Special Functions

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1 \hypertarget{v11-coupled_lab_full-exact-laboratory-and-certified-evaluation}{%
2 \chapter[Exact laboratory and certified evaluation]{Exact laboratory and certified
3 evaluation}\label{app:laboratory}\label{v11-coupled_lab_full-exact-laboratory-and-certified-evaluation}}
4 
5 The reusable module \path{coupled_lab.py} uses Python's standard
6 library and imports the existing exact coefficient, phase and tap
7 operations from \path{family_lab.py}. The figure builder additionally
8 uses NumPy and Matplotlib. Neither script contacts an external service.
9 The generic integer and signed catalog rows are formal power series,
10 with no claim of analytic convergence.
11 
12 \hypertarget{v11-coupled_lab_full-normalization-and-recurrence}{%
13 \section{Normalization and
14 recurrence}\label{v11-coupled_lab_full-normalization-and-recurrence}}
15 
16 Write
17 
18 \[
19 U(z)=\sum_{n\ge0}A_n\frac{z^{2n+1}}{(2n+1)!},\qquad
20 V(z)=\sum_{n\ge0}B_n\frac{z^{2n+1}}{(2n+1)!}.
21 \]
22 
23 The exact recurrence is
24 
25 \[
26 A_n=\sum_{i+j=n-1}\binom{2n}{2i+1}q^{2i+1}r^{2j+1}A_iB_j,
27 \qquad
28 B_n=\sum_{i+j=n-1}\binom{2n}{2i+1}a^{2i+1}b^{2j+1}A_iB_j,
29 \]
30 
31 with \(A_0=B_0=1\). Coefficients with even derivative index vanish.
32 Swapping the components sends the parameter tuple \((q,r,a,b)\) to
33 \((b,a,r,q)\). Identical rows \((a,b)=(q,r)\), or exchanged rows
34 \((a,b)=(r,q)\), yield the scalar reduction \(U=V=T_{q,r}\).
35 
36 Both derivative arrays are polynomials in four scale variables with
37 nonnegative integer coefficients. Their total degree is at most
38 \(n(n+1)\). Indeed, a recurrence term with \(i+j=n-1\) has degree at
39 most \(2n+i(i+1)+j(j+1)\le n(n+1)\). If all four denominators divide
40 \(D\), then \(D^{n(n+1)}A_n\) and \(D^{n(n+1)}B_n\) are integers. The
41 clearing factor is a convenient uniform choice and need not be minimal.
42 
43 \hypertarget{v11-coupled_lab_full-an-elementary-exact-evaluation-certificate}{%
44 \section{An elementary exact evaluation
45 certificate}\label{v11-coupled_lab_full-an-elementary-exact-evaluation-certificate}}
46 
47 This argument uses \(0<q,r,a,b<1\) and does not require the order
48 theorem. Let \(Q=\max(q,r,a,b)\). All ordinary odd Taylor coefficients
49 are positive and at most one: induction in the ordinary recurrence
50 bounds the coefficient of \(z^{2n+1}\) by \(n/(2n+1)\) for \(n\ge1\).
51 Thus both series converge on the unit disk and, for \(0\le x<1\),
52 
53 \[
54 \max(U(x),V(x))\le \frac{x}{1-x^2}.
55 \]
56 
57 Analytic continuation by the integral equations expands each known disk
58 by the factor \(1/Q\), so both solutions are entire. Positive
59 coefficients give \(\max_{|z|\le x}|F(z)|=F(x)\) for either component.
60 The integral equations therefore imply
61 
62 \[
63 M(x):=\max(U(x),V(x))\le x\bigl(1+M(Qx)^2\bigr).
64 \]
65 
66 Start at the first member \(Q^kx<1\) of this finite contraction ladder,
67 use \(x/(1-x^2)\) there, and repeatedly apply \(B(x)=x(1+B(Qx)^2)\)
68 backward. This defines the rational \path{disk_majorant} bound for
69 rational parameters and radius. The implementation has explicit
70 recursion and integer size guards, which raise errors rather than
71 returning a weakened result.
72 
73 If \(F_N\) retains odd indices \(0,\ldots,N\), then for \(S>R\ge0\),
74 
75 \[
76 \max_{|z|\le R}|F(z)-F_N(z)|
77 \le B(S)\left(\frac RS\right)^{2N+3}.
78 \]
79 
80 For every omitted exponent \(m\ge2N+3\), use \(R^m\le (R/S)^{2N+3}S^m\),
81 sum positive terms, and bound their sum by \(F(S)\le B(S)\). No extra
82 geometric denominator is needed. For rational \(x\ge0\),
83 \path{certified_evaluation} returns the exact interval
84 \([F_N(x),F_N(x)+B(S)(x/S)^{2N+3}]\) for each component.
85 
86 \Needspace{23\baselineskip}
87 \hypertarget{v11-coupled_lab_full-worked-calls}{%
88 \section{Worked calls}\label{v11-coupled_lab_full-worked-calls}}
89 
90 \begin{verbatim}
91 from fractions import Fraction as Q
92 import coupled_lab as lab
93 
94 p = (Q(1,5), Q(2,5), Q(3,5), Q(7,10))
95 A, B = lab.coupled_odd_derivatives(*p, last=12)
96 certificate = lab.certified_evaluation(*p, Q(1), last=30, outer_radius=2)
97 root = lab.pf_order(*p)  # floating approximation, not interval arithmetic
98 
99 derivatives = lab.full_egf_coefficients(B)
100 phases = lab.phase_lanes(derivatives, 3)
101 matrix = lab.polyphase_tap_matrix([1, -2, 1], 3)
102 length = len(derivatives) + 2
103 output_phases = lab.apply_polyphase_matrix(phases, matrix, length)
104 output = lab.merge_lanes(output_phases, length)
105 \end{verbatim}
106 
107 At \(x=1\), with \(N=30\) and \(S=2\), the common tail bound is below
108 \(1.600\times10^{-13}\). The partial values are approximately
109 \(U(1)=1.027044341827337\) and \(V(1)=1.1467074957737624\). These
110 displayed decimals are rounded illustrations. The returned fractions are
111 the actual certificate.
112 
113 Run the independent checks and regenerate figures from the package\textquotesingle s \texttt{research/} directory:
114 
115 \begin{verbatim}
116 python3 checks/coupled_v11.py
117 python3 build_coupled_figures.py
118 \end{verbatim}
119 
120 The first command passes 1,559 exact assertions, 10 numerical
121 comparisons, and 13 separate input/resource guards. Independent
122 constructions include dense ordinary Taylor Picard iteration,
123 sine/cosine division for tangent, the exact sinh formula, component
124 symmetry, and direct convolution versus phase carry. The test counts
125 describe bounded implementation checks, not universal proofs.

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