---
title: Barycentric Rational Representation
url: https://www.emergentmind.com/topics/barycentric-rational-representation
type: topic
---

# Barycentric Rational Representation

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) [1711.00707, 2412.18813, 1705.10132, 2101.07949, 2410.02000, 2601.10667]. 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 $p$ a degree-$m$ polynomial, one retrieves $r(z) \equiv p(z)$ provided the weights $w_j$ satisfy the linear relations
$$
\sum_{j=0}^m w_j z_j^k = 0, \qquad k=0,\ldots,m-1,\qquad \sum_{j=0}^m w_j z_j^m \ne 0
$$
[2601.10667]. For rational interpolants of type $[m,n]$, the same structure applies with two sets of nodes and weights, maintaining interpolation at up to $m+n+1$ points [1307.4245].

## 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 $w_j/(z-z_j)$ is well-behaved away from $z_j$; near $z \approx z_k$ 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 [1711.00707, 1705.10132].
- **Choice of Weights:** In many contexts, barycentric weights can be chosen as
$$
w_j = \frac{1}{\prod_{i \neq j} (z_j - z_i)}
$$
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 [2412.18813, 1705.10132].

## 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 $z_j$ and computes weights $w_j$ by solving least-squares problems on the Loewner matrix:
$$
A^{(j)}_{i,k} = \frac{f(Z^{(j)}_i) - f(z_k)}{Z^{(j)}_i - z_k}
$$
with $w$ as the right singular vector for the smallest singular value. The barycentric form at each step is
$$
b_j(z) = \frac{\sum_{k=1}^j w_k f(z_k)/(z-z_k)}{\sum_{k=1}^j w_k/(z-z_k)}
$$
The process continues until the maximal residual falls below a specified tolerance (default $10^{-13}$) [2412.18813].

### 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 $\ell^\infty$ 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 [1705.10132].

#### Applications Table

| Algorithm         | Node Selection        | Error Metric   |
|-------------------|----------------------|---------------|
| AAA               | Greedy (residual)    | least-squares |
| Remez             | Alternants           | $\ell^\infty$ minimax |
| AAA-Lawson        | Fixed, IRLS          | weighted $\ell^\infty$ |
| 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:
$$
\sum_{j=0}^m w_j\,f_j\,z_j^\ell = 0, \ \ell=0,\dots,\mu-1; \quad \sum_{j=0}^m w_j\,z_j^\ell = 0, \ \ell=0,\dots,\nu-1,
$$
with $\nu-\mu = $ desired relative degree. This enables high-frequency extrapolation that is consistent with the system’s physical structure and robust model selection for extrapolation [2410.02000].

### 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:
$$
\hat{H}(s) = \frac{\sum_{i=1}^r h_i\,w_i\,(s-s_i)(s-\omega_i)}{1 + \sum_{i=1}^r w_i\,(s-s_i)(s-\omega_i)}
$$
with $\omega_i$ as additional parameter support points, aligning the approximation with the mechanical system’s structure and permitting efficient identification via Loewner-type algorithms [2303.12576].

## 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 $r_j(z) = \frac{w_j/(z-z_j)}{\sum_i w_i/(z-z_i)}$ satisfy a recursion, yielding a pencil $(Z_m \underline{W_m}, \underline{W_m})$ 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 [1711.00707].

### Efficient Evaluation and Stable Derivative Computation

Recent developments provide $O(n)$ 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 $O(n^2)$ double-sum approaches) ensures forward stability for high-precision differentiation [2601.10667].
- 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 $100$–$1000 \times$ faster than MaxEnt, and the method is robust to moderate noise and under-resolved data [2412.18813].

### Minimax and Padé-Type Approximation

In challenging minimax rational approximation (e.g., type $(80,80)$ for $|x|$ on $[-1,1]$), barycentric Remez algorithms (as implemented in Chebfun MINIMAX) attain $10^{-12}$ accuracy in double precision—achieving results previously feasible only with $200$-digit extended precision [1705.10132]. Barycentric Padé representations simplify the algebra, allow node adaptation, and offer strong numerical robustness [1307.4245].

### 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 [2101.08589].

### Practical Guidelines and Limitations

- **Node Selection:** Adaptive strategies (e.g., via equilibrium potential, AAA, Remez alternants) cater to singularities and nonuniform function behavior [2303.15222].
- **Weight Computation:** Linear or SVD-based solvers for the Loewner matrix, possibly subject to nullspace constraints for degree control or structure preservation [2410.02000].
- **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 [2101.07949, 2601.10667].
- **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 [1705.10132, 2101.07949].

## References

- Explicit conversion, stability theory, and applications: [1711.00707]
- Adaptive rational interpolation and analytic continuation: [2412.18813]
- Minimax and AAA-Lawson rational approximation: [1705.10132]
- Fast, stable evaluation and derivative algorithms: [2601.10667]
- Prescribed relative degree and model selection: [2410.02000]
- Structured barycentric forms for system identification: [2303.12576]
- Barycentric coordinates in positive geometry: [2101.08589]
- Equilibrium potential-based node placement: [2303.15222]
- Padé approximation in barycentric form: [1307.4245]

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.

Source: https://www.emergentmind.com/topics/barycentric-rational-representation