Second-Order Steps, Spectral Accuracy: Discrete Parallel Transport of Invariant Subspaces
Abstract
Eigensolvers return orthonormal bases of invariant subspaces, but each basis is determined only up to a unitary factor, so bases computed at different parameter values cannot, in general, be interpolated directly. Discrete parallel transport resolves this ambiguity by aligning consecutive bases through the polar factors of their overlaps, but approximates the continuous parallel-transported frame only to second order. We show that the resulting frames can nevertheless be interpolated with spectral accuracy. For analytic families of invariant subspaces sampled at N+1 Chebyshev nodes, the computed frames are exact samples of an analytic frame related to the parallel-transported frame by a gauge that remains within O(N−2) of the identity throughout a fixed complex neighborhood, uniformly in N. Consequently, Chebyshev interpolation approximates this analytic frame with geometric convergence. Moreover, the continuous parallel-transported frame can be recovered to spectral accuracy by solving a small matrix differential equation, without additional eigensolves or derivatives of the underlying matrix family. The results do not require the family to be Hermitian and remain valid through eigenvalue crossings and branch points, provided the selected invariant subspace depends analytically on the parameter. Numerical examples confirm the predicted rates.
Related
- Noncommutative commutant lifting, interpolation, and shift-invariant subspaces
- Numerical range and invariant subspaces of weighted composition operators on the Hardy space of Dirichlet series
- Projections and minimal invariant subspaces in the Hardy space over the bidisk
- Invariant subspaces for free linearizations of Lipschitz maps
- Fermat's Last Theorem in star-invariant subspaces II: small boundary spectrum, projective plane curves, and the cubic Fermat equation
BibTeX
@misc{zhang2026secondorder,
title = {{Second-Order Steps, Spectral Accuracy: Discrete Parallel Transport of Invariant Subspaces}},
author = {Hanwen Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34494},
eprint = {2610.34494},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34494},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Second-Order Steps, Spectral Accuracy: Discrete Parallel Transport of Invariant Subspaces
AU - Hanwen Zhang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34494
UR - https://arxiv.org/abs/2610.34494
AB - Eigensolvers return orthonormal bases of invariant subspaces, but each basis is determined only up to a unitary factor, so bases computed at different parameter values cannot, in general, be interpolated directly. Discrete parallel transport resolves this ambiguity by aligning consecutive bases through the polar factors of their overlaps, but approximates the continuous parallel-transported frame only to second order. We show that the resulting frames can nevertheless be interpolated with spectral accuracy. For analytic families of invariant subspaces sampled at N+1 Chebyshev nodes, the computed frames are exact samples of an analytic frame related to the parallel-transported frame by a gauge that remains within O(N^−2) of the identity throughout a fixed complex neighborhood, uniformly in N. Consequently, Chebyshev interpolation approximates this analytic frame with geometric convergence. Moreover, the continuous parallel-transported frame can be recovered to spectral accuracy by solving a small matrix differential equation, without additional eigensolves or derivatives of the underlying matrix family. The results do not require the family to be Hermitian and remain valid through eigenvalue crossings and branch points, provided the selected invariant subspace depends analytically on the parameter. Numerical examples confirm the predicted rates.
ER -