AAA-Lawson Rational Approximation

• AAA-Lawson is a two-phase algorithm that first applies adaptive AAA barycentric fitting to generate a near-minimax rational approximant.
• The Lawson phase uses iteratively reweighted least squares to refine the solution toward Chebyshev optimality by minimizing the ℓ∞ norm.
• The method is robust for real and complex data across diverse domains, making it central to modern rational minimax approximation toolkits.

AAA-Lawson is a two-phase numerical algorithm for uniform-norm rational approximation of functions on real intervals or complex domains. It combines the adaptive Antoulas–Anderson (AAA) barycentric rational fitting, which quickly generates a near-minimax approximant, with a nonconvex Lawson-style iteratively reweighted least squares (IRLS) refinement phase that drives the approximation toward true Chebyshev optimality (minimizing the ℓ_∞ norm of the error) (Nakatsukasa et al., 2019, Zhang et al., 2023, Driscoll et al., 2023, Antoulas et al., 6 Nov 2025). The method is applicable to both real and complex data, arbitrary degrees, and diverse geometric domains, and is central to modern rational minimax approximation toolkits such as Chebfun.

1. Problem Setting and Motivation

Given sample points $\{z_j\}_{j=1}^M \subset \mathbb{C}$, data values $f_j = f(z_j)$, and a target rational type (typically $(n,n)$ or $(m,n)$), the objective is

$$\min_{r \in \mathcal R_{m,n}} \max_{1 \leq j \leq M} |f_j - r(z_j)|$$

where $\mathcal R_{m,n}$ denotes the set of rational functions $r(z)=p(z)/q(z)$ of numerator degree $\leq m$ and denominator degree $\leq n$ with $q(z_j)\neq 0$.

Traditional Remez-type algorithms are not robust for complex data or arbitrary domains. AAA-Lawson overcomes these limitations by leveraging the numerically stable barycentric AAA representation and enhancing it with IRLS to approximate the minimax solution even for highly ill-conditioned or nonclassical geometries (Nakatsukasa et al., 2019).

2. Algorithmic Framework: AAA Phase and Lawson Phase

AAA Phase

The adaptive Antoulas–Anderson (AAA) algorithm constructs a rational function in barycentric form

$f_j = f(z_j)$0

by greedily selecting “support points” $f_j = f(z_j)$1 at maximum residuals and solving a sequence of least-squares problems using the SVD of a Loewner/Cauchy-type matrix. This phase yields a rational approximant that is generally within 1–2 digits of the true minimax error, but typically not equioscillatory, and so not certifiably optimal in the Chebyshev sense (Nakatsukasa et al., 2019, Zhang et al., 2023).

Lawson Phase (IRLS)

With the support set $f_j = f(z_j)$2 and initial weights $f_j = f(z_j)$3 fixed, the AAA-Lawson phase addresses

$f_j = f(z_j)$4

via iteratively reweighted least squares: at each step, solve a weighted least-squares problem minimizing the sum of squared residuals penalized by inverse magnitudes of the current residuals, iterate weights as $f_j = f(z_j)$5, and continue until the error stabilizes or a prescribed maximum iteration count is reached (Driscoll et al., 2023, Antoulas et al., 6 Nov 2025).

Pseudocode excerpt (Driscoll et al., 2023, Zhang et al., 2023): $$\min_{r \in \mathcal R_{m,n}} \max_{1 \leq j \leq M} |f_j - r(z_j)|$$4

3. Mathematical Structure and Variants

The AAA-Lawson approach can be analyzed from multiple perspectives:

• Weighted Least Squares: Each IRLS step solves

$f_j = f(z_j)$6

where $f_j = f(z_j)$7 is the barycentric Loewner/Cauchy matrix and $f_j = f(z_j)$8 penalizes large errors (Nakatsukasa et al., 2019, Driscoll et al., 2023).

