArbFloat(0.0))

```

the program simply collapse and quit.

Is there any other packages which provides Airy functions (or more generally, special functions) with arbitrary precision?

---

<div class="post-metadata">

**Author:** ![Dan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dan/32/42581_2.png) [@Dan](https://discourse.julialang.org/u/Dan)\
**Post date:** [January 9, 2024, 4:30am UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546/2 "2024-01-09T04:30:38Z")

</div>

It seems to take BigFloat alright in 1.10

```julia
 | | |_| | | | (_| | | Version 1.10.0 (2023-12-25)
 _/ |\ __'_|_|_|\__'_| | Official https://julialang.org/ release
|__/ |

julia> using SpecialFunctions

julia> airyai(BigFloat(0.5))
0.2316936064808334897691252545099217396183864753577499817922638265232339844931842

```

```julia
(@v1.10) pkg> status SpecialFunctions
[276daf66] SpecialFunctions v2.3.1

```

---

<div class="post-metadata">

**Author:** ![baizhan](https://avatars.discourse-cdn.com/v4/letter/b/ecae2f/32.png) [@baizhan](https://discourse.julialang.org/u/baizhan)\
**Post date:** [January 9, 2024, 4:58am UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546/3 "2024-01-09T04:58:00Z")

</div>

Thank you for your reply.

sorry I made a mistake.  
The `airyai` does accept `BigFloat`.  
However, I also need to calculate its derivative, using `airyaiprime`.  
It is `airyaiprime` that cannot calculate BigFloat.

I have already updated to julia v1.10 and SpecialFunction v2.3.1,  
but it still doesn’t work.

I have modified the main post.

---

<div class="post-metadata">

**Author:** ![Dan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dan/32/42581_2.png) [@Dan](https://discourse.julialang.org/u/Dan)\
**Post date:** [January 9, 2024, 5:33am UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546/4 "2024-01-09T05:33:39Z")

</div>

Works with ArbNumerics (perhaps the `setprecision` is needed):

```julia
julia> using ArbNumerics

julia> setprecision(ArbFloat, precision(BigFloat))
256

julia> a = ArbFloat(0.0)
0

julia> result = airyaiprime(a)
-0.25881940379280679840518356018920396347909113835493458221000181385610277267679

```

