Global Padé-Based Fits

• Global Padé-based fits are rational approximations that combine local series and asymptotic constraints to achieve enhanced accuracy and analytic stability.
• They employ methods such as barycentric forms, SVD-based doublet removal, and Tikhonov regularization to improve numerical robustness and mitigate overfitting.
• Applications span physics, engineering, and fractional calculus, enabling reliable extraction of parameters and efficient regression for noisy or complex datasets.

A global Padé-based fit is a numerical technique in which rational functions—Padé approximants—are constructed to reproduce the behavior of a target function or dataset across an extended domain, often by matching both local (e.g., Taylor/Maclaurin series) and global (e.g., asymptotic) expansions or by interpolating measured data with analytic constraints. This approach yields analytic, stable, and often physically meaningful parametrizations for complex problems in physics, engineering, and applied mathematics, going significantly beyond local polynomial approximations in both flexibility and accuracy.

1. Mathematical Formulation of Global Padé-Based Fits

Let $f(z)$ admit a power series representation, $f(z) = \sum_{k=0}^\infty c_k z^k$. A Padé approximant of type $[m/n]$ is a rational function of the form

$$P_{m,n}(z) = \frac{p(z)}{q(z)} = \frac{a_0 + a_1 z + \cdots + a_m z^m}{1 + b_1 z + \cdots + b_n z^n}$$

constructed such that its Maclaurin expansion matches $f(z)$ up to order $m + n$; i.e.,

$q(z)f(z) - p(z) = O(z^{m+n+1}).$

This matrix linear problem—matching coefficients in $z$—enables efficient determination of the Padé coefficients for analytic and numerical applications (Brezinski et al., 2014, Herrmann, 2024).

For global accuracy, constraints from the small-$z$ series can be augmented with asymptotic (large-$z$) expansions, resulting in a linear or block-linear system that incorporates more information about the target function (Sarumi et al., 2019). Alternatively, data fitting with Padé forms can be recast as a weighted least-squares problem, allowing the rational function to interpolate or regress noisy or incomplete datasets while potentially subject to regularization (Yevkin et al., 2022).

2. Algorithmic Approaches and Representations

Global Padé-based fits employ variants of the standard Padé construction to enhance numerical stability, enforce analytic constraints, or facilitate practical computation:

A. Barycentric and Partial Fraction Forms:

Padé and Padé-type approximants can be recast in barycentric forms, where the placement of nodes (poles/zeros) is explicitly parameterized, often greatly improving conditioning. In partial fraction (Prony) form, poles are first determined via roots of the denominator, then residues are solved via a smaller Vandermonde system (Brezinski et al., 2014). This separation improves stability for high-order or wide-range problems.

B. Robust/Froissart Doublet Removal:

Singular value decomposition (SVD) on the Hankel or Sylvester matrix generated by the series-matching conditions isolates the rank corresponding to true signal components, discarding unstable or numerically spurious pole-zero (Froissart doublet) pairs (Hippel, 2023).

C. Regularization Techniques:

Tikhonov (ridge) regularization penalizes large coefficients in both numerator and denominator to suppress oscillatory instabilities or overfitting (Yevkin et al., 2022).

D. Treatment of Branch Points and Cuts:

For functions with nontrivial analytic structure (e.g., Stieltjes functions, functions with branch cuts), D-Log Padé constructions fit the log-derivative and reconstruct the original function by exponentiation and integration, ensuring physical analyticity constraints are met (P. et al., 2024).

3. Applications in Physics and Computational Mathematics

A. Masses and Matrix Elements from Multi-Exponential Data

Global Padé-based fits are foundational in the postprocessing of lattice correlator data, where one reconstructs masses ($E_n$) and amplitudes ($A_n$) from a sum of exponentials contaminated by noise:

$$C(ka) = \sum_{n=1}^N A_n e^{-E_n k a} + \text{noise}.$$

The Z-transform maps this to a rational function whose poles encode the $E_n$, so matching the series via Padé yields direct extraction of physical parameters (Hippel, 2023). Laplace-domain variants require accurate quadrature rules, motivating the development of nonlinear integration schemes tailored for exponentials.

B. Spacelike Form Factor Representation in Hadron Phenomenology

Padé-based global fits underpin the construction of analytic, stable, and asymptotically controlled representations of electromagnetic form factors (e.g., nucleon electric/magnetic form factors as functions of momentum transfer $t$), uniformly fitting world data up to $Q^2 \sim 6\,\mathrm{GeV}^2$ without spurious oscillation and with fewer parameters than alternative models (Vaziri et al., 28 Dec 2025).

C. Regression and Extrapolation in Physics and Reliability Theory

Global Padé fits are applied to regression problems where the underlying physical law is rational or exhibits asymptotic scaling, outperforming polynomials in extrapolation and robustness to noise. Examples include fits to resonance curves, cumulative failure probabilities (Weibull models), and reaction kinetics (Yevkin et al., 2022).

D. Special Functions in Fractional Calculus

For entire or meromorphic special functions such as the two-parametric Mittag-Leffler function $E_{\alpha,\beta}(-x)$, global Padé fits constructed by matching both low-order series and high-order asymptotics yield mesh-free, spectrally-accurate approximations and their inverses, enabling rapid evaluation in scalar and matrix arguments required in fractional differential equations (Sarumi et al., 2019, Herrmann, 2024).

4. Error Analysis and Stability

The global Padé procedure, matching both power-series and asymptotic coefficients, produces rational functions whose error decays as $O(x^k)$ for small arguments and $O(x^{-m})$ for large arguments, depending on the matching orders (Sarumi et al., 2019). The addition formula for the Mittag-Leffler function, when combined with Padé—applying the rational fit to a transformed argument and reconstructing the original function via analytic recomposition—achieves relative errors of $10^{-12}$–$10^{-14}$ for moderate Padé orders $n \sim 12{-}15$ (Herrmann, 2024).

For data-driven fits, statistical errors on Padé coefficients are propagated from data covariances; systematic uncertainties are assessed by varying the order or comparing independent sequences (Padé vs D-Log), monitoring pole structure, and validating with synthetic data (P. et al., 2024, Vaziri et al., 28 Dec 2025).

5. Best Practices and Practical Guidelines

• Order Selection: Begin with low orders; increment only to achieve target accuracy and avoid overfitting.
• Regularization: Tikhonov/ridge regularization is crucial for ill-conditioned or noisy data.
• Pole Analysis: After fitting, inspect denominators for physical pole placement (e.g., positive real axis for Laplace-space fits), rejecting fits with complex or unstable poles.
• Robust Construction: Always employ SVD or similar robust solvers for Hankel/Sylvester systems.
• Physical Constraints: Encode known asymptotic fall-off, normalization, or analytic properties as side constraints in the fit (e.g., enforcing power-law decrease consistent with QCD, normalizing to sum rules) (Vaziri et al., 28 Dec 2025).
• Uncertainty Quantification: Use bootstrapping, order variation, and cross-validation to quantify fit stability.
• Special Function Acceleration: Use addition/recomposition formulas and global matching for functions with slow convergence or challenging asymptotics (Herrmann, 2024).

6. Extensions, Representations, and Computational Aspects

Alternative Representations: Padé and Padé-type rational approximants admit representations in barycentric and partial-fraction (Prony) forms, concise for matrix and spectral computations. These forms facilitate both analytic manipulation (e.g., enforcing known poles/zeros) and numerical stability, particularly when nodes (support points) are chosen near singularities or equispaced across fitting intervals (Brezinski et al., 2014).

Efficient Implementation:

State-of-the-art codes implement global Padé construction with symbolic algebra tools or high-performance linear algebra libraries (Eigen for C++, Mathematica for prototyping) (Herrmann, 2024). For special functions, highly efficient evaluation is achieved via partial-fraction decompositions with precomputed roots and residues, and inversion of the rational function for inverse problems (Sarumi et al., 2019).

Comparative Performance:

Global Padé-based fits typically require fewer parameters than high-degree polynomials for equivalent accuracy, yield monotonic and bounded extrapolation, and are robust against Runge phenomena and overfitting. Extension to matrix arguments is direct in the partial-fraction representation, crucial for the numerical solution of fractional-in-time evolution equations (Sarumi et al., 2019).

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References:

• (Brezinski et al., 2014) "New representations of Padé and Padé-type approximants"
• (Sarumi et al., 2019) "Highly Accurate Global Padé Approximations of Generalized Mittag-Leffler Function and its Inverse"
• (Yevkin et al., 2022) "On regression analysis with Padé approximants"
• (Hippel, 2023) "Padé and Padé-Laplace Methods for masses and matrix elements"
• (Herrmann, 2024) "Combining arbitrary order global Padé approximation of the Mittag-Leffler function with its addition formula for a significant accuracy boost"
• (P. et al., 2024) "The role of Padé and D-Log Padé approximants in the context of the MUonE Experiment"
• (Vaziri et al., 28 Dec 2025) "From QCD-Based Descriptions to Direct Fits: A Unified Study of Nucleon Electromagnetic Form Factors"