This package is a thin, efficient wrapper around Arb - a C library for arbitrary-precision ball arithmetic. Since 2023 Arb is part of Flint.
using Pkg
pkg"add Arblib"From the Arb documentation:
Arb is a C library for rigorous real and complex arithmetic with arbitrary precision. Arb tracks numerical errors automatically using ball arithmetic, a form of interval arithmetic based on a midpoint-radius representation. On top of this, Arb provides a wide range of mathematical functionality, including polynomials, power series, matrices, integration, root-finding, and many transcendental functions. Arb is designed with efficiency as a primary goal, and is usually competitive with or faster than other arbitrary-precision packages.
The following table indicates how Arb C-types can be translated to the Julia side.
Note that all Julia structs additionally contain a precision field storing the precision
used.
Julia types with Ref in their name are similar to the Ref type in
base Julia. They contain a pointer to an object of the corresponding
type, as well as a reference to it parent object to protect it from
garbage collection.
| Arb | Julia |
|---|---|
mag_t |
Mag or MagRef |
arf_t |
Arf or ArfRef |
arb_t |
Arb or ArbRef |
acb_t |
Acb or AcbRef |
arb_t* |
ArbVector or ArbRefVector |
acb_t* |
AcbVector or AcbRefVector |
arb_mat_t |
ArbMatrix or ArbRefMatrix |
acb_mat_t |
AcbMatrix or AcbRefMatrix |
arb_poly_t |
ArbPoly or ArbSeries |
acb_poly_t |
AcbPoly or AcbSeries |
Indexing of an ArbMatrix returns an Arb whereas indexing ArbRefMatrix returns an ArbRef.
An ArbMatrix A can also be index using the ref function , e.g, ref(A, i, j) to obtain
an ArbRef.
Additionally, there are multiple union types defined to capture a Ref and non-Ref version.
For example Arb and ArbRef are subtypes of ArbLike. Similarly, we provide
MagLike, ArfLike, ArbLike, AcbLike, ArbVectorLike, AcbVectorLike, ArbMatrixLike,
AcbMatrixLike.
Both ArbPoly and ArbSeries wrap the arb_poly_t type. The
difference is that ArbSeries has a fixed length and is therefore
suitable for use when Taylor series are computed using the _series
functions in Arb. Similar for AcbPoly and AcbSeries.
Example:
julia> A = ArbMatrix([1 2; 3 4]; prec=64)
2×2 ArbMatrix:
1.000000000000000000 2.000000000000000000
3.000000000000000000 4.000000000000000000
julia> a = A[1,2]
2.000000000000000000
julia> Arblib.set!(a, 12)
12.00000000000000000
# Memory in A not changed
julia> A
2×2 ArbMatrix:
1.000000000000000000 2.000000000000000000
3.000000000000000000 4.000000000000000000
julia> b = ref(A, 1, 2)
2.000000000000000000
julia> Arblib.set!(b, 12)
12.00000000000000000
# Memory in A also changed
julia> A
2×2 ArbMatrix:
1.000000000000000000 12.00000000000000000
3.000000000000000000 4.000000000000000000Arb functions are wrapped by parsing the Arb documentation and applying the following set of rules to "Juliafy" the function names:
- The parts of a function name which only refer to the type of input are removed since Julia has multiple dispatch to deal with this problem.
- Functions which modify the first argument get an
!appened. - For functions which take a precision argument this arguments becomes a
preckeyword argument which is by default set to the precision of the first argument (if applicable). - For functions which take a rounding mode argument this arguments becomes a
rndkeyword argument which is by default set toRoundNearest.
Example: The function
arb_add_si(arb_t z, const arb_t x, slong y, slong prec)`becomes
add!(z::ArbLike, x::ArbLike, y::Int; prec = precision(z))Arb defines a number of functions for setting something to a specific
value, for example void arb_set_si(arb_t y, slong x). All of these
are renamed to set! and rely on multiple dispatch to choose the
correct one. In addition to the ones defined in Arb there is a number
of methods of set! added in Arblib to make it more convenient to
work with. For example there are setters for Rational and all
irrationals defined in Base.MathConstants. For Arb there is also a
setter which takes a tuple (a, b) representing an interval and
returns a ball containing this interval.
