Exact real and complex numbers

Exact real and complex numbers are provided by Calcium. Internally, a number $z$ is represented as an element of an extension field of the rational numbers. That is,

\[z \in \mathbb{Q}(a_1,\ldots,a_n)\]

where $a_1, \ldots, a_n$ are symbolically defined algebraic or transcendental real or complex numbers such as $\pi$, $\sqrt{2}$ or $e^{\sqrt{2} \pi i}$. The user does not normally need to worry about the details of the internal representation; Calcium constructs extension numbers and fields automatically as needed to perform operations.

The user must create a CalciumField instance which represents the mathematical domain $\mathbb{C}$. This parent object holds a cache of extension numbers and fields used to represent individual elements. It also stores various options for evaluation (documented further below).

Library	Element type	Parent type
Calcium	`CalciumFieldElem`	`CalciumField`

Please note the following:

It is in the nature of exact complex arithmetic that some operations must be implemented using incomplete heuristics. For example, testing whether an element is zero will not always succeed. When Calcium is unable to perform a task, Nemo will throw an exception. This ensures that Calcium fields behave exactly and never silently return wrong results.
Calcium elements can optionally hold special non-numerical values:
- Unsigned infinity $\hat \infty$
- Signed infinities ($\pm \infty$, $\pm i \infty$, and more generally $e^{i \theta} \cdot \infty$)
- Undefined
- Unknown
By default, such special values are disallowed so that a CalciumField represents the mathematical field $\mathbb{C}$, and any operation that would result in a special value (for example, $1 / 0 = \hat \infty$) will throw an exception. To allow special values, pass extended=true to the CalciumField constructor.
CalciumField instances only support single-threaded use. You must create a separate parent object for each thread to do parallel computation.
When performing an operation involving two CalciumFieldElem operands with different parent objects, Nemo will arbitrarily coerce the operands (and hence the result) to one of the parents.

Calcium field options

The CalciumField parent stores various options that affect simplification power, performance, or appearance. The user can override any of the default values using C = CalciumField(options=dict) where dict is a dictionary with Symbol => Int pairs. To retrieve the option values as a dictionary (including any default values not set by the user), call options(C).

The following options are supported:

Option	Explanation
`:verbose`	Enable debug output
`:print_flags`	Flags controlling print style
`:mpoly_ord`	Monomial order for polynomials
`:prec_limit`	Precision limit for numerical evaluation
`:qqbar_deg_limit`	Degree limit for algebraic numbers
`:low_prec`	Initial precision for numerical evaluation
`:smooth_limit`	Factor size limit for smooth integer factorization
`:lll_prec`	Precision for integer relation detection
`:pow_limit`	Maximum exponent for in-field powering
`:use_gb`	Enable Gröbner basis computation
`:gb_length_limit`	Maximum ideal basis length during Gröbner basis computation
`:gb_poly_length_limit`	Maximum polynomial length during Gröbner basis computation
`:gb_poly_bits_limit`	Maximum bit size during Gröbner basis computation
`:gb_vieta_limit`	Maximum degree to use Vieta's formulas
`:trig_form`	Default form of trigonometric functions

An important function of these options is to control how hard Calcium will try to find an answer before it gives up. For example:

Setting :prec_limit => 65536 will allow Calcium to use up to 65536 bits of precision (instead of the default 4096) to prove inequalities.
Setting :qqbar_deg_limit => typemax(Int) (instead of the default 120) will force most calculations involving algebraic numbers to run to completion, no matter how long this will take.
Setting :use_gb => 0 (instead of the default 1) disables use of Gröbner bases. In general, this will negatively impact Calcium's ability to simplify field elements and prove equalities, but it can speed up calculations where Gröbner bases are unnecessary.

For a detailed explanation, refer to the following section in the Calcium documentation: https://fredrikj.net/calcium/ca.html#context-options

Basic examples

julia> C = CalciumField()
Exact complex field

julia> exp(C(pi) * C(1im)) + 1
0

julia> log(C(-1))
3.14159*I {a*b where a = 3.14159 [Pi], b = I [b^2+1=0]}

julia> log(C(-1)) ^ 2
-9.86960 {-a^2 where a = 3.14159 [Pi], b = I [b^2+1=0]}

julia> log(C(10)^23) // log(C(100))
11.5000 {23/2}

julia> 4*atan(C(1)//5) - atan(C(1)//239) == C(pi)//4
true

julia> Cx, x = polynomial_ring(C, "x")
(Univariate polynomial ring in x over exact complex field, x)

julia> (a, b) = (sqrt(C(2)), sqrt(C(3)))
(1.41421 {a where a = 1.41421 [a^2-2=0]}, 1.73205 {a where a = 1.73205 [a^2-3=0]})

julia> (x-a-b)*(x-a+b)*(x+a-b)*(x+a+b)
x^4 + -10*x^2 + 1

Conversions and numerical evaluation

Calcium numbers can created from integers (ZZ), rationals (QQ) and algebraic numbers (algebraic_closure(QQ)), and through the application of arithmetic operations and transcendental functions.

