pith. sign in

arxiv: 2502.10665 · v2 · pith:DQ6BRVHYnew · submitted 2025-02-15 · 🧮 math.NA · cs.NA

Interpolation constrained rational minimax approximation with barycentric representation

Lei-Hong Zhang , Ya-Nan Zhang This is my paper

Pith reviewed 2026-05-23 03:29 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords rational minimax approximationbarycentric representationLawson's methodinterpolation constraintsdual formulationnumerical stabilityminimax approximation
0
0 comments X

The pith

The b-d-Lawson method solves rational minimax approximation under interpolation constraints by pairing barycentric representations with a dual max-min formulation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a dual-based Lawson's iteration, called b-d-Lawson, that computes the rational function of given degree minimizing the maximum deviation from a target while exactly matching prescribed values at given points. Barycentric weights serve as the decision variables so that the interpolation conditions hold by construction and the representation avoids the conditioning problems of polynomial bases. The dual framework recasts the original nested min-max problem as an unconstrained max-min problem over a single level, allowing standard Lawson iteration to be applied directly. The resulting procedure is shown to produce stable, accurate approximations on several test problems.

Core claim

By expressing the rational approximant in barycentric form and converting the constrained minimax task into an equivalent max-min dual problem, the b-d-Lawson iteration computes the desired rational function while automatically satisfying the interpolation conditions and avoiding the numerical instability associated with monomial bases.

What carries the argument

The dual framework that converts the bi-level min-max problem into a single-level max-min problem solvable by Lawson's iteration, together with barycentric weights that enforce interpolation by construction.

If this is right

  • Interpolation conditions are satisfied exactly for any choice of barycentric weights.
  • The method avoids forming or inverting Vandermonde-type matrices.
  • Lawson's iteration applies directly to the transformed dual problem without additional projection steps.
  • The same framework can be used for any target function and any prescribed interpolation set of compatible cardinality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may extend to rational approximation on contours or in the complex plane once suitable barycentric bases are defined.
  • Replacing Lawson iteration with other first-order solvers could yield faster variants for large-scale problems.
  • The dual construction may apply to related problems such as constrained Chebyshev approximation or model-order reduction with fixed poles.

Load-bearing premise

The dual formulation remains equivalent to the original minimax problem and can be solved efficiently by Lawson iteration while exactly preserving the required interpolation values.

What would settle it

A concrete data set and interpolation nodes for which b-d-Lawson either fails to converge or returns an approximant that violates one or more interpolation conditions.

Figures

Figures reproduced from arXiv: 2502.10665 by Lei-Hong Zhang, Ya-Nan Zhang.

Figure 7.1
Figure 7.1. Figure 7.1: The top row: error curves from b-d-Lawson(40), d-Lawson(40) and AAA(40) of the approximants of type (6, 6), respectively. Note that there are 11 = 2n + 2 − ℓ extreme points in the error curve from b-d-Lawson(40), while 14 = 2n + 2 extreme points for d-Lawson(40) and AAA(40). The bottom row: (bottom-left) the function f(x) in (7.1); (bottom-middle) the dual objective function values and maximum errors ver… view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Error curves from b-d-Lawson(40), d-Lawson(40) and AAA(40) of the approximants of type (8, 8) for Example 7.3. There are 18 = 2n + 2 extreme points for d-Lawson(40) and AAA(40), whereas 15 = 2n + 2 − ℓ extreme points for b-d-Lawson(40). Example 7.4. We next apply b-d-Lawson to the following discontinuous function f(x) = ® 1 − 1 2 sin(πx), x ∈ (0, 1), 0, x = 0, 1. (7.2) 19 [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: The right three subfigures are error curves from b-d-Lawson(40), d-Lawson(40) and AAA(40) of ξ ∈ R(6) in approximating the function (7.2), respectively. Example 7.5. We approximate the Riemann zeta function f(z) = ζ(z) by a rational function using the data in the critical line L : z = 0.5 + it, t ∈ [−50, 50] where i = √ −1. In this testing, we use m = 200 equally spaced points in L as samples data {xj , … view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: Approximation of the Riemann zeta function [PITH_FULL_IMAGE:figures/full_fig_p021_7_4.png] view at source ↗
Figure 7.5
Figure 7.5. Figure 7.5: The (1,1)-subfigure: the sets E and F; the (1,2)-subfigure: errors associated with the samples from b-d-Lawson(40); the (1,3)-subfigure: errors associated with the samples from AAA without the option ‘sign’; the (1,4)-subfigure: errors associated with the samples from AAA￾Lawson(40) with the option ‘sign’; the (2,2)-subfigure: the sets E and F with two interpolation nodes; the (2,3)-subfigure: errors ass… view at source ↗
read the original abstract

