Barycentric Rational Representation

• Barycentric Rational Representation is an explicit, numerically stable formulation for rational interpolants, defined by interpolation nodes, data values, and barycentric weights.
• It enables efficient algorithms for minimax and least-squares rational approximation while preserving system identification structures and high-order spectral properties.
• Adaptive techniques such as the AAA algorithm optimize node and weight selection, ensuring robust performance in high-dimensional interpolation and spectral derivative computations.

A barycentric rational representation is an explicit, numerically stable form for rational interpolants, widely used in approximation theory, model reduction, and computational mathematics. It characterizes a rational function by its interpolation nodes, data values, and a set of nonzero barycentric weights, with the rational function evaluated as the quotient of two weighted sums of reciprocal differences. This approach achieves superior numerical conditioning, flexibility in node choice, and enables powerful algorithms for minimax and least-squares rational approximation, structure-preserving system identification, and spectral methods.

1. Canonical Barycentric Rational Formulation

Given $m+1$ distinct nodes $(z_0, f_0),\ldots,(z_m, f_m)$ and nonzero weights $w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$, the barycentric rational interpolant of type $[m,m]$ is

$$r(z) = \frac{\displaystyle\sum_{j=0}^m \frac{w_j f_j}{z-z_j}}{\displaystyle\sum_{j=0}^m \frac{w_j}{z-z_j}},$$

which satisfies $r(z_j) = f_j$ for $j=0,\ldots,m$ and generically possesses at most $m$ finite poles (locations where the denominator vanishes) (Elsworth et al., 2017, Huang et al., 2024, Filip et al., 2017, Kong et al., 2021, Pradovera et al., 2024, Driscoll et al., 15 Jan 2026). This form is algebraically equivalent to a polynomial quotient $p(z)/q(z)$, but the use of barycentric weights and node localization avoids the severe cancellation and ill-conditioning associated with direct evaluation in the monomial basis.

Polynomial and Rational Exactness

If $f_j = p(z_j)$ at the nodes for $(z_0, f_0),\ldots,(z_m, f_m)$0 a degree-$(z_0, f_0),\ldots,(z_m, f_m)$1 polynomial, one retrieves $(z_0, f_0),\ldots,(z_m, f_m)$2 provided the weights $(z_0, f_0),\ldots,(z_m, f_m)$3 satisfy the linear relations

$(z_0, f_0),\ldots,(z_m, f_m)$4

(Driscoll et al., 15 Jan 2026). For rational interpolants of type $(z_0, f_0),\ldots,(z_m, f_m)$5, the same structure applies with two sets of nodes and weights, maintaining interpolation at up to $(z_0, f_0),\ldots,(z_m, f_m)$6 points (Brezinski et al., 2013).

2. Derivation, Conditioning, and Stability

The barycentric form derives as a quotient of two weighted Lagrange-type sums, each with poles at the interpolation nodes. The rationale for its widespread adoption is twofold:

• Numerical Conditioning: Each term $(z_0, f_0),\ldots,(z_m, f_m)$7 is well-behaved away from $(z_0, f_0),\ldots,(z_m, f_m)$8; near $(z_0, f_0),\ldots,(z_m, f_m)$9 the dominant numerator and denominator terms cancel, promoting numerical stability.
• Avoidance of Ill-Conditioning: The form does not require explicit polynomial expansion or monomial division at high degrees, avoiding large condition numbers and potential loss of significance for large node sets (Elsworth et al., 2017, Filip et al., 2017).
• Choice of Weights: In many contexts, barycentric weights can be chosen as

$w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$0

for classical polynomial interpolation, but are often computed adaptively (e.g., AAA algorithm, Remez minimax, vector fitting, Loewner approaches) to balance approximation error and conditioning (Huang et al., 2024, Filip et al., 2017).

3. Adaptive Weight/Node Selection and Algorithmic Variants

AAA Algorithm

The Adaptive Antoulas-Anderson (AAA) algorithm is a prominent method for constructing barycentric rational interpolants. It greedily selects nodes $w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$1 and computes weights $w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$2 by solving least-squares problems on the Loewner matrix:

$w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$3

with $w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$4 as the right singular vector for the smallest singular value. The barycentric form at each step is

$w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$5

The process continues until the maximal residual falls below a specified tolerance (default $w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$6) (Huang et al., 2024).

Rational Remez, AAA-Lawson, and Differential Correction

Variants include Remez-type minimax solvers (iterative alternation using barycentric representations for minimax optimality), AAA-Lawson’s iterated reweighted least squares (refining fits in weighted $w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$7 norms), and differential correction algorithms (linear program for piecewise optimal rational correction). Each maintains the barycentric structure but differs in the error metric and convergence properties (Filip et al., 2017).

Applications Table

Algorithm	Node Selection	Error Metric
AAA	Greedy (residual)	least-squares
Remez	Alternants	$w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$8 minimax
AAA-Lawson	Fixed, IRLS	weighted $w_0,\ldots,w_m \in \mathbb{C} \setminus \{0\}$9
Diff. Correction	Fixed	LP-minimax

4. Structural Variations and Generalizations

Relative Degree Constrained Barycentric Form

For dynamical models with known or targeted relative degree (difference in numerator/denominator degree), the barycentric weights can be constrained to enforce the desired asymptotic scaling via linear constraints:

$[m,m]$0

with $[m,m]$1 desired relative degree. This enables high-frequency extrapolation that is consistent with the system’s physical structure and robust model selection for extrapolation (Pradovera et al., 2024).

Structured Barycentric Forms for Second-Order Systems

Modeling second-order systems from frequency measurement data, the barycentric structure can be embedded into the transfer function’s form:

$[m,m]$2

with $[m,m]$3 as additional parameter support points, aligning the approximation with the mechanical system’s structure and permitting efficient identification via Loewner-type algorithms (Gosea et al., 2023).

5. Connections, Transformations, and Computational Aspects

Conversion to RKFUN and Newton Forms

The barycentric form can be systematically converted into the rational Krylov (RKFUN) representation or Newton basis form:

• The rational basis functions $[m,m]$4 satisfy a recursion, yielding a pencil $[m,m]$5 for efficient root and pole finding via small QZ problems.
• In the Newton form, the barycentric basis translates to a recurrence with parameters read off from barycentric data, facilitating structured linearizations for nonlinear eigenvalue problems (Elsworth et al., 2017).

Efficient Evaluation and Stable Derivative Computation

Recent developments provide $[m,m]$6 cost algorithms for evaluation of rational functions and all derivatives in barycentric form, avoiding cancellation instabilities near nodes:

• The stable single-sum algorithm (as opposed to previous $[m,m]$7 double-sum approaches) ensures forward stability for high-precision differentiation (Driscoll et al., 15 Jan 2026).
• For higher derivatives, recursive formulae in the barycentric basis are implemented efficiently, making barycentric rational representations suitable for high-order spectral methods and model reduction.

6. Applications and Performance Benchmarks

Analytic Continuation in Quantum Many-Body Physics

Barycentric rational approximants constructed via AAA enable high-accuracy analytic continuation of Matsubara Green’s functions. The resulting “BarRat+AAA” approach competes favorably with the maximum entropy method (MaxEnt), especially for noisy, discrete, or matrix-valued spectral data. CPU evaluation times are over $[m,m]$8–$[m,m]$9 faster than MaxEnt, and the method is robust to moderate noise and under-resolved data (Huang et al., 2024).

Minimax and Padé-Type Approximation

In challenging minimax rational approximation (e.g., type $$r(z) = \frac{\displaystyle\sum_{j=0}^m \frac{w_j f_j}{z-z_j}}{\displaystyle\sum_{j=0}^m \frac{w_j}{z-z_j}},$$0 for $$r(z) = \frac{\displaystyle\sum_{j=0}^m \frac{w_j f_j}{z-z_j}}{\displaystyle\sum_{j=0}^m \frac{w_j}{z-z_j}},$$1 on $$r(z) = \frac{\displaystyle\sum_{j=0}^m \frac{w_j f_j}{z-z_j}}{\displaystyle\sum_{j=0}^m \frac{w_j}{z-z_j}},$$2), barycentric Remez algorithms (as implemented in Chebfun MINIMAX) attain $$r(z) = \frac{\displaystyle\sum_{j=0}^m \frac{w_j f_j}{z-z_j}}{\displaystyle\sum_{j=0}^m \frac{w_j}{z-z_j}},$$3 accuracy in double precision—achieving results previously feasible only with $$r(z) = \frac{\displaystyle\sum_{j=0}^m \frac{w_j f_j}{z-z_j}}{\displaystyle\sum_{j=0}^m \frac{w_j}{z-z_j}},$$4-digit extended precision (Filip et al., 2017). Barycentric Padé representations simplify the algebra, allow node adaptation, and offer strong numerical robustness (Brezinski et al., 2013).

Geometric and High-Dimensional Interpolation

Positive geometry frameworks generalize barycentric coordinates to convex polytopes, Grassmannians, and spline spaces, yielding barycentric weights as ratios of canonical rational forms. Rational interpolation over arbitrary positive geometries offers new theoretical and computational paradigms for geometry-based data fitting and projective-invariant methods (Vaitkus, 2021).

Practical Guidelines and Limitations

• Node Selection: Adaptive strategies (e.g., via equilibrium potential, AAA, Remez alternants) cater to singularities and nonuniform function behavior (Zhao et al., 2023).
• Weight Computation: Linear or SVD-based solvers for the Loewner matrix, possibly subject to nullspace constraints for degree control or structure preservation (Pradovera et al., 2024).
• Numerical Stability: Well-conditioned, free of spurious poles on the real axis if weights alternate (e.g., Chebyshev-Jacobi choices), and suitable for high-precision tasks even with large degrees (Kong et al., 2021, Driscoll et al., 15 Jan 2026).
• Limits: In cases of highly clustered or coincident nodes, or when the underlying function lacks appropriate analytic continuation, barycentric representations may suffer from node-dependent loss of accuracy or require mesh refinement or regularization (Filip et al., 2017, Kong et al., 2021).

• Explicit conversion, stability theory, and applications: (Elsworth et al., 2017)
• Adaptive rational interpolation and analytic continuation: (Huang et al., 2024)
• Minimax and AAA-Lawson rational approximation: (Filip et al., 2017)
• Fast, stable evaluation and derivative algorithms: (Driscoll et al., 15 Jan 2026)
• Prescribed relative degree and model selection: (Pradovera et al., 2024)
• Structured barycentric forms for system identification: (Gosea et al., 2023)
• Barycentric coordinates in positive geometry: (Vaitkus, 2021)
• Equilibrium potential-based node placement: (Zhao et al., 2023)
• Padé approximation in barycentric form: (Brezinski et al., 2013)

The barycentric rational representation constitutes a foundational method in contemporary rational approximation, enabling robust, efficient interpolation, minimax approximation, spectral computation, and data-driven surrogate modeling across mathematics, engineering, and physics.