Numerical derived algebra

The algebra of a finite module depends on which vectors are independent, which are sent to zero, and which represent the same class after taking a quotient. With floating coefficients, these questions require a numerical rank decision. This guide explains how tolerances affect derived computations on the retained finite module and how to assess their results. The category guide specifies what its Hom, Ext, and Tor calculations mean; the numerical choices here do not change that category.

RealField computes numerical kernels, images and quotient coordinates using the tolerances stored in the field. It does not supply the exact-rank guarantee of QQField or a prime field. On well-conditioned data whose nonzero singular values are separated from the rank cutoff, Ext and Tor should nevertheless have the mathematically expected dimensions and satisfy their map and product identities within the declared tolerance.

using TamerOp

field = RealField(Float64; atol=1e-12, rtol=1e-10)

Use the same field, including its tolerances, throughout a derived computation. Changing tolerances can change a numerical quotient and its coordinates. Complexes, homology/cohomology data, and spectral subquotients retain this field through construction, lazy representatives, shifts, cones, and induced maps. Raw matrix constructors accept field=...; omitting it infers the default field from the coefficient type. Summaries and provenance expose the stored field so a custom tolerance remains inspectable.

For numerical cancellation, choose a positive absolute tolerance appropriate to the units of the data. The default RealField has atol=0: a mathematically zero product represented by tiny rounding residuals can then fail a purely relative consistency check. Retaining a user-supplied positive tolerance is essential; it does not justify silently loosening that tolerance.

Independent computations can choose different bases even when they represent the same homology group. Compare them using a coordinate transport or a basis-independent mathematical quantity, rather than assuming their matrices must be entrywise equal.

Rank and solve decisions

The numerical rank cutoff for a coefficient matrix A is

tau(A) = atol + rtol * opnorm(A, 1).

The absolute term matters for small matrices or small units of measurement; the relative term scales with the matrix. Tolerances and matrix entries must be finite, and tolerances must be nonnegative. Rescale data if forming the cutoff or a factorization would overflow.

The field-aware linear-algebra owner exposes operation-specific backends:

backend=:auto selects the operation's normal backend. RREF uses left-to-right columns and partial row pivoting; QR selects independent columns; dense SVD compares singular values with the cutoff. These rank statistics need not agree near a cutoff. An explicit backend is useful for checking that a computation lies away from that ambiguous regime.

Sparse QR receives the field cutoff during factorization. Its reported rank and column permutation are used together when constructing the image and kernel. Factoring with a different cutoff and then merely counting accepted diagonal entries is invalid: the accepted columns need not form the prefix assumed by the subsequent triangular solve.

A solve has a separate consistency test. For each right-hand side y_j and computed solution x_j, the full-column solver checks a backward-error bound of the form

norm(B*x_j - y_j) <= atol + rtol * (norm(B)*norm(x_j) + norm(y_j)).

Each column must pass; a large, accurate right-hand side cannot hide an incompatible smaller one. Scaling a compatible right-hand side should not cause rejection merely because the coefficient matrix stayed small. This residual criterion checks compatibility with the selected numerical image; it does not restore a direction discarded by the rank decision. The explicit check_rhs=false option bypasses this consistency check.

Checking numerical results

Dimensions alone do not establish that a numerical derived computation has preserved the expected maps. Check the relevant matrix identities too: for example, two successive differentials should compose to zero, and a proposed module morphism should commute with the structure maps. For an identity X=Y, a tolerance-based comparison can use

norm(X-Y) <= atol + rtol * max(norm(X), norm(Y)).

Here atol and rtol are the tolerances chosen for the comparison. Inspect the residual relative to max(1, norm(X), norm(Y)) together with the condition numbers of any changes of basis. For a rank-deficient differential, the ordinary condition number is infinite; inspect its nonzero singular values and their separation from the rank cutoff instead. A small residual describes that computation, not numerical stability for arbitrary input scales.

Numerical Hom forms naturality constraints and computes their field-aware nullspace. The tolerance prevents rounding residuals in dependent equations from being treated automatically as new independent constraints. Tor multiplication requires explicit compatible product data; numerical tolerance does not create a canonical product on arbitrary additive Tor groups.

Cached full-column solves require an unchanged coefficient matrix. A change to the field's effective rank cutoff triggers refactorization. Use cache=false throughout workflows that reuse a matrix object while changing its entries.

Near-singular inputs

A singular value or elimination pivot close to tau(A) makes numerical rank sensitive to perturbations, scaling and backend choice. Different operations can then select different ranks, so the library cannot promise one coherent numerical quotient across an entire derived computation. A downstream solve, chain-map check or quotient construction may reject incompatible numerical decisions. An ambiguous rank is not mathematically exact, and computation does not silently switch to rational algebra. When that distinction matters, inspect the singular-value gap, compare backends or tolerances, and retain the original rational input for an exact computation.

An approximate kernel vector must still satisfy its residual contract, and an image basis must span the selected numerical image. Likewise, a solve checks its right-hand sides, and a product on a quotient must be independent of the chosen representatives. Inspect these properties alongside the numerical rank; a dimension alone does not determine the resulting algebra.