In this paper, we propose a novel dual-based Lawson's method, termed {b-d-Lawson}, designed for addressing the rational minimax approximation under specific interpolation conditions. The {b-d-Lawson} approach incorporates two pivotal components that have been recently gained prominence in the realm of the rational approximations: the barycentric representation of the rational function and the dual framework for tackling minimax approximation challenges. The employment of barycentric formulae enables a streamlined parameterization of the rational function, ensuring natural satisfaction of interpolation conditions while mitigating numerical instability typically associated with Vandermonde basis matrices when monomial bases are utilized. This enhances both the accuracy and computational stability of the method. To address the bi-level min-max structure, the dual framework effectively transforms the challenge into a max-min dual problem, thereby facilitating the efficient application of Lawson's iteration. The integration of this dual perspective is crucial for optimizing the approximation process. We will discuss several applications of interpolation-constrained rational minimax approximation and illustrate numerical results to evaluate the performance of the {b-d-Lawson} method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes the b-d-Lawson method, a dual-based variant of Lawson's iteration for rational minimax approximation subject to interpolation conditions. It uses barycentric representations of the rational function to enforce the interpolation constraints by construction and to avoid Vandermonde-type instability, while a dual reformulation converts the original bi-level min-max problem into a max-min problem that Lawson's iteration can solve directly. The manuscript discusses applications and presents numerical results to illustrate performance.

Significance. If the dual transformation is shown to preserve the interpolation conditions and the convergence properties of Lawson's method without introducing additional parameters or hidden constraints, the approach would provide a stable, direct algorithmic route to constrained rational minimax problems. The explicit use of barycentric forms and the dual framework are timely given recent interest in these tools; reproducible numerical examples would strengthen the contribution.

major comments (2)
  1. [Section 2 or 3 (dual formulation)] The abstract states that the dual framework 'effectively transforms the bi-level min-max structure into a max-min dual problem' while preserving interpolation conditions, but without the explicit statement of the dual problem (e.g., the Lagrangian or the saddle-point formulation) it is impossible to verify that the interpolation constraints remain feasible after the transformation. A concrete derivation or theorem establishing equivalence is required.
  2. [Section 2 (barycentric parameterization)] The claim that barycentric representation 'ensures natural satisfaction of interpolation conditions' needs to be accompanied by the precise weight or node selection rule that enforces the conditions; if this rule depends on the data or on an auxiliary optimization step, the method is no longer parameter-free as implied.
minor comments (2)
  1. [Abstract] The abstract contains several grammatical issues ('have been recently gained prominence', 'the challenge into a max-min dual problem') that should be corrected for readability.
  2. [Numerical experiments] Numerical results are mentioned but no tables, error metrics, or comparison baselines are referenced in the provided abstract; the full manuscript should include quantitative tables with at least one standard method (e.g., linearized or Remez-type) for each application.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We address each point below and will revise the manuscript accordingly to strengthen the presentation of the dual formulation and barycentric details.

read point-by-point responses

  1. Referee: [Section 2 or 3 (dual formulation)] The abstract states that the dual framework 'effectively transforms the bi-level min-max structure into a max-min dual problem' while preserving interpolation conditions, but without the explicit statement of the dual problem (e.g., the Lagrangian or the saddle-point formulation) it is impossible to verify that the interpolation constraints remain feasible after the transformation. A concrete derivation or theorem establishing equivalence is required.

    Authors: We agree that an explicit derivation is needed for verification. In the revised manuscript we will add, in Section 3, the Lagrangian of the constrained minimax problem, the resulting saddle-point formulation, and a theorem establishing equivalence between the original bi-level problem and the max-min dual. The proof will explicitly track the interpolation constraints through the dual variables to confirm they remain feasible and are preserved by construction. revision: yes

  2. Referee: [Section 2 (barycentric parameterization)] The claim that barycentric representation 'ensures natural satisfaction of interpolation conditions' needs to be accompanied by the precise weight or node selection rule that enforces the conditions; if this rule depends on the data or on an auxiliary optimization step, the method is no longer parameter-free as implied.

