Chebyshev Inertial Iteration

Updated 3 April 2026

Chebyshev Inertial Iteration is a method that leverages Chebyshev polynomials to compute optimal, iteration-dependent inertial factors for fixed-point schemes.
It extends classical acceleration techniques like SOR and Krasnosel’skiĭ–Mann by minimizing worst-case error amplification over a prescribed spectral interval.
The approach is widely applied in inverse problems, distributed consensus, and inertial navigation, yielding significant reductions in iterations and error rates.

Chebyshev inertial iteration is a methodology that employs Chebyshev polynomials to optimize and accelerate the convergence rate of fixed-point iterations, including both linear and nonlinear cases. By linking the selection of inertial (relaxation) factors across iterations to the roots of affine-transformed Chebyshev polynomials, this approach achieves minimax suppression of error amplification over a prescribed spectral interval. Chebyshev inertial iteration encompasses and extends classical acceleration schemes such as successive over-relaxation (SOR) and Krasnosel’skiĭ–Mann (KM) types. Its theoretical underpinning is the mapping between the worst-case error propagation and optimal minimax polynomial approximation, yielding provably optimal rates under broad spectral and smoothness conditions. Its algorithmic principles underlie a family of accelerated solvers across inverse problems, distributed consensus, and precision inertial navigation.

1. Mathematical Formulation and Optimal Inertial Factors

Consider a generic fixed-point iteration $x^{(k+1)} = T(x^{(k)})$ converging locally to $x^* = T(x^*)$ . The Chebyshev inertial iteration modifies this process via an iteration-dependent inertial factor:

$x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$

In the linear setting with $T(x) = A x + b$ , this reduces to SOR when $\omega_k \equiv \omega$ is constant.

To derive an inertial sequence $\{\omega_k\}$ that optimally suppresses the spectral amplification of error, the process is linearized around $x^*$ :

$e^{(k+1)} \approx (I - \omega_k B) e^{(k)}, \quad B := I - \frac{\partial T}{\partial x}|_{x^*}.$

The product $\prod_{k=0}^{N-1}(I - \omega_k B)$ over a cycle of length $N$ yields an error polynomial in $x^* = T(x^*)$ 0 whose worst-case norm over the spectral interval $x^* = T(x^*)$ 1 is minimized by the degree- $x^* = T(x^*)$ 2 affine-transformed Chebyshev polynomial:

$x^* = T(x^*)$ 3

where $x^* = T(x^*)$ 4 is the Chebyshev polynomial of the first kind.

The iteration-dependent inertial factors are set to the inverses of the $x^* = T(x^*)$ 5 real roots of $x^* = T(x^*)$ 6,

$x^* = T(x^*)$ 7

and are applied in a periodic cycle.

2. Relation to Classical Schemes and Chebyshev Polynomial Acceleration

Chebyshev inertial iteration generalizes several existing acceleration paradigms:

SOR (Successive Over-Relaxation): Uses a constant $x^* = T(x^*)$ 8, optimizing the spectral radius over a single value. Chebyshev iteration’s non-constant, periodic $x^* = T(x^*)$ 9 minimizes the spectral radius globally over the entire interval, always outperforming SOR for the same spectral data (Wadayama et al., 2020).
Krasnosel’skiĭ–Mann (KM) Iteration: Typically restricts $x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$ 0 with convergence via summability conditions. Chebyshev inertial iteration permits factors exceeding unity (over-relaxation) dictated strictly by polynomial optimality, not diminishing-step rules.
Distributed Consensus: In distributed linear averaging, the Chebyshev-inertial approach translates to a second-order three-term recurrence derived from the Chebyshev polynomial recurrence, accelerating convergence in both fixed and switching topologies by shrinking the worst-case contraction factor beyond spectral radius limitations (Montijano et al., 2011).

3. Local and Global Convergence Guarantees

For differentiable $x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$ 1 with real eigenvalues of $x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$ 2 strictly inside $x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$ 3 so $x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$ 4, the local convergence rate is governed by the worst-case Chebyshev contraction parameter

$x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$ 5

leading to

$x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$ 6

This rate strictly improves on constant-relaxation methods and can be tuned via cycle length $x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$ 7. If all eigenvalues of the Jacobian are real, Chebyshev inertial iteration provably accelerates the convergence of virtually any locally contractive fixed-point scheme. In finite Hilbert-space settings (e.g., the Landweber algorithm for linear inverse problems), global convergence rates are similarly dominated by the $x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$ 8 factor reflecting the Chebyshev-minimax contraction (Wadayama et al., 2020).

