Real balls

Real balls

Arbitrary precision real ball arithmetic is supplied by Arb which provides a ball representation which tracks error bounds rigorously. Real numbers are represented in mid-rad interval form $[m \pm r] = [m-r, m+r]$.

The Arb real field is constructed using the ArbField constructor. This constructs the parent object for the Arb real field.

However, we define

RealField = ArbField

so that one can construct the Arb real field parent object using RealField instead of ArbField.

The types of real balls in Nemo are given in the following table, along with the libraries that provide them and the associated types of the parent objects.

LibraryFieldElement typeParent type
Arb$\mathbb{R}$ (balls)arbArbField

All the real field types belong to the Field abstract type and the types of elements in this field, i.e. balls in this case, belong to the FieldElem abstract type.

Real ball functionality

Real balls in Nemo implement the full AbstractAlgebra.jl field interface.

https://nemocas.github.io/AbstractAlgebra.jl/latest/fields

Below, we document the additional functionality provided for real balls.

Constructors

In order to construct real balls in Nemo, one must first construct the Arb real field itself. This is accomplished with the following constructor.

ArbField(prec::Int)

Return the Arb field with precision in bits prec used for operations on interval midpoints. The precision used for interval radii is a fixed implementation-defined constant (30 bits).

We define

RealField = ArbField

so that one can use RealField in place of ArbField.

Here is an example of creating an Arb real field and using the resulting parent object to coerce values into the resulting field.

Examples

RR = RealField(64)

a = RR("0.25")
b = RR("0.1")
c = RR(0.5)
d = RR(12)

Note that whilst one can coerce double precision floating point values into an Arb real field, unless those values can be represented exactly in double precision the resulting ball can't be any more precise than the double precision supplied.

If instead, values can be represented precisely using decimal arithmetic then one can supply them to Arb using a string. In this case, Arb will store them to the precision specified when creating the Arb field.

If the values can be stored precisely as a binary floating point number, Arb will store the values exactly. See the function isexact below for more information.

Real ball constructors

ball(mid::arb, rad::arb)

Constructs an arb enclosing the range $[m-|r|, m+|r|]$, given the pair $(m, r)$.

Examples

RR = RealField(64)

c = ball(RR(3), RR("0.0001"))

Conversions

convert(::Type{Float64}, x::arb)

Return the midpoint of $x$ rounded down to a machine double.

Basic manipulation

isnonzero(x::arb)

Return true if $x$ is certainly not equal to zero, otherwise return false.

isfinite(x::arb)

Return true if $x$ is finite, i.e. having finite midpoint and radius, otherwise return false.

isexact(x::arb)

Return true if $x$ is exact, i.e. has zero radius, otherwise return false.

isint(x::arb)

Return true if $x$ is an exact integer, otherwise return false.

ispositive(x::arb)

Return true if $x$ is certainly positive, otherwise return false.

isnonnegative(x::arb)

Return true if $x$ is certainly nonnegative, otherwise return false.

isnegative(x::arb)

Return true if $x$ is certainly negative, otherwise return false.

isnonpositive(x::arb)

Return true if $x$ is certainly nonpositive, otherwise return false.

midpoint(x::arb)

Return the midpoint of the ball $x$ as an Arb ball.

radius(x::arb)

Return the radius of the ball $x$ as an Arb ball.

accuracy_bits(x::arb)

Return the relative accuracy of $x$ measured in bits, capped between typemax(Int) and -typemax(Int).

Examples

RR = RealField(64)

a = RR("1.2 +/- 0.001")
b = RR(3)

ispositive(a)
isfinite(b)
isint(b)
isnegative(a)
d = midpoint(b)
f = accuracy_bits(a)

Containment

It is often necessary to determine whether a given exact value or ball is contained in a given real ball or whether two balls overlap. The following functions are provided for this purpose.

overlaps(x::arb, y::arb)

Returns true if any part of the ball $x$ overlaps any part of the ball $y$, otherwise return false.

contains(x::arb, y::arb)

Returns true if the ball $x$ contains the ball $y$, otherwise return false.

contains(x::arb, y::Integer)

Returns true if the ball $x$ contains the given integer value, otherwise return false.

contains(x::arb, y::fmpz)

Returns true if the ball $x$ contains the given integer value, otherwise return false.

contains(x::arb, y::fmpq)

Returns true if the ball $x$ contains the given rational value, otherwise return false.

contains(x::arb, y::Rational{T}) where {T <: Integer}

Returns true if the ball $x$ contains the given rational value, otherwise return false.

contains(x::arb, y::BigFloat)

Returns true if the ball $x$ contains the given floating point value, otherwise return false.

The following functions are also provided for determining if a ball intersects a certain part of the real number line.

contains_zero(x::arb)

Returns true if the ball $x$ contains zero, otherwise return false.

contains_negative(x::arb)

Returns true if the ball $x$ contains any negative value, otherwise return false.

contains_positive(x::arb)

Returns true if the ball $x$ contains any positive value, otherwise return false.

contains_nonnegative(x::arb)

Returns true if the ball $x$ contains any nonnegative value, otherwise return false.

contains_nonpositive(x::arb)

Returns true if the ball $x$ contains any nonpositive value, otherwise return false.

Examples

RR = RealField(64)
x = RR("1 +/- 0.001")
y = RR("3")

overlaps(x, y)
contains(x, y)
contains(y, 3)
contains(x, ZZ(1)//2)
contains_zero(x)
contains_positive(y)

Comparison

Nemo provides a full range of comparison operations for Arb balls. Note that a ball is considered less than another ball if every value in the first ball is less than every value in the second ball, etc.

In addition to the standard comparison operators, we introduce an exact equality. This is distinct from arithmetic equality implemented by ==, which merely compares up to the minimum of the precisions of its operands.

isequal(x::arb, y::arb)

Return true if the balls $x$ and $y$ are precisely equal, i.e. have the same midpoints and radii.

We also provide a full range of ad hoc comparison operators. These are implemented directly in Julia, but we document them as though isless and == were provided.

Function
==(x::arb, y::Integer)
==(x::Integer, y::arb)
==(x::arb, y::fmpz)
==(x::fmpz, y::arb)
==(x::arb, y::Float64)
==(x::Float64, y::arb)
isless(x::arb, y::Integer)
isless(x::Integer, y::arb)
isless(x::arb, y::fmpz)
isless(x::fmpz, y::arb)
isless(x::arb, y::Float64)
isless(x::Float64, y::arb)
isless(x::arb, y::BigFloat)
isless(x::BigFloat, y::arb)
isless(x::arb, y::fmpq)
isless(x::fmpq, y::arb)

Examples

RR = RealField(64)
x = RR("1 +/- 0.001")
y = RR("3")
z = RR("4")

isequal(x, deepcopy(x))
x == 3
ZZ(3) < z
x != 1.23

Absolute value

abs(x::arb)

Return the absolute value of $x$.

Examples

RR = RealField(64)
x = RR("-1 +/- 0.001")

a = abs(x)

Shifting

ldexp(x::arb, y::Int)

Return $2^yx$. Note that $y$ can be positive, zero or negative.

ldexp(x::arb, y::fmpz)

Return $2^yx$. Note that $y$ can be positive, zero or negative.

Examples

RR = RealField(64)
x = RR("-3 +/- 0.001")

a = ldexp(x, 23)
b = ldexp(x, -ZZ(15))

Miscellaneous operations

trim(x::arb)

Return an arb interval containing $x$ but which may be more economical, by rounding off insignificant bits from the midpoint.

unique_integer(x::arb)

Return a pair where the first value is a boolean and the second is an fmpz integer. The boolean indicates whether the interval $x$ contains a unique integer. If this is the case, the second return value is set to this unique integer.

setunion(x::arb, y::arb)

Return an arb containing the union of the intervals represented by $x$ and $y$.