    Authors: The barycentric weights at the prescribed interpolation nodes are set directly from the given data values so that the rational function interpolates by construction; no auxiliary optimization is performed. We will state this explicit (data-dependent but non-iterative) weight rule in the revised Section 2 and clarify that the overall algorithm remains free of additional tunable parameters beyond the standard Lawson iteration. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is algorithmic construction

full rationale

The paper proposes an algorithmic method (b-d-Lawson) that combines barycentric rational representation with a dual reformulation to enable Lawson's iteration under interpolation constraints. No equations, fitted parameters, or predictions are exhibited that reduce by construction to the inputs. The abstract describes the components as recently prominent in the literature without invoking self-citation chains as load-bearing justification for the central claim. The derivation is presented as a direct, self-contained construction rather than a renaming or self-referential fit, making the method independent of the patterns that would trigger circularity flags.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, invented entities, or non-standard axioms are stated. The method is described as building on existing barycentric and dual frameworks.

axioms (1)
  • domain assumption Standard assumptions of rational approximation theory and convergence of Lawson iteration hold under the stated interpolation constraints.
    The proposal relies on these background results without re-deriving them.

pith-pipeline@v0.9.0 · 5709 in / 1132 out tokens · 20266 ms · 2026-05-23T03:29:34.371817+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages

  1. [1]

    Allison and N

    A. Allison and N. Young, Numerical algorithms for the Nevanlinna-P ick problem, Numer. Math., 42 (1983), 125–145

    work page 1983

  2. [2]

    Barrodale, M

    I. Barrodale, M. J. D. Powell and F. D. K. Roberts, The different ial correction algorithm for rational ℓ∞ approximation, SIAM J. Numer. Anal. , 9 (1972), 493–504. 22

    work page 1972

  3. [3]

    Beckermann, The condition number of real Vandermonde, Kr ylov and positive definite Hankel matrices, Numer

    B. Beckermann, The condition number of real Vandermonde, Kr ylov and positive definite Hankel matrices, Numer. Math. , 85 (2000), 553–577

    work page 2000

  4. [4]

    Berljafa and S

    M. Berljafa and S. G¨ uttel, The RKFIT algorithm for nonlinear rat ional approximation, SIAM J. Sci. Comput. , 39 (2017), A2049–A2071, URL https://doi.org/10.1137/15M1025426

    work page doi:10.1137/15m1025426 2017

  5. [5]

    Berrut and H

    J.-P. Berrut and H. D. Mittelmann, Matrices for the direct deter mination of the barycentric weights of rational interpolation, J. Comput. Appl. Math. , 78 (1997), 355–370

    work page 1997

  6. [6]

    Berrut and H

    J.-P. Berrut and H. Mittelmann, Lebesgue constant minimizing linea r rational interpolation of continuous functions over the interval, Computers Math. Appl. , 33 (1997), 77–86

    work page 1997

  7. [7]

    Berrut and L

    J.-P. Berrut and L. N. Trefethen, Barycentric Lagrange inter polation, SIAM Rev., 46 (2004), 501–517

    work page 2004

  8. [8]

    Berrut and G

    J.-P. Berrut and G. Klein, Recent advances in linear barycentric r ational interpolation, J. Comput. Appl. Math. , 259 (2014), 95–107

    work page 2014

  9. [9]

    Boyd and L

    S. Boyd and L. Vandenberghe, Convex Optimization , Cambridge University Press, 2004

    work page 2004

  10. [10]

    Byrnes, T

    C. Byrnes, T. Georgiou and A. Lindquist, A generalized entropy criterion for Nevanlinna-Pick interpolation with degree constraint, IEEE T. Automat. Contr. , 46 (2001), 822–839

    work page 2001

  11. [11]

    E. W. Cheney and H. L. Loeb, Two new algorithms for rational ap proximation, Numer. Math., 3 (1961), 72–75

    work page 1961

  12. [12]

    Cheney and W

    W. Cheney and W. Light, A Course in Approximation Theory , Pacific Grove, CA: Brooks/Cole, 2000

    work page 2000

  13. [13]

    A. K. Cline, Rate of convergence of Lawson’s algorithm, Math. Comp. , 26 (1972), 167–176

    work page 1972

  14. [14]

    Coelho, J

    C. Coelho, J. Phillips and L. Silveira, Passive constrained rational approximation algorithm using nevanlinna-pick interpolation, in Proceedings 2002 Design, Automation and Test in Europe Conference and Exhibition , 2002, 923–930

    work page 2002

  15. [15]

    T. A. Driscoll, Y. Nakatsukasa and L. N. Trefethen, AAA ration al approximation on a continuum, SIAM J. Sci. Comput. , 46 (2024), A929–A952

    work page 2024

  16. [16]

    G. H. Elliott, The construction of Chebyshev approximations in the comple x plane, PhD thesis, Faculty of Science (Mathematics), University of London, 1978

    work page 1978

  17. [17]

