Certification Pipeline

This page describes the full two-script pipeline that certifies the first 50 eigenvalues of the GKW transfer operator $L_1 : H^2(D_1) \to H^2(D_1)$ and computes the spectral coefficients $\ell_j(1)$ with rigorous error bounds. For the discretization of $L_1$ and the definitions of $A_K$, $\varepsilon_K$, and $C_2$ see Discretization.

Overview

The pipeline is split into two scripts that communicate through serialized checkpoints in data/:

A third script, scripts/bigfloat_spectral_K1024.jl, uses $K = 1024$ and 2048-bit arithmetic to compute tighter spectral coefficients; it loads Script 1's $M_{\infty,j}$ data and applies a different strategy for $\ell_j(1)$ (see K = 1024 Variant).

All intermediate results are cached as .jls files so that each script can be interrupted and resumed from any completed phase.

Phase 0 — Constants

Both scripts begin by computing two $K$-independent constants rigorously in Arb arithmetic:

• $C_2$: an operator norm bound for $L_1 : H^2(D_1) \to H^2(D_{3/2})$, computed by summing a finite part and bounding the tail via a Hurwitz zeta estimate (splitting parameter $N = 5000$, giving $C_2 \lesssim 10.06$).

• $\varepsilon_K = C_2 \cdot (2/3)^{K+1}$: the truncation error bound $\|L_1 - A_K\|_{H^2(D_1)} \leq \varepsilon_K$ (Corollary 4.1).

The results are converted to rigorous Float64 upper bounds via _arb_to_float64_upper.

Script 1 — Resolvent Certification

The small-gain condition

The link between the finite matrix $A_K$ and the infinite-dimensional operator $L_1$ is the small-gain condition: if

$

\alpha = \varepsilonK \cdot \sup{z \in \Gammaj} \|R{A_K}(z)\| < 1 $

on a simple closed contour $\Gamma_j$ encircling $\lambda_j$ (and no other eigenvalue of $A_K$), then $L_1$ has exactly one eigenvalue inside $\Gamma_j$ (Theorem 2.3). When this holds, the bound

$

M{\infty,j} = \frac{\sup{z \in \Gammaj}\|R{A_K}(z)\|}{1 - \alpha} $

is a rigorous upper bound on $\sup_{z \in \Gamma_j}\|R_{L_1}(z)\|$.

The resolvent norm $\|R_{A_K}(z)\|$ is bounded at each sample point $z$ by certifying a lower bound on $\sigma_{\min}(zI - A_K)$ via the CertifScripts subpackage of BallArithmetic.jl (adaptive arcs + Weyl propagation between samples).

Phase 1a — Standard resolvent scan at $K = 48$ (eigenvalues 1–20)

For the first 20 eigenvalues (largest in magnitude), the full $(K_{\text{low}}+1) \times (K_{\text{low}}+1)$ Float64 BallMatrix $A_{48}$ is assembled and a circle scan is run directly:

circle_j = CertificationCircle(λ_j, r_j; samples=256)
cert     = run_certification(A_low, circle_j)
α        = ε_{K_low} * cert.resolvent_original   # RoundUp
M_∞,j    = cert.resolvent_original / (1 - α)     # RoundUp/RoundDown

The circle radius $r_j$ is chosen as $\min(0.01\,|\lambda_j|,\; d_j/3)$ where $d_j$ is the distance from $\lambda_j$ to the nearest other eigenvalue of $A_{48}$.

Phase 1a succeeds for eigenvalues 1–20 because:

• \[|\lambda_j|\]

  is not too small, so $r_j$ can be moderate;

• \[\varepsilon_{48} \approx 10^{-10}\]

  is small enough to close the small-gain condition even when $\|R_{A_{48}}\|$ is a few hundred.

Phase 1b — Schur-direct certification at $K = 256$ (eigenvalues 21–50)

For eigenvalues 21–50, Phase 1a fails because the eigenvalues are small ($|\lambda_j| \sim 10^{-3}$ or smaller), forcing $r_j$ to be tiny and $\|R_{A_{48}}(z)\| \approx 1/r_j$ to be huge, so $\alpha \geq 1$.

Increasing $K$ to 256 gives $\varepsilon_{256} \approx 10^{-46}$, but running a full circle scan on the 257×257 BallMatrix can still produce a large resolvent bound. Instead, Phase 1b exploits the natural Schur ordering to bound the resolvent via a block structure, avoiding ordschur entirely.

Natural block partition

GenericSchur.jl returns eigenvalues in decreasing order of magnitude. For eigenvalue at Schur position $p$ (i.e., $|\lambda_p| \leq |\lambda_{p-1}|$), the Schur matrix $T$ of $A_{256}$ splits into three natural blocks without any reordering:

$

T = \begin{pmatrix} T{11} & T{12} \ 0 & T{22} \end{pmatrix}, \qquad T{11} \in \mathbb{C}^{(p-1)\times(p-1)},\quad T_{22} \in \mathbb{C}^{(n-p+1)\times(n-p+1)}. $

• \[T_{11}\]

  : eigenvalues larger in magnitude than $\lambda_p$ (positions $1, \ldots, p-1$).