```julia
(@v1.10) pkg> status ArbNumerics
Status `~/.julia/environments/v1.10/Project.toml`
  [7e558dbc] ArbNumerics v1.3.3

```

---

<div class="post-metadata">

**Author:** ![baizhan](https://avatars.discourse-cdn.com/v4/letter/b/ecae2f/32.png) [@baizhan](https://discourse.julialang.org/u/baizhan)\
**Post date:** [January 9, 2024, 12:31pm UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546/6 "2024-01-09T12:31:26Z")

</div>

Thank you.

After more test, I found that `ArbNumerics` works fine on my Mac.  
However, on my win11 laptop, `airyaiprime(ArbFloat(0.0))` will lead to crash.

I think this is a bug of the package.

---

<div class="post-metadata">

**Author:** ![Dan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dan/32/42581_2.png) [@Dan](https://discourse.julialang.org/u/Dan)\
**Post date:** [January 9, 2024, 11:21pm UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546/7 "2024-01-09T23:21:18Z")

</div>

> [@baizhan](#):
>
> lead to crash

Yeah, probably is a bug. Can you show the printout of the crash? Opening an issue is definitely needed.

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**Author:** ![Oscar\_Smith](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/oscar_smith/32/25343_2.png) [@Oscar\_Smith](https://discourse.julialang.org/u/Oscar_Smith)\
**Post date:** [January 9, 2024, 11:31pm UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546/8 "2024-01-09T23:31:08Z")

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My guess is that this is likely related to differences in `Int` type lengths in C (specifically, windows makes `long` and thus Julia’s `CInt` 32 bits, while on mac and linux, `long` is 64 bits).

---

<div class="post-metadata">

**Author:** ![Oscar\_Smith](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/oscar_smith/32/25343_2.png) [@Oscar\_Smith](https://discourse.julialang.org/u/Oscar_Smith)\
**Post date:** [January 9, 2024, 11:32pm UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546/9 "2024-01-09T23:32:15Z")

</div>

Also it looks like this issue is at least reported:

> <https://github.com/JeffreySarnoff/ArbNumerics.jl/issues/61>
>
> Maybe I'm doing it wrong, but it looks like there is a problem with airyaiprime …and BigFloats:
> \`\`\`
> \_ \_ \_(\_)\_ | Documentation: https://docs.julialang.org
> (\_) | (\_) (\_) |
> \_ \_ \_| |\_ \_\_ \_ | Type "?" for help, "\]?" for Pkg help.
> | | | | | | |/ \_\` | |
> | | |\_| | | | (\_| | | Version 1.9.0 (2023-05-07)
> \_/ |\\\_\_'\_|\_|\_|\\\_\_'\_| | Official https://julialang.org/ release
> |\_\_/ |
> 
> julia @v1.9\> using ArbNumerics
> \[Info: Precompiling ArbNumerics \[7e558dbc-694d-5a72-987c-6f4ebed21442\]
> 
> julia @v1.9\> ArbNumerics.airyai(BigFloat(0))
> 0.3550280538878172392600631860041831763979791741991772405833265103008100424501266
> 
> julia @v1.9\> ArbNumerics.airyaiprime(BigFloat(0))
> ERROR: MethodError: no method matching airyaiprime(::Complex{BigFloat})
> 
> Closest candidates are:
> airyaiprime(::Complex{\<:AbstractFloat})
> @ SpecialFunctions ~/julia-versions/dot\_julia/packages/SpecialFunctions/sKqs4/src/bessel.jl:168
> airyaiprime(::Complex)
> @ SpecialFunctions ~/julia-versions/dot\_julia/packages/SpecialFunctions/sKqs4/src/bessel.jl:165
> airyaiprime(::ArbComplex{P}) where P
> @ ArbNumerics ~/julia-versions/dot\_julia/packages/ArbNumerics/OJfxd/src/float/airy.jl:53
> ...
> 
> Stacktrace:
> \[1\] airyaiprime(z::Complex{BigFloat})
> @ SpecialFunctions ~/julia-versions/dot\_julia/packages/SpecialFunctions/sKqs4/src/bessel.jl:168
> \[2\] airyaiprime(x::BigFloat)
> @ SpecialFunctions ~/julia-versions/dot\_julia/packages/SpecialFunctions/sKqs4/src/bessel.jl:173
> \[3\] top-level scope
> @ REPL\[3\]:1
> \`\`\`

---

<div class="post-metadata">

**Author:** ![baizhan](https://avatars.discourse-cdn.com/v4/letter/b/ecae2f/32.png) [@baizhan](https://discourse.julialang.org/u/baizhan)\
**Post date:** [January 10, 2024, 2:19am UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546/10 "2024-01-10T02:19:50Z")

</div>

The problem is, there is no printout at all.

The julia REPL simply quit without any warning.

 ![image](https://global.discourse-cdn.com/julialang/original/3X/4/c/4c02cbac27ffb3b7dbf0666a5c73a5b081481c47.png)

---

<div class="post-metadata">

**Author:** ![joeldahne](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/joeldahne/32/45861_2.png) [@joeldahne](https://discourse.julialang.org/u/joeldahne)\
**Post date:** [January 16, 2024, 8:31pm UTC](https://discourse.julialang.org/t/special-function-airy-function-with-arbitrary-precision/108546/11 "2024-01-16T20:31:34Z")

</div>

If you can’t get around the crash you could also consider using [Arblib.jl](https://github.com/kalmarek/Arblib.jl/) instead. In that case you could also compute the value and derivative (and higher order derivatives) simultaneously using `ArbSeries`.

```julia-repl
julia> using Arblib, SpecialFunctions

julia> airyai(Arb(0))
[0.35502805388781723926006318600418317639797917419917724058332651030081004245013 +/- 7.76e-78]

julia> airyaiprime(Arb(0))
[-0.25881940379280679840518356018920396347909113835493458221000181385610277267679 +/- 7.31e-78]

julia> airyai(ArbSeries((0, 1)))
[0.35502805388781723926006318600418317639797917419917724058332651030081004245013 +/- 7.76e-78] + [-0.25881940379280679840518356018920396347909113835493458221000181385610277267679 +/- 7.31e-78]⋅x + 𝒪(x^2)

```

If you don’t need the ball enclosure provided by the `Arb` type you can convert it to `BigFloat` after computing the Airy-function of course!