Examples

RR = RealField(64)
x = RR("-3 +/- 0.001")
y = RR("2 +/- 0.5")

a = trim(x)
b, c = unique_integer(x)
d = setunion(x, y)

Constants

const_pi(r::ArbField)

Return $\pi = 3.14159\ldots$ as an element of $r$.

const_e(r::ArbField)

Return $e = 2.71828\ldots$ as an element of $r$.

const_log2(r::ArbField)

Return $\log(2) = 0.69314\ldots$ as an element of $r$.

const_log10(r::ArbField)

Return $\log(10) = 2.302585\ldots$ as an element of $r$.

const_euler(r::ArbField)

Return Euler's constant $\gamma = 0.577215\ldots$ as an element of $r$.

const_catalan(r::ArbField)

Return Catalan's constant $C = 0.915965\ldots$ as an element of $r$.

const_khinchin(r::ArbField)

Return Khinchin's constant $K = 2.685452\ldots$ as an element of $r$.

const_glaisher(r::ArbField)

Return Glaisher's constant $A = 1.282427\ldots$ as an element of $r$.

Examples

RR = RealField(200)

a = const_pi(RR)
b = const_e(RR)
c = const_euler(RR)
d = const_glaisher(RR)

Mathematical and special functions

floor(x::arb)

Compute the floor of $x$, i.e. the greatest integer not exceeding $x$, as an Arb.

ceil(x::arb)

Return the ceiling of $x$, i.e. the least integer not less than $x$, as an Arb.

sqrt(x::arb)

Return the square root of $x$.

rsqrt(x::arb)

Return the reciprocal of the square root of $x$, i.e. $1/\sqrt{x}$.

sqrt1pm1(x::arb)

Return $\sqrt{1+x}-1$, evaluated accurately for small $x$.

sqrtpos(x::arb)

Return the sqrt root of $x$, assuming that $x$ represents a nonnegative number. Thus any negative number in the input interval is discarded.

log(x::arb)

Return the principal branch of the logarithm of $x$.

log1p(x::arb)

Return $\log(1+x)$, evaluated accurately for small $x$.

exp(x::arb)

Return the exponential of $x$.

expm1(x::arb)

Return $\exp(x)-1$, evaluated accurately for small $x$.

sin(x::arb)

Return the sine of $x$.

cos(x::arb)

Return the cosine of $x$.

sinpi(x::arb)

Return the sine of $\pi x$.

cospi(x::arb)

Return the cosine of $\pi x$.

tan(x::arb)

Return the tangent of $x$.

cot(x::arb)

Return the cotangent of $x$.

tanpi(x::arb)

Return the tangent of $\pi x$.

cotpi(x::arb)

Return the cotangent of $\pi x$.

sinh(x::arb)

Return the hyperbolic sine of $x$.

cosh(x::arb)

Return the hyperbolic cosine of $x$.

tanh(x::arb)

Return the hyperbolic tangent of $x$.

coth(x::arb)

Return the hyperbolic cotangent of $x$.

atan(x::arb)

Return the arctangent of $x$.

asin(x::arb)

Return the arcsine of $x$.

acos(x::arb)

Return the arccosine of $x$.

atanh(x::arb)

Return the hyperbolic arctangent of $x$.

asinh(x::arb)

Return the hyperbolic arcsine of $x$.

acosh(x::arb)

Return the hyperbolic arccosine of $x$.

gamma(x::arb)

Return the Gamma function evaluated at $x$.

lgamma(x::arb)

Return the logarithm of the Gamma function evaluated at $x$.

rgamma(x::arb)

Return the reciprocal of the Gamma function evaluated at $x$.

digamma(x::arb)

Return the logarithmic derivative of the gamma function evaluated at $x$, i.e. $\psi(x)$.

zeta(x::arb)

Return the Riemann zeta function evaluated at $x$.

sincos(x::arb)

Return a tuple $s, c$ consisting of the sine $s$ and cosine $c$ of $x$.

sincospi(x::arb)

Return a tuple $s, c$ consisting of the sine $s$ and cosine $c$ of $\pi x$.

sinpi(x::fmpq, r::ArbField)

Return the sine of $\pi x$ in the given Arb field.

cospi(x::fmpq, r::ArbField)

Return the cosine of $\pi x$ in the given Arb field.

sincospi(x::fmpq, r::ArbField)

Return a tuple $s, c$ consisting of the sine and cosine of $\pi x$ in the given Arb field.

sinhcosh(x::arb)