• Convex Duality: The IRLS process is interpreted as a block-coordinate ascent on the dual of a linearized minimax problem. Specifically, for complex rational minimax, the dual variable $f_j = f(z_j)$9 lies in a simplex, and maximizing the dual functional under Ruttan’s condition certifies global optimality (Zhang et al., 2023).
• Lawson Parameter ($(n,n)$0): Modifies weight update as $(n,n)$1. Choosing $(n,n)$2 yields classical Lawson; smaller $(n,n)$3 adds damping and may improve stability (Zhang et al., 2023).
• Continuum AAA-Lawson: In domains requiring dense adaptive refinement (e.g., near singularities on $(n,n)$4 or $(n,n)$5), AAA-Lawson operates on adaptively discretized grids and employs high-resolution sampling at each step to stabilize the error curve (Driscoll et al., 2023).

4. Numerical Performance and Comparative Analysis

AAA-Lawson’s performance has been thoroughly benchmarked against the Loewner framework, RKFIT, and traditional AAA:

Method	Achievable $(n,n)$6 error	Computational Cost	Artifacts
AAA	Fast, not near-minimax	Mild, nearly linear	Spurious poles
AAA-Lawson	Near-minimax	$(n,n)$7, costly at high $(n,n)$8	Spurious complex structure possible
Loewner Framework	Near-minimax or better	Cost flat in $(n,n)$9	Structure-preserving

• AAA-Lawson achieves uniform errors down to $(m,n)$0–$(m,n)$1 in double precision for moderate degrees and grids of size $(m,n)$2, outperforming AAA alone (Nakatsukasa et al., 2019, Antoulas et al., 6 Nov 2025).
• For high-degree rational approximants ($(m,n)$3), the Loewner framework can be both faster and more accurate, especially with respect to computational cost, and avoids introducing non-physical complex coefficients in real-data problems.
• For applications demanding equioscillation or strict Chebyshev optimality, AAA-Lawson provides a practical compromise between direct optimization and numerical tractability.

5. Convergence, Limitations, and Certificates of Optimality

• Convergence: Empirically, AAA-Lawson shows rapid, often linear convergence of the maximum residual, typically achieving one additional digit of accuracy every 5–10 steps until stagnation at machine precision (Nakatsukasa et al., 2019, Driscoll et al., 2023).
• Certification: While AAA-Lawson reliably pushes the error toward equioscillation, only “dual-Lawson” (convex dual) variants as in (Zhang et al., 2023) can rigorously certify attainment of the true minimax error, as monitored by the duality gap.
• Non-monotonicity: If support set selection or weight damping ($(m,n)$4) is suboptimal, the error or dual objective may momentarily worsen before improving.
• Artifacts: For high-degree or large-scale problems, spurious complex-conjugate pole/zero pairs may appear in problems with real data due to lack of symmetry constraints in the Lawson phase.
• Computational cost: The Lawson phase is $(m,n)$5 for $(m,n)$6 steps, where $(m,n)$7 is the degree and $(m,n)$8 the grid size; for large $(m,n)$9 this becomes prohibitive.

6. Implementation and Applications

AAA-Lawson is implemented in open-source software suites such as Chebfun. Maximal practical benefit is achieved when:

• The initial AAA approximant is already within a factor $$\min_{r \in \mathcal R_{m,n}} \max_{1 \leq j \leq M} |f_j - r(z_j)|$$0–$$\min_{r \in \mathcal R_{m,n}} \max_{1 \leq j \leq M} |f_j - r(z_j)|$$1 of minimax, ensuring a well-conditioned support set.
• The number of Lawson steps is chosen modest (e.g., $$\min_{r \in \mathcal R_{m,n}} \max_{1 \leq j \leq M} |f_j - r(z_j)|$$2–$$\min_{r \in \mathcal R_{m,n}} \max_{1 \leq j \leq M} |f_j - r(z_j)|$$3).
• Applications include spectrally accurate function approximation, model reduction, physical system identification, and numerical solution of inverse Zolotarev problems (Antoulas et al., 6 Nov 2025, Driscoll et al., 2023).

In summary, AAA-Lawson is a robust, widely utilized approach for rational minimax approximation, combining the adaptivity of AAA barycentric sampling with the uniform-norm refinement of Lawson IRLS. It is most suitable for medium-scale or moderate-degree problems where near-optimal uniform error and equioscillation are critical, while alternative methods such as Loewner or convex-dual approaches may be preferable in large-scale or certifiably minimax settings.