Numerical computations

A numerical computation begins with a choice of coefficient field, just as an exact computation does. This page explains the scalar model used by TensorCategories.jl: arbitrary-precision ball arithmetic, comparison at a chosen working precision, and the mathematical conclusions that ball enclosures can support. These points do not depend on a particular category model.

The later chapter on numerical fusion categories applies this scalar model to $F$- and $R$-symbols, coherence, unitarity, and modularity. After defining Drinfeld centers and half-braidings, the center chapter applies the same model to numerical center computations.

Arbitrary-precision ball arithmetic

ArbField(p) and AcbField(p) provide real and complex ball arithmetic with a user-selected working precision of $p$ bits. A ball records a midpoint together with an error radius, and arithmetic propagates the enclosure. This is arbitrary-precision arithmetic in the sense that $p$ may be chosen as large as needed. A particular field, category, and computation nevertheless use one chosen working precision. The implementation does not automatically increase that precision unless an algorithm explicitly says so. The arithmetic model is described by Johansson (2017). Its use for numerical center and $F$- and $R$-symbol computations in TensorCategories.jl is described in Mäurer (2026), §5.2.3.

For example, this constructs a real ball field with $128$ bits of working precision:

using Oscar
R = ArbField(128)
x = sqrt(R(2))
@assert contains_zero(x^2 - R(2))
precision(R)

This differs from ordinary Float64 arithmetic in two ways: the working precision is not restricted to 53 binary digits, and uncertainty is part of the scalar. The common numerical comparisons recognize ArbField, AcbField, and ComplexField as ball coefficient fields; a category-specific algorithm can have narrower coefficient-field methods. In an ArbField(p) or AcbField(p), the chosen precision belongs to the field and therefore to the category over that field. ComplexField() instead uses Nemo's mutable global ball precision. For reproducible category computations, prefer AcbField(p); conversions produced by numeric use it.

Exact and numerical models

Exact scalars and numerical balls answer different questions. Exact fields support algebraic equality and symbolic field operations. Numerical fields support high-precision analytic and linear-algebra computations and retain an enclosure for every result. When exact source data are available, keep them and construct a new numerical model for each numerical computation. The exact source records the algebraic object; the numerical model records one approximation at one chosen working precision.

For a supported object E, numeric(E,p) requests a numerical realization at approximately $p$ bits. Conversions can use additional internal precision (guard bits), so the precision of the resulting base field is authoritative. For a skeletal fusion category over a number field, this conversion uses the complex embedding stored with the category; the abstract number field alone does not determine a numerical realization. The numerical $F$-symbol workflow explains how to supply an embedding explicitly when one is not already stored. Increasing the precision of a field cannot recover information already lost through decimal or low-precision input.

Equality and overlap

Structural equality == remains an equivalence relation. It is not replaced by ball overlap, because overlap is not transitive. Numerical algorithms instead use explicit tests such as overlaps(x,y) or contains_zero(x) at the point where the mathematics asks whether two enclosures are compatible or whether a quantity may vanish.

Rigorous enclosures versus exact algebra

Ball arithmetic makes the enclosure rigorous, provided the input balls rigorously enclose the intended coefficients. Suppose that a ball $B$ contains an exact quantity $b$. Then

If $0\notin B$, then the exact quantity satisfies $b\neq0$;
disjoint balls containing $b$ and $c$ prove $b\neq c$; but
if $0\in B$, or if two balls overlap, this does not prove equality.

Thus a numerical calculation can produce an exact mathematical conclusion. For example, a determinant ball excluding zero certifies nonvanishing. This does not make every ball-valued computation an exact symbolic computation: the scalar is still represented by an enclosure rather than by an exact algebraic expression.

How this distinction applies to categorical identities and structural predicates is explained under Numerical fusion categories.

Continue with A first computation.