Return a tuple $s, c$ consisting of the hyperbolic sine and cosine of $x$.

atan2(x::arb, y::arb)

Return atan2$(b,a) = \arg(a+bi)$.

agm(x::arb, y::arb)

Return the arithmetic-geometric mean of $x$ and $y$

zeta(s::arb, a::arb)

Return the Hurwitz zeta function $\zeta(s,a)$.

hypot(x::arb, y::arb)

Return $\sqrt{x^2 + y^2}$.

root(x::arb, n::Int)

Return the $n$-th root of $x$. We require $x \geq 0$.

fac(x::arb)

Return the factorial of $x$.

fac(n::Int, r::ArbField)

Return the factorial of $n$ in the given Arb field.

binom(x::arb, n::UInt)

Return the binomial coefficient ${x \choose n}$.

binom(n::UInt, k::UInt, r::ArbField)

Return the binomial coefficient ${n \choose k}$ in the given Arb field.

fib(n::fmpz, r::ArbField)

Return the $n$-th Fibonacci number in the given Arb field.

fib(n::Int, r::ArbField)

Return the $n$-th Fibonacci number in the given Arb field.

gamma(x::fmpz, r::ArbField)

Return the Gamma function evaluated at $x$ in the given Arb field.

gamma(x::fmpq, r::ArbField)

Return the Gamma function evaluated at $x$ in the given Arb field.

zeta(n::Int, r::ArbField)

Return the Riemann zeta function $\zeta(n)$ as an element of the given Arb field.

bernoulli(n::Int, r::ArbField)

Return the $n$-th Bernoulli number as an element of the given Arb field.

risingfac(x::arb, n::Int)

Return the rising factorial $x(x + 1)\ldots (x + n - 1)$ as an Arb.

risingfac(x::fmpq, n::Int, r::ArbField)

Return the rising factorial $x(x + 1)\ldots (x + n - 1)$ as an element of the given Arb field.

risingfac2(x::arb, n::Int)

Return a tuple containing the rising factorial $x(x + 1)\ldots (x + n - 1)$ and its derivative.

polylog(s::Union{arb,Int}, a::arb)

Return the polylogarithm Li$_s(a)$.

chebyshev_t(n::Int, x::arb)

Return the value of the Chebyshev polynomial $T_n(x)$.

chebyshev_u(n::Int, x::arb)

Return the value of the Chebyshev polynomial $U_n(x)$.

chebyshev_t2(n::Int, x::arb)

Return the tuple $(T_{n}(x), T_{n-1}(x))$.

chebyshev_u2(n::Int, x::arb)

Return the tuple $(U_{n}(x), U_{n-1}(x))$

bell(n::fmpz, r::ArbField)

Return the Bell number $B_n$ as an element of $r$.

bell(n::Int, r::ArbField)

Return the Bell number $B_n$ as an element of $r$.

numpart(n::fmpz, r::ArbField)

Return the number of partitions $p(n)$ as an element of $r$.

numpart(n::Int, r::ArbField)

Return the number of partitions $p(n)$ as an element of $r$.

Examples

RR = RealField(64)

a = floor(exp(RR(1)))
b = sinpi(QQ(5,6), RR)
c = gamma(QQ(1,3), RealField(256))
d = bernoulli(1000, RealField(53))
f = polylog(3, RR(-10))

Linear dependence

lindep(A::Array{arb, 1}, bits::Int)

Find a small linear combination of the entries of the array $A$ that is small (using LLL). The entries are first scaled by the given number of bits before truncating to integers for use in LLL. This function can be used to find linear dependence between a list of real numbers. The algorithm is heuristic only and returns an array of Nemo integers representing the linear combination.

Examples

RR = RealField(128)

a = RR(-0.33198902958450931620250069492231652319)

V = [RR(1), a, a^2, a^3, a^4, a^5]
W = lindep(V, 20)
  simplest_rational_inside(x::arb)

Return the simplest fraction inside the ball x. A canonical fraction a_1/b_1 is defined to be simpler than a_2/b_2 iff b_1 < b_2 or b_1 = b_2 and a_1 < a_2.

Examples

RR = RealField(64)
simplest_rational_inside(const_pi(RR))