• \[T_{22}\]

  : $\lambda_p$ at position $(1,1)$ followed by all smaller eigenvalues.

On the circle $\Gamma_j$ centered at $\lambda_p$ with radius $r_j$:

• \[(zI - T_{22})\]

  has $\lambda_p$ at distance $r_j$ from $z$, and all other $T_{22}$ eigenvalues at distance $\geq$ gap to the next eigenvalue — so $\|R_{T_{22}}(z)\|$ is finite and moderate.

• \[(zI - T_{11})\]

  has all eigenvalues of $T_{11}$ separated from $\lambda_p$ by construction; a lower bound on $\sigma_{\min}(z_0 I - T_{11})$ at the circle centre $z_0 = \lambda_p$ is obtained via BigFloat SVD + Miyajima certification, and then Weyl propagation gives $\sigma_{\min}(zI - T_{11}) \geq \sigma_{\min}(z_0 I - T_{11}) - r_j$.

Block resolvent formula

$

\|RT(z)\| \leq \|R{T{11}}(z)\| \cdot \bigl(1 + \|T{12}\|F \cdot \|R{T_{22}}(z)\|\bigr)

• \|R{T{22}}(z)\|,

$

where $\|R_{T_{11}}(z)\| \leq 1/(\sigma_{\min}(z_0 I - T_{11}) - r_j)$ is constant on $\Gamma_j$, and $\|R_{T_{22}}(z)\|$ is bounded by a CertifScripts scan (with svdbox fallback) on the smaller matrix $T_{22}$.

Schur bridge

Since $A_{256} = Q T Q^*$ (up to residual $\|E\| \leq E_{\text{Schur}}$, bounded by compute_schur_and_error):

$

\|R{A{256}}(z)\| \leq \|Q\| \cdot \|R_T(z)\| \cdot \|Q^{-1}\|, $

where $\|Q\|$ and $\|Q^{-1}\|$ are the rigorous bounds returned by compute_schur_and_error.

Why no ordschur is needed

The critical insight is that the natural Schur ordering (decreasing $|\lambda|$) already places $\lambda_p$ at the border of the two blocks. No Givens swaps are required. This avoids the need to track ordschur rounding errors in the resolvent bound (ordschur error tracking is only needed for $\ell_j(1)$, in Script 2).

Phase 1c — Fallback

If any eigenvalue in positions 1–20 is not certified by Phase 1a (rare: happens when the Schur ordering of $A_{48}$ places a near-degenerate eigenvalue at position $i$ while the true $|\lambda_i| < |\lambda_{i+1}|$ by more than a threshold), Phase 1c reruns certify_eigenvalue_schur_direct at $K = 256$ for those positions.

Phase 2 — Transfer bridge and Riesz projector errors

Once $M_{\infty,j}$ is established at $K_{\text{low}}$ (or $K_{\text{high}}$ for Phase 1b), the resolvent bound is transferred to any higher truncation $K'$ via the reverse transfer formula:

$

\|R{A{K'}}(z)\| \leq \frac{M{\infty,j}}{1 - M{\infty,j} \cdot \varepsilon{K'}} \quad \text{(valid when } M{\infty,j} \cdot \varepsilon_{K'} < 1\text{)}. $

The Riesz projector approximation error (the distance between the infinite-dimensional Riesz projector $P_{L_1}(\Gamma_j)$ and its finite approximation $P_{A_{K'}}(\Gamma_j)$) is then bounded by:

$

\|P{L1}(\Gammaj) - P{A{K'}}(\Gammaj)\| \leq \frac{|\Gammaj|}{2\pi} \cdot \frac{\|R{A{K'}}(\Gammaj)\|^2 \cdot \varepsilon{K'}} {1 - \|R{A{K'}}(\Gammaj)\| \cdot \varepsilon_{K'}}. $

Phase 3 — Tail resolvent on the separating circle

To bound the contribution of all eigenvalues $\lambda_j$ with $j > 50$, a separating circle of radius $\rho_{\text{tail}} = (|\lambda_{50}| + |\lambda_{51}|)/2$ is introduced around the origin. The resolvent of $L_1$ on this circle is bounded by the same block formula applied to the full 50/remainder split of the BigFloat Schur form of $A_{256}$, giving $M_{\infty,\text{tail}}$.

Script 2 — NK Refinement and Spectral Coefficients

Phase 1 — Newton–Kantorovich certification

For each eigenvalue $j \leq 50$ that passes the small-gain condition, a Newton–Kantorovich argument gives a much tighter enclosure radius $r_{\mathrm{NK},j}$:

$

r{\mathrm{NK},j} = \frac{2y}{(1 - q0) + \sqrt{(1-q0)^2 - 4qk y}} $

where $y$ is a residual bound, $q_k$ is a discrete defect, and $q_0$ accounts for the infinite-dimensional tail ($q_0 = C_2 \cdot \varepsilon_K$). This is computed by certify_eigenpair_nk with BigFloat arithmetic at $K = 256$.

