bibkey: johansson2018ballintegration authors: Fredrik Johansson year: 2018 title: Numerical integration in arbitrary-precision ball arithmetic doi: null url: https://arxiv.org/abs/1802.07942v1 claim: Validated Petras integration combines adaptive subdivision and Gaussian quadrature with complex analytic magnitude bounds. Its bounded-path and holomorphic-callback contract supplies a reusable integration method, not a sign certificate for the mixed theta matrix. strata_touched: [] license: bibliographic-reference-only triage: anchor
Validated integration supplier and callback obligations
The inspected primary text is
arXiv:1802.07942v1, with SHA-256
64abe78ce44e513bacf764e205f5f4eb728e6c6a0c79dc1a114b2cbfb44aae78.
Printed pp.2–3 describe the Petras strategy and its callback contract;
pp.6–7 describe handling singularities by splitting, transformations and
separately bounded tails. The integration method is existing work and
is reused without an originality claim.
The accompanying API contract was inspected in
FLINT v3.3.1, doc/source/acb_calc.rst,
with SHA-256
c57859a8a3b05fd0dc95a4f8a3f2448f04d70e5655a83bf944faab1d6fcde171.
This pins the inspected documentation; it does not identify the FLINT
version linked by any separately installed Python runtime.
Finite directed enclosures
acb_calc_integrate encloses a straight-segment integral supplied by a
complex-ball callback. Finite endpoints and a bounded integrand on the
path are required for a finite result. Improper integrals therefore need
a regularizing substitution or a finite retained path together with an
independent tail bound.
An order-zero callback supplies a pointwise enclosure without a regularity assumption. An order-one callback must also verify holomorphicity on the requested complex enclosure, returning nonfinite values when this fails. Meromorphic field expressions automatically reject enclosed poles; branch-cut functions need explicit analytic checks. A callback using a truncated theta series must enclose its omitted summands uniformly on that same complex domain.
The documented Gauss rule on , for an integrand bounded by on a Bernstein ellipse of parameter , has error at most
The algorithm combines this rule with direct interval enclosures and adaptive bisection. Absolute and relative tolerances are targets; the final enclosure must itself be inspected. Evaluation limits, cancellation and accumulated subdivision errors can prevent the requested final accuracy. An API success or a nominal working precision does not independently certify a desired matrix sign.
Use in the mixed theta form
The actual theta matrix assembly must regularize both same-cell diagonal cancellation and adjacent-cell jump corners before invoking the bounded-path supplier. One-sided rational branches preserve the jump terms. Complex callbacks use polarized products of those real branches; complex absolute squares are not holomorphic extensions.
For a bounded holomorphic integrand on a product of rescaled Bernstein ellipses, successive application of the documented one-dimensional rule gives a tensor rule on with error at most . This is a standard application of the existing quadrature bound. An adaptively evaluated inner integral does not by itself satisfy the analytic contract of an outer callback; tensor rules, parameter-uniform inner enclosures or interval-box integration are needed instead.
Quadrature supplies entry enclosures. A simultaneous Loewner lower matrix additionally needs correctly directed truncation, full-probability centering and a coefficient-vector error bound. These separate obligations are retained in the linked model note. No numerical theta matrix, positive-semidefinite sign, computational feasibility, Lean certification, RH or full Robin conclusion follows from this source alone.