Calcium numbers can be converted to integers, rational and algebraic fields provided that the values are integer, rational or algebraic. An exception is thrown if the value does not belong to the target domain, if Calcium is unable to prove that the value belongs to the target domain, or if Calcium is unable to compute the explicit value because of evaluation limits.

julia> QQ(C(1))
1

julia> algebraic_closure(QQ)(sqrt(C(2)) // 2)
Root 0.707107 of 2x^2 - 1

julia> QQ(C(pi))
ERROR: unable to convert to a rational number
[...]

julia> QQ(C(10) ^ C(10^9))
ERROR: unable to convert to a rational number
[...]

To compute arbitrary-precision numerical enclosures, convert to ArbField or AcbField:

julia> CC = AcbField(64)
Complex Field with 64 bits of precision and error bounds

julia> CC(exp(C(1im)))
[0.54030230586813971740 +/- 9.37e-22] + [0.84147098480789650665 +/- 2.51e-21]*im

The constructor

(R::AcbField)(a::CalciumFieldElem; parts::Bool=false)

returns an enclosure of the complex number a. It attempts to obtain a relative accuracy of prec bits where prec is the precision of the target field, but it is not guaranteed that this goal is achieved.

If parts is set to true, it attempts to achieve the target accuracy for both real and imaginary parts. This can be significantly more expensive if one part is smaller than the other, or if the number is nontrivially purely real or purely imaginary (in which case an exact proof attempt is made).

julia> x = sin(C(1), form=:exponential)
0.841471 + 0e-24*I {(-a^2*b+b)/(2*a) where a = 0.540302 + 0.841471*I [Exp(1.00000*I {b})], b = I [b^2+1=0]}

julia> AcbField(64)(x)
[0.84147098480789650665 +/- 2.51e-21] + [+/- 4.77e-29]*im

julia> AcbField(64)(x, parts=true)
[0.84147098480789650665 +/- 2.51e-21]

The constructor

(R::ArbField)(a::CalciumFieldElem; check::Bool=true)

returns a real enclosure. If check is set to true (default), the number a is verified to be real, and an exception is thrown if this cannot be determined. With check set to false, this function returns an enclosure of the real part of a without checking that the imaginary part is zero. This can be significantly faster.

Comparisons and properties

Except where otherwise noted, predicate functions such as iszero, ==, < and isreal act on the mathematical values of Calcium field elements. For example, although evaluating $x = \sqrt{2} \sqrt{3}$ and $y = \sqrt{6}$ results in different internal representations ($x \in \mathbb{Q}(\sqrt{3}, \sqrt{2})$ and $y \in \mathbb{Q}(\sqrt{6})$), the numbers compare as equal:

julia> x = sqrt(C(2)) * sqrt(C(3))
2.44949 {a*b where a = 1.73205 [a^2-3=0], b = 1.41421 [b^2-2=0]}

julia> y = sqrt(C(6))
2.44949 {a where a = 2.44949 [a^2-6=0]}

julia> x == y
true

julia> iszero(x - y)
true

julia> isinteger(x - y)
true

Predicate functions return true if the property is provably true and false if the property if provably false. If Calcium is unable to prove the truth value, an exception is thrown. For example, with default settings, Calcium is currently able to prove that $e^{e^{-1000}} \ne 1$, but it fails to prove $e^{e^{-3000}} \ne 1$:

julia> x = exp(exp(C(-1000)))
1.00000 {a where a = 1.00000 [Exp(5.07596e-435 {b})], b = 5.07596e-435 [Exp(-1000)]}

julia> x == 1
false

julia> x = exp(exp(C(-3000)))
1.00000 {a where a = 1.00000 [Exp(1.30784e-1303 {b})], b = 1.30784e-1303 [Exp(-3000)]}

julia> x == 1
ERROR: Unable to perform operation (failed deciding truth of a predicate): isequal
[...]

In this case, we can get an answer by allowing a higher working precision:

julia> C2 = CalciumField(options=Dict(:prec_limit => 10^5));

julia> exp(exp(C2(-3000))) == 1
false

Real numbers can be ordered and sorted the usual way. We illustrate finding square roots that are well-approximated by integers:

julia> sort([sqrt(C(n)) for n=0:10], by=x -> abs(x - floor(x + C(1)//2)))
11-element Vector{CalciumFieldElem}:
 0
 1
 2
 3
 3.16228 {a where a = 3.16228 [a^2-10=0]}
 2.82843 {2*a where a = 1.41421 [a^2-2=0]}
 2.23607 {a where a = 2.23607 [a^2-5=0]}
 1.73205 {a where a = 1.73205 [a^2-3=0]}
 2.64575 {a where a = 2.64575 [a^2-7=0]}
 1.41421 {a where a = 1.41421 [a^2-2=0]}
 2.44949 {a where a = 2.44949 [a^2-6=0]}

As currently implemented, order comparisons involving nonreal numbers yield false (in both directions) rather than throwing an exception:

julia> C(1im) < C(1im)
false

julia> C(1im) > C(1im)
false

This behavior may be changed or may become configurable in the future.

Interface

iszero — Method

iszero(a::CalciumFieldElem)

Return whether a is the number 0.