Phase 2 — Spectral coefficient $\ell_j(1)$

The spectral coefficient $\ell_j(1)$ is the pairing of the constant function $1$ with the left eigenvector corresponding to $\lambda_j$, i.e.,

$

\ellj(1) = e1^* \Pij e1, $

where $\Pi_j$ is the Riesz projector and $e_1$ is the first basis vector (the constant function in the shifted monomial basis). The computation proceeds as:

1. ordschur_ball: Givens rotations with BallArithmetic radius tracking move $\lambda_j$ to position $(1,1)$ in the Schur factor $T$, giving reordered factors $Q_{\mathrm{ord}}$ and $T_{\mathrm{ord}}$ as BallMatrix objects (radii propagated rigorously).

2. Sylvester solve: the block projector $\Pi_j$ in Schur coordinates has the form $P_S = \begin{pmatrix} I & Y \\ 0 & 0 \end{pmatrix}$ where $Y$ satisfies the Sylvester equation $T_{22}^* Y - Y T_{11}^* = T_{12}^*$ (with $Y = -X^*$ and $X$ from triangular_sylvester_miyajima_enclosure).

3. Formula:

   q      = conj(Q_ord[1, :])          # = Q_ord^H e₁
   ℓ_j(1) = real(q[1] - dot(X_mid, q[2:end]))

   Note: dot(a,b) = aᴴb in Julia, so dot(X_mid, q_rest) = Xᴴ q_rest which equals $-Y q_{\mathrm{rest}}$, giving the correct formula.

4. Error budget:

   • Sylvester error: componentwise propagation from BallMatrix radii of $X$.
   • Projector perturbation: from the resolvent bounds via spectral_projector_error_bound.
   • NK eigenvector correction: $\|\Pi_S\|_2 \cdot 2 r_{\mathrm{NK},j}$ where $\|\Pi_S\|_2 = \sqrt{1 + \|Y\|_2^2}$.

Phase 3 — Tail bound

The tail of the spectral expansion is bounded by:

$

\|R{50}^{(n)}\|{H^2(D1)} \leq \rho{\mathrm{tail}}^{n+1} \cdot M{\infty,\mathrm{tail}} \cdot \|Q{50}(L1) \cdot 1\|{H^2}, $

where $Q_{50}(L_1) = I - \sum_{j=1}^{50} \Pi_j$ is the spectral complement projector. The norm $\|Q_{50} \cdot 1\|$ is bounded by computing the Galerkin approximation

$

Q{50}(AK) e1 = e1 - \sum{j=1}^{50} \ellj(1) \cdot v_j $

(using the certified $\ell_j(1)$ and eigenvectors $v_j$) and adding the projector approximation errors $\vartheta_j$ from Phase 2.

K = 1024 Variant

The script scripts/bigfloat_spectral_K1024.jl refines the spectral data using $K = 1024$ and 2048-bit precision ($\varepsilon_{1024} \approx 10^{-181}$). Two key differences from Script 2:

Direct triangular Sylvester solve (no Miyajima)

The Miyajima enclosure for Sylvester equations fails for $K \geq 513$ because the condition number of the eigenvector matrix of $T_{22}^*$ grows with the eigenvalue spread (see BugReport_MiyajimaSylvesterLargeMatrix.md). Instead, $\ell_j(1)$ is computed by a direct triangular solve:

$

z = (T{22} - \lambdaj I)^{-1} q{\mathrm{rest}}, \qquad \ellj(1) = \operatorname{Re}!\bigl(q1 - \textstyle\sumk [T{12}]k \, z_k\bigr). $

Here sum(T12 .* z) (not dot) is used to avoid Julia's implicit conjugation of the first argument in dot.

The solve error (residual $\|(T_{22} - \lambda_j I) z - q_{\mathrm{rest}}\|$) is $\sim 10^{-309}$ at 2048-bit precision — negligible. The error budget is dominated by the Schur perturbation bound:

$

|\hat\ellj(1) - \ellj(1)| \leq \varrho_j $

where $\varrho_j$ is computed from the ordschur rounding (tracked by ordschur_ball radii) and the Schur decomposition residual $\|A - QTQ^*\|_2$, using a Neumann-series / resolvent resolvent hybrid bound on the spectral projector (taking the minimum of the two estimates).

No NK certification

At $K = 1024$, NK certification (certify_eigenpair_nk) cannot be run because it internally uses the Miyajima Sylvester solve (same failure mode). Instead, eigenvalue enclosures are obtained by transferring $M_{\infty,j}$ (from Script 1) to $K = 1024$ and applying the projector error formula:

$

|\lambdaj^* - \hat\lambdaj| \leq \frac{\varepsilon{1024}(1 + \varthetaj) + 2\|A{1024}\|2 \cdot \varthetaj}{1 - \varthetaj}, \qquad \varthetaj = \frac{|\Gammaj|}{2\pi} \cdot \frac{\|R{A{1024}}\|^2 \varepsilon{1024}} {1 - \|R{A{1024}}\|\varepsilon{1024}}. $

Summary of Rigorous Bounds Produced