ArbNumerics.jl
Copyright © 20152019 by Jeffrey Sarnoff.
This work is released under The MIT License.
For multiprecision numerical computing using values with 25..2,500 digits. With arithmetic and higher level mathematics, this package offers you the best balance of performance and accuracy.
This package uses the Arb C Library, and adapts some C library interface work from Nemo (see below). Here is a presentation by the designer and architect of the Arb C library.
Introduction
ArbNumerics exports three types: ArbFloat
, ArbReal
, ArbComplex
. ArbFloat
is an extended precision floating point type. Math using ArbFloat
is expected to be very near the veridical value, and often is the closest value for the precision in use. ArbReal
is an intervalvalued quantity formed of an ArbFloat
(the midpoint) and a radius
. Math functions with ArbReal
are assured to enclose the veridical value. This assurance extends to multiple function applications. ArbComplex
is an ArbReal
pair (real, imaginary). The same enclosure assurance applies.
While the bounds of an ArbReal
or ArbComplex
are available, the default is to show these values as digit sequences which almost assuredly are accurate, in a round to nearest sense, to the precision displayed. Math with ArbFloat
does not provide the assurance one gets using ArbReal
, as an ArbFloat
is a point value. While some effort has been taken to provide you with more reliable results from math with ArbFloat
values than would be the case using the underlying library itself, ArbReal
or ArbComplex
are suggested for work that is important to you. ArbFloat
is appropriate when exactness is not required during development, or with applications that are approximating something at increasing precisions.
Installation
pkg> add Readables
pkg> add ArbNumerics
Important
When updating ArbNumerics, do pkg> gc
to prevent accruing a great deal of unused diskspace.
StartUp
using ArbNumerics
or, if you installed Readables,
using ArbNumerics, Readables
Precision
If you want to work with bitlevel precision, first do setextrabits(0)
.
Otherwise, some extra bits are used to assist with printing values rounded to the last digit displayed. You can find out how many extra bits are used with extrabits()
. If you want to change the number of extra bits used, call setextrabits
with the desired number of extra bits.
You can set the internal working precision (which is the same as the displayed precision with setextrabits(0)
) to a given number of bits or a given number of decimal digits:
setworkingprecision(ArbFloat, bits=250)
, setworkingprecision(ArbReal, digits=100)
The type can be any of ArbFloat
, ArbReal
, ArbComplex
. All types share the same precision so interconversion makes sense.
You can set the external displayed precision (which is the the same as the internal precision with setextrabits(0)
) to a given
number of bits or a given number of decimal digits:
setprecision(ArbFloat, bits=250)
, setworkingprecision(ArbReal, digits=100)
The type can be any of ArbFloat
, ArbReal
, ArbComplex
. All types share the same precision so interconversion makes sense.
Using ArbFloat
Reading the sections that follow gives you a good platform from which to develop.
 consider
using ArbNumerics, Readables
julia> ArbFloat(pi, digits=30, base=10)
3.14159265358979323846264338328
julia> readable(ans)
3.14159_26535_89793_23846_26433_8328
information about using Readables
Constructors
Initially, the default precision is set to 106 bits. All ArbNumeric types use the same default precision. You can change this to e.g. 750 bits: setprecision(ArbFloat, 750)
or setprecision(ArbReal, 750)
or setprecision(ArbComplex, 750)
. Change one default the others follow automatically. This is done to preserve internal consistency. Sometimes more than one type is used within a function. The minimal precision allowed is 24 bits. There is no maximum. The underlying C library calculates more rapidly than BigFloat at any precision.
The precision in use may be set globally, as with BigFloats, or it may be given with the constructor. For most purposes, you should work with a type at one, two, or three precisions. It is helps clarity to convert precisions explicitly, however, it is not necessary.
Constructors using the default precision
julia> a = ArbFloat(3)
3.0000000000000000000000000000000
julia> b = ArbReal(pi)
3.1415926535897932384626433832795
julia> c = one(ArbComplex)
1.0000000000000000000000000000000 + 0im
Constructors using a specified precision
julia> BITS = 53;
julia> a = sqrt(ArbFloat(2, BITS))
1.414213562373095
julia> b = ArbReal(pi, BITS)
3.141592653589793
julia> c = ArbComplex(a, b, BITS)
1.414213562373095 + 3.141592653589793im
julia> DIGITS = 78;
julia> ArbFloat(pi, bits4digits(DIGITS))
3.14159265358979323846264338327950288419716939937510582097494459230781640628621
julia> DIGITS == length(string(ans))  1 # (1 for the decimal point)
true
changing precision
julia> a = ArbFloat(2, 25)
2.000000
julia> a = ArbFloat(a, 50)
2.00000000000000
julia> precision = 25
julia> a = ArbFloat(2, precision)
2.000000
julia> precision = 50
julia> a = ArbFloat(a, precision)
2.00000000000000
interconversion
julia> a = sqrt(ArbFloat(2))
1.414213562373095048801688724210
julia> b = ArbReal(a)
1.414213562373095048801688724210
julia> c = ArbComplex(a, b)
1.414213562373095048801688724210 + 1.414213562373095048801688724210im
julia> Float64(a)
1.4142135623730951
julia> Float32(b)
1.4142135f0
julia> Float16(c)
Float16(1.414)
Consider using ArbReals instead of ArbFloats if you want your results to be rock solid. That way you can examine the enclosures for your results with radius(value)
or bounds(value)
. This is strongly suggested when working with precisions that you are increasing dynamically.
Math
arithmetic functions
+
,
,*
,/
square
,cube
,sqrt
,cbrt
,hypot
pow(x,i)
,root(x,i)
where i is an integer > 0factorial
,doublefactorial
,risingfactorial
binomial
elementary functions
exp
,expm1
,log
,log1p
sin
,cos
,tan
,csc
,sec
,cot
asin
,acos
,atan
,atan(y,x)
sinh
,cosh
,tanh
,csch
,sech
,coth
asinh
,acosh
,atanh
arithmeticgeometric mean
agm
,agm1
gamma functions
gamma
,lgamma
rgamma
,digamma
error functions
erf
,erfc
,erfi
Bessel functions
besselj
,besselj0
,besselj1
bessely
,bessely0
,bessely1
besseli
,besselk
Airy functions
airyai
,airyaiprime
airybi
,airybiprime
elliptic integrals
elliptic_e
,elliptic_k
elliptic_p
,elliptic_pi
elliptic_zeta
,elliptic_sigma
elliptic integrals of squared modulus
elliptic_e2
,elliptic_k2
elliptic_p2
,elliptic_pi2
elliptic_zeta2
,elliptic_sigma2
elliptic functions
weierstrass_p
,weierstrass_invp
weierstrass_zeta
,weierstrass_sigma
hypergeometric functions
hypgeom0f1
,hypgeom1f1
,hypgeom1f2
hypgeom0f1reg
,hypgeom1f1reg
,hypgeom1f2reg
(regularized)
other special functions
ei
,si
,ci
shi
,chi
zeta
,eta
,xi
# Reimannlambertw
,polylog
linear algebra
dot
(vectors)det
,tr
,inv
(matrix)*
(matrix multiply) see docs for more functions
transforms
dft
,inverse_dft
 see docs for use
Intervals
parts
 midpoint, radius
 upperbound, lowerbound, bounds
 upperbound_abs, lowerbound_abs, bounds_abs
construction
setball(midpoint, radius)
setinterval(lobound, hibound)
retrieval
midpoint, radius = ball(x::ArbReal)
lobound, hibound = interval(x::ArbReal)
working with intervals
The radii are kept using an Arb C library internal structure that has a 30 bit unsigned significand and a powerof2 exponent that is, essentially, a BigInt. All radii are nonnegative. From the Arb documentation:
The mag_t type holds an unsigned floatingpoint number with a fixedprecision mantissa (30 bits) and an arbitraryprecision exponent ..., suited for representing magnitude bounds. The special values zero and positive infinity are supported, but not NaN. Operations that involve rounding will always produce a valid bound, For performance reasons, no attempt is made to compute the best possible bounds: in general, a bound may be several ulps larger/smaller than the optimal bound.
When constructing intervals , you should scale the radius to be as small as possible while preserving enclosure.
a caution for BigFloat
julia> p=64;setprecision(BigFloat,p);
julia> ArbFloat(pi,p+8)
3.14159265358979323846
julia> ArbFloat(pi,p),BigFloat(pi)
(3.141592653589793238, 3.14159265358979323851)
julia> [ArbFloat(pi,p), BigFloat(pi)]
2element Array{ArbFloat{88},1}:
3.141592653589793238
3.141592653589793239
The Arb C library

Arb is a C library for rigorous real and complex arithmetic with arbitrary precision. Fredrik Johansson is Arb's designer and primary author., with contributions from others.

Arb tracks numerical errors automatically using the midpointradius representation of an interval.

Arb is designed to provide evaluands that contain the veridical numerical result.

Arb uses algorithms with provable error bounds foe multiprecision mathematical functions.

The code is threadsafe, portable, and extensively tested. The library outperforms others.
Acknowledgements
This work develops parts the Arb C library within Julia. It is entirely dependent on Arb by Fredrik Johansson and would not exist without the good work of William Hart, Tommy Hofmann and the Nemo.jl team. The libraries for Arb
and Flint
, and build file are theirs, used with permission.
Alternatives
For a numeric types like Float64
and ComplexF64
with about twice their precision, Quadmath.jl exports Float128
and ComplexF128
. For almost as much precision with better performance, DoubleFloats.jl exports Double64
and ComplexDF64
. ValidatedNumerics.jl and other packages available at JuliaIntervals provide an alternative approach to developing correctly contained results. Those packages are very good and worthwhile when you do not require multiprecision numerics.
notes
 To propose internal changes, please use pull requests.
 To discuss improvements, please raise a GitHub issue.