    Filip, Y

    S.-I. Filip, Y. Nakatsukasa, L. N. Trefethen and B. Beckerman n, Rational minimax approxima- tion via adaptive barycentric representations, SIAM J. Sci. Comput. , 40 (2018), A2427–A2455

    work page 2018

  18. [18]

    Fischer and R

    B. Fischer and R. Freund, On the constrained Chebyshev appr oximation problem on ellipses, J. Approx. Theory , 62 (1990), 297–315

    work page 1990

  19. [19]

    M. S. Floater and K. Hormann, Barycentric rational interpolat ion with no poles and high rates of approximation, Numer. Math. , 107 (2007), 315–331

    work page 2007

  20. [20]

    G. H. Golub and C. F. Van Loan, Matrix Computations, 4th edition, Johns Hopkins University Press, Baltimore, Maryland, 2013

    work page 2013

  21. [21]

    I. V. Gosea and S. G¨ uttel, Algorithms for the rational approx imation of matrix-valued functions, SIAM J. Sci. Comput. , 43 (2021), A3033–A3054, URL https://doi.org/10.1137/20M1324727. 23

    work page doi:10.1137/20m1324727 2021

  22. [22]

    Gustavsen and A

    B. Gustavsen and A. Semlyen, Rational approximation of frequ ency domain responses by vector fitting, IEEE Trans. Power Deliv. , 14 (1999), 1052–1061

    work page 1999

  23. [23]

    M. H. Gutknecht, On complex rational approximation. Part I: T he characterization problem, in Computational Aspects of Complex Analysis (H. Werneret at. ,eds.). Dordrecht : Reidel , 1983, 79–101

    work page 1983

  24. [24]

    G¨ uttel and G

    S. G¨ uttel and G. Klein, Convergence of linear barycentric rat ional interpola- tion for analytic functions, SIAM J. Numer. Anal. , 50 (2012), 2560–2580, URL https://doi.org/10.1137/120864787

    work page doi:10.1137/120864787 2012

  25. [25]

    Henrici, Elements of Numerical Analysis , Wiley, New York, 1962

    P. Henrici, Elements of Numerical Analysis , Wiley, New York, 1962

    work page 1962

  26. [26]

    Henrici, Barycentric formulas for interpolating trigonometr ic polynomials and their conju- gates, Numer

    P. Henrici, Barycentric formulas for interpolating trigonometr ic polynomials and their conju- gates, Numer. Math. , 33 (1979), 225–234

    work page 1979

  27. [27]

    Herglotz, I

    G. Herglotz, I. Schur, G. Pick, R. Nevanlinna and H. Weyl, Ausgew¨ ahlte arbeiten zu den urspr¨ ungen der Schur-analysis, vol. 16 of Teubner-Archiv zur Mathematik, B.G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991

    work page 1991

  28. [28]

    N. J. Higham, Accuracy and Stability of Numerical Algorithms, Second edi tion, SIAM, Philadephia, USA, 2002

    work page 2002

  29. [29]

    N. J. Higham, The numerical stability of barycentric Lagrange in terpolation, IMA J. Numer. Anal., 24 (2004), 547–556

    work page 2004

  30. [30]

    J. M. Hokanson, Multivariate rational approximation using a sta bilized Sanathanan-Koerner iteration, 2020, URL arXiv:2009.10803v1

    work page arXiv 2020

  31. [31]

    A. C. Ionit ¸˘ a,Lagrange Rational Interpolation and its Applications to Ap proximation of Large- Scale Dynamical Systems , PhD thesis, Rice University, USA, 2013

    work page 2013

  32. [32]

    Istace and J.-P

    M.-P. Istace and J.-P. Thiran, On computing best Chebyshev co mplex rational approximants, Numer. Algorithms , 5 (1993), 299–308

    work page 1993

  33. [33]

    Karlsson and A

    J. Karlsson and A. Lindquist, Stability-preserving rational app roximation subject to interpo- lation constraints, IEEE T. Automat. Contr. , 53 (2008), 1724–1730

    work page 2008

  34. [34]

    C. L. Lawson, Contributions to the Theory of Linear Least Maximum Approxi mations, PhD thesis, UCLA, USA, 1961

    work page 1961

  35. [35]

    Li and G.-L

    J.-K. Li and G.-L. Xu, On the problem of best rational approxima tion with interpolating constraints (I), J. Comput. Math. , 8 (1990), 233–240, URL http://global-sci.org/intro/article_detail/jcm/9436 .html

    work page 1990

  36. [36]

    Li and G.-L

    J.-K. Li and G.-L. Xu, On the problem of best rational approxima tion with interpolating constraints (II), J. Comput. Math. , 8 (1990), 241–251, URL http://global-sci.org/intro/article_detail/jcm/9437 .html

    work page 1990

  37. [37]

    Li, Vandermonde matrices with Chebyshev nodes, Linear Algebra Appl

    R.-C. Li, Vandermonde matrices with Chebyshev nodes, Linear Algebra Appl. , 428 (2008), 1803–1832

    work page 2008

  38. [38]

    Liang, R.-C

    X. Liang, R.-C. Li and Z. Bai, Trace minimization principles for posit ive semi-definite pencils, Linear Algebra Appl. , 438 (2013), 3085–3106. 24

    work page 2013

  39. [39]

    Lietaert, K

    P. Lietaert, K. Meerbergen, J. P´ erez and B. Vandereycken , Automatic rational approximation and linearization of nonlinear eigenvalue problems, IMA J. Numer. Anal. , 42 (2022), 1087– 1115

    work page 2022

  40. [40]

    H. L. Loeb, On rational fraction approximations at discrete points , Technical report, Convair Astronautics, 1957, Math. Preprint #9

    work page 1957

  41. [41]

    L. A. Luxemburg and P. R. Brown, The scalar Nevanlinna–Pick int erpolation prob- lem with boundary conditions, J. Comput. Appl. Math. , 235 (2011), 2615–2625, URL https://www.sciencedirect.com/science/article/pii/S0377042710006308

    work page 2011

  42. [42]

    Monz´ on, W

    L. Monz´ on, W. Johns, S. Iyengar, M. Reynolds, J. Maack and K. Prabakar, A multi-function AAA algorithm applied to frequency dependent line modeling, in 2020 IEEE Power & Energy Society General Meeting (PESGM) , 2020, 1–5

    work page 2020

  43. [43]

    Nakatsukasa, O

    Y. Nakatsukasa, O. S` ete and L. N. Trefethen, The AAA algor ithm for rational approximation, SIAM J. Sci. Comput. , 40 (2018), A1494–A1522

    work page 2018

  44. [44]

    Nakatsukasa and L

    Y. Nakatsukasa and L. N. Trefethen, An algorithm for real an d complex rational minimax approximation, SIAM J. Sci. Comput. , 42 (2020), A3157–A3179

    work page 2020

  45. [45]

    Nakatsukasa, O

    Y. Nakatsukasa, O. S` ete and L. N. Trefethen, The first five years of the AAA algorithm, Intl. Cong. Basic Sci. , URL https://arxiv.org/pdf/2312.03565, To appear

    work page arXiv

  46. [46]

    Nevanlinna, ¨Uber beschr¨ ankte funktionen, die in gegebenen punkten vorges chrieben werte annehmen, Ann

    R. Nevanlinna, ¨Uber beschr¨ ankte funktionen, die in gegebenen punkten vorges chrieben werte annehmen, Ann. Acad. Sci. Fenn. Sel A. , 13 (1919), 1–72

    work page 1919

  47. [47]

    Nocedal and S

    J. Nocedal and S. Wright, Numerical Optimization , 2nd edition, Springer, New York, 2006

    work page 2006

  48. [48]

    Pach´ on and L

    R. Pach´ on and L. N. Trefethen, Barycentric-Remez algorith ms for best polynomial approxi- mation in the chebfun system, BIT, 49 (2009), 721–741

    work page 2009

  49. [49]

    V. Y. Pan, How bad are Vandermonde matrices?, SIAM J. Matrix Anal. Appl. , 37 (2016), 676–694

    work page 2016

  50. [50]

    Pick, ¨Uber die beschr¨ ankungen analytischer funktionen, welche durch vorgegebene funk- tionswerte bewirkt werden, Math

    G. Pick, ¨Uber die beschr¨ ankungen analytischer funktionen, welche durch vorgegebene funk- tionswerte bewirkt werden, Math. Ann. , 77 (1916), 7–23

    work page 1916

  51. [51]

    T. J. Rivlin and H. S. Shapiro, A unified approach to certain proble ms of approximation and minimization, J. Soc. Indust. Appl. Math. , 9 (1961), 670–699

    work page 1961

  52. [52]

    Rutishauser, Vorlesungen ¨ uber numerische Mathematik, Vol

    H. Rutishauser, Vorlesungen ¨ uber numerische Mathematik, Vol. 1, Birkh¨ au ser, Basel, Stuttgart, 1976; English translation, Lectures on Numeric al Mathematics , