Padé Table¶
A two-dimensional array whose entry at degree pair (m,n) is the Padé rational approximant with numerator degree at most m and denominator degree at most n matching a given formal power series to the highest prescribed order.
Core Idea¶
The Padé table turns one power series into a landscape of rational approximations. Moving across the two axes changes how much numerator and denominator complexity is available while preserving the same local coefficient-matching principle.
Its value lies in structure among cells: diagonals, staircases, repeated entries, defects, poles, and continued-fraction paths. A visually stable pole or convergence claim still needs the theorem and function class appropriate to that path.
Scope of Application¶
- Complex approximation. Studies rational continuation beyond polynomial truncation.
- Continued fractions. Relates convergents to paths through the table.
- Numerical analysis. Builds approximants and diagnoses spurious poles.
- Special functions and perturbation theory. Extracts information from formal or asymptotic series cautiously.
Clarity¶
State source series and expansion point, table-axis orientation, numerator and denominator bounds, normalization, coefficient-matching order, defective-cell convention, selected path, exact or floating arithmetic, convergence theorem, and pole-zero diagnostics. Inclusion test: Require a two-index organization of rational Padé approximants derived from the same formal power series under declared degree and normalization conventions. Exclusion test: Exclude a table of arbitrary rational fits, one isolated Padé approximant with no array context, a Taylor coefficient table, and a continued fraction treated as the table itself. Nearest boundary: A Padé approximant is one cell; the Padé table is the degree-indexed family and exposes relations among many cells. Exit condition: Cells can be defective, nonunique in polynomial representation, or repeated as rational functions, so table analysis must retain normality and existence conditions. Common misclassifications: One Padé approximant is not the whole table. Entries are not arbitrary rational fits to sampled data. Every cell need not be normal or distinct. Local series matching does not guarantee uniform global accuracy. Nearest named distinctions: Padé approximant: Is one rational function at a chosen degree pair. Taylor table: Would organize polynomial coefficients rather than rational degree pairs. Rational interpolation: Matches sampled values rather than necessarily a local power series. Continued fraction: Can generate a path of convergents but is not the whole table.
Manages Complexity¶
A doubly indexed family exposes how algebraic degree allocation changes analytic behavior. The compact grid contains linear-system solvability, singularity approximation, repeated rational functions, and path-dependent convergence.
Abstract Reasoning¶
- Fix the formal power series, coefficient field, and expansion point.
- For each selected (m,n), solve the Padé matching equations under one normalization.
- Reduce and record defective or repeated cells without losing their degree position.
- Analyze rows, diagonals, or staircase paths using appropriate convergence results.
- Test apparent poles and zeros for numerical stability and cancellation.
Knowledge Transfer¶
Degree-indexed rational approximation transfers to many series problems, but convergence and pole interpretation depend on the analytic class and selected table path. Sample-based rational fitting does not inherit Padé theory automatically.
Relationships to Other Abstractions¶
Current abstraction Padé Table Domain-specific
Parents (1) — more general patterns this builds on
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Padé Table presupposes Approximation Prime
Padé Table presupposes Approximation because every table entry is a rational approximation matching a power series to a prescribed order.
Hierarchy path (1) — routes to 1 parentless root
- Padé Table → Approximation → Representation → Abstraction
Neighborhood in Abstraction Space¶
Padé Table sits in a crowded region of the domain-specific corpus (32nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Polynomial Rings & Rational Approximants (15 abstractions)
Nearest neighbors
- Humbert Polynomials — 0.91
- Continued Fraction — 0.91
- Peters Polynomials — 0.90
- Log-Sum Inequality — 0.88
- Grey Relational Analysis — 0.87
Computed from structural-signature embeddings · 2026-10-08