4. Algorithmic Realizations and Pseudocode

A canonical periodic Chebyshev inertial iteration is:

$T(x) = A x + b$ 9 (Wadayama et al., 2020)

For Chebyshev-inertial Landweber, the update at iteration $x^{(k+1)} = x^{(k)} + \omega_k (T(x^{(k)}) - x^{(k)}) = (1-\omega_k)x^{(k)} + \omega_k T(x^{(k)}).$ 9 is

$T(x) = A x + b$ 0

with $T(x) = A x + b$ 1 determined from the spectral interval of $T(x) = A x + b$ 2 (Wadayama et al., 2020).

5.1 Proximal Gradient and Inverse Problems

Chebyshev inertial iteration delivers marked acceleration for proximal gradient algorithms (e.g., ISTA), reducing iteration count by an order of magnitude or more compared to standard and even Nesterov-accelerated variants in compressed sensing and LASSO settings. In signal recovery with the Landweber algorithm, such as Hilbert-space deconvolution or MIMO detection, the Chebyshev schedule yields 3–5× or even one to two orders of magnitude reduction in iterations and symbol error rate, respectively (Wadayama et al., 2020, Wadayama et al., 2020).

5.2 Distributed Consensus Acceleration

The Chebyshev-inertial consensus algorithm, implemented as a second-order recurrence derived directly from the Chebyshev polynomial three-term relation, enables large reductions in consensus iteration count. Empirically, in networks with 100 nodes and geometric random graphs, iteration counts drop from 700–900 (standard) to 60–100 (Chebyshev-inertial) for typical tolerances, representing order-of-magnitude speedup (Montijano et al., 2011).

In the context of inertial navigation, Chebyshev inertial iteration underpins Picard–Chebyshev-type methods for continuous-time strapdown mechanization (attitude, velocity, position) (Wu et al., 2018, Wu, 2019, Jiang et al., 2022). These methods fit sensor data to Chebyshev polynomials, propagate function iterates as truncated Chebyshev series, and employ closed-form integration within each Picard update. Matrix formulations (iNavFIter-M) further enhance computational tractability, enabling real-time, ultra-precise navigation on standard hardware or FPGAs. Non-commutativity errors (coning, sculling, scrolling) are reduced from meters (traditional methods) to micrometers or below without any classical correction terms.

6. Guidelines for Parameter Selection, Spectral Estimation, and Limitations

Optimal performance of Chebyshev inertial iteration relies on accurate estimation of the spectral interval $T(x) = A x + b$ 3 for $T(x) = A x + b$ 4. In practical settings, these are estimated via power/inverse power methods, Marchenko–Pastur laws, or design-time analysis. Cycle lengths $T(x) = A x + b$ 5 of 5–20 usually balance per-iteration complexity and rapid acceleration. The key sufficient condition is the reality of the Jacobian's spectrum—the method is specifically optimal for maps $T(x) = A x + b$ 6 with symmetric $T(x) = A x + b$ 7 and non-negative $T(x) = A x + b$ 8. For inertial navigation, Chebyshev truncation orders are chosen so the last retained coefficient is at or below the required accuracy, and the smoothness of measured data ensures rapid coefficient decay (Wadayama et al., 2020, Jiang et al., 2022).

7. Computational Complexity and Implementation Considerations

The use of Chebyshev inertial iteration, particularly with matrix formulations, enables dramatic reductions in both per-iteration cost and total computational effort. For example, iNavFIter-M achieves the near-machine-precision navigation of full functional iteration at a runtime comparable to the classic two-sample coning-sculling algorithm and demonstrates real-time feasibility at high IMU sampling rates on both CPUs and FPGAs. In consensus and inverse problems, the precomputation of inertial factors ensures that the increase in storage or arithmetic is negligible relative to realized speedup (Jiang et al., 2022, Wadayama et al., 2020, Montijano et al., 2011).

References:

(Wadayama et al., 2020, Wadayama et al., 2020, Montijano et al., 2011, Wu et al., 2018, Wu, 2019, Jiang et al., 2022)