Almost all of the constructors are simple wrappers around these
setters. This means that it's usually more informative to look at the
methods for set! than for say Arb to figure out what constructors
exists. Both Arb and Acb are constructed in such a way that the
result will always enclose the input.
Example:
x = Arblib.set!(Arb(), π)
y = Arb(π)
x = Arblib.set!(Arb(), 5//13)
y = Arb(5//13)
x = Arblib.set!(Arb(), (0, π))
y = Arb((0, π))Arb is made for rigorous numerics and any functions which do not produce rigorous results are clearly marked as such. This is not the case with Julia in general and you therefore have to be careful when interacting with the ecosystem if you want your results to be completely rigorous. Below are three things which you have to be extra careful with.
Julia automatically promotes types in many cases and in particular you
have to watch out for temporary non-rigorous values. For example
2(π*(Arb(ℯ))) is okay, but not 2π*Arb(ℯ)
julia> 2(π*(Arb(ℯ)))
[17.079468445347134130927101739093148990069777071530229923759202260358457222314 +/- 9.19e-76]
julia> 2π*Arb(ℯ)
[17.079468445347133465140073658536286170170195258393831755094914544308087031794 +/- 7.93e-76]
julia> Arblib.overlaps(2(π*(Arb(ℯ))), 2π*Arb(ℯ))
falseIn many cases this is obvious, for example Julias built in methods for solving linear systems will not produce rigorous results.
TODO: Come up with more examples
In some cases the implementation in Julia implicitly makes certain
assumptions to improve performance and this can lead to issues. For
example, prior to Julia version 1.8 the minimum and maximum
methods in Julia checked for NaN results (on which is short fuses)
using x == x, which works for most numerical types but not for Arb
(x == x is only true if the radius is zero). See
JuliaLang/julia#36287 and in particular
JuliaLang/julia#45932 for more details.
Since Julia version 1.8 the minimum and maximum methods work
correctly for Arb, for earlier versions of Julia it only works
correctly in some cases.
These types of problems are the hardest to find since they are not
clear from the documentation but you have to read the implementation,
@which and @less are your friends in these cases.
Here is the naive sine compuation example form the Arb documentation in Julia:
using Arblib
function sin_naive!(res::Arb, x::Arb)
s, t, u = zero(x), zero(x), zero(x)
tol = one(x)
Arblib.mul_2exp!(tol, tol, -precision(tol))
k = 0
while true
Arblib.pow!(t, x, UInt(2k + 1))
Arblib.fac!(u, UInt(2k + 1))
Arblib.div!(t, t, u)
Arblib.abs!(u, t)
if u ≤ tol
Arblib.add_error!(s, u)
break
end
if iseven(k)
Arblib.add!(s, s, t)
else
Arblib.sub!(s, s, t)
end
k += 1
end
Arblib.set!(res, s)
end
let prec = 64
while true
x = Arb("2016.1"; prec = prec)
y = zero(x)
y = sin_naive!(y, x)
print("Using $(lpad(prec, 5)) bits, sin(x) = ")
println(Arblib.string_nice(y, 10))
y < zero(y) && break
prec *= 2
end
endArblib extends the methods from
SpecialFunctions.jl
with versions from Arb. In some cases the Arb version is more general
than the version in SpecialFunctions, for example ellipk is not
implemented for complex arguments in SpecialFunctions but it is in
Arb. We refer to the Arb documentation for details about the
Arb-versions.
Some methods from SpecialFunctions are however not implemented in Arb and does are not extended, these are mostly scaled version of methods. Arb does however implement many special functions that are not in SpecialFunction and at the moment there is no user friendly interface for most of them.
Enabling a threaded version of flint can be done by setting the
environment variable NEMO_THREADED=1. Note that this should be
set before Arblib.jl is loaded. To set the actual number of threads,
use Arblib.flint_set_num_threads($numberofthreads).
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