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Radial Interval-sampled Build And Extents

Pseudocode

  1. Start from one RadialBasisSpec. The front-door radial recipe chooses:

    • family
    • one public extent: rmax or count
    • mapping
    • reference spacing
    • seed-window parameters
    • optional xgaussian supplement

    Code: src/bases.jl

  2. Separate the public extent from the internal extents. The current contract is:

    • rmax: public last-center extent
    • build_umax: internal setup/build extent in mapped u
    • quadrature_umax: internal active quadrature extent in mapped u

    Code: src/bases.jl, src/quadrature.jl

  3. Determine the internal build target in reference/mapped space. The construction logic derives:

    • odd-count seed window
    • internal build/setup extent
    • setup grid extent large enough for localization and cleanup

    Code: src/bases.jl

  4. Build the setup-grid sampled layer with interval support, not dense full columns. For each shifted seed gausslet and each injected xgaussian, construct a setup-grid object only on the interval where it is numerically active. Code: src/bases.jl

  5. Use the runtime machine-significant family support for that setup layer. The runtime family tables are trimmed after the last coefficient with abs(c) >= 1e-17, while the full high-precision family tables remain in the repo as internal reference data. Code: src/families.jl, src/internal/families_high_prec.jl

  6. Assemble the seed overlap and position data by overlap-range weighted dot products. The interval-sampled layer supplies:

    • seed Gram matrices
    • seed position matrices
    • seed-to-xgaussian cross terms

    without storing long dense zero-tail columns. Code: src/bases.jl

  7. Keep the later dense seed-space linear algebra unchanged. The present radial build still does:

    • odd/even cleanup
    • delta-direction removal
    • COMX localization
    • final QR/sign-fix flow
    • primitive contraction into the public RadialBasis

    Code: src/bases.jl

  8. Build the final public basis in the same form as before. The interval-sampled objects are construction-time helpers only. Once the final localized basis is formed, they are discarded and the public result is the ordinary RadialBasis. Code: src/bases.jl

  9. Derive the default quadrature extent from retained support, not setup padding. The current automatic rule takes:

    • retained radial centers
    • runtime family support width
    • retained xgaussian support

    and builds quadrature_umax from those active retained objects. Code: src/bases.jl, src/quadrature.jl

  10. Build the physical quadrature grid from that retained-support extent. The automatic default path maps quadrature_umax to physical space and refines until the public-quality quadrature checks are satisfied. quadrature_rmax remains only as an expert override. Code: src/quadrature.jl, src/diagnostics.jl

References

  • Supporting notes:
    • docs/radial_extent_policy_note.md
    • docs/radial_trust_milestone_note.md
    • docs/radial_default_path_cache_study.md

What This Builds

This page records the current trusted radial construction route:

  • fast interval-sampled basis construction
  • public rmax semantics separated from internal numerical extents
  • automatic quadrature extent derived from retained support

It is the current source-of-truth algorithm page for the radial front door.

Current Repo Status

The repo now has the radial line in the state described on this page:

  • runtime family tables use machine-significant tails
  • full high-precision family tables are preserved internally for reference
  • basis construction uses interval-sampled setup-grid objects
  • the standard radial path is no longer dominated by build_basis
  • rmax, build_umax, and quadrature_umax are now distinct concepts in both code and docs

The normal public story is therefore:

  • users choose rmax
  • the library owns build_umax
  • the library owns quadrature_umax

Why This Matters

This algorithm page replaces two older confusions:

  • treating rmax as if it directly set every internal extent
  • treating the old dense setup-grid build as if it were the defining radial algorithm rather than an implementation bottleneck

The present route is both faster and conceptually cleaner.

Code Map

The current trusted radial path is split across three files:

  • src/bases.jl
    • _select_construction_data(...) build-grid selection and refinement framing for steps 3 and 7
    • _seed_scalar_integrals(...) setup-grid seed sampling used by the interval-sampled layer in steps 4 and 6
    • _xgaussian_sample_matrix(...) setup-grid xgaussian sampling in steps 4 and 6
    • _build_radial_coefficients(...) main radial construction path for steps 3-8
    • _radial_overlap_deviation(...) construction-quality gate used during grid refinement
    • build_basis(spec::RadialBasisSpec; ...) public front door for the whole build route
  • src/quadrature.jl
    • radial_quadrature(basis::RadialBasis; ...) automatic retained-support quadrature construction for steps 9 and 10
  • src/diagnostics.jl
    • basis_diagnostics(basis::RadialBasis; ...) public diagnostics path sharing the same automatic quadrature policy

The code-comment linkage on this line is still lighter than on the QW route. This page is the main source-of-truth for tightening it further.

Implementation Notes

Recommended code-comment style:

# Alg Radial-Build step 4: Sample shifted seeds and xgaussians only on their
# active setup-grid intervals.
# See docs/src/algorithms/radial_interval_sampled_build_and_extents.md.

Guidelines:

  • keep step numbers aligned with this page
  • use this page for the trusted radial build/extents contract
  • keep cache-policy or milestone numerics in supporting notes rather than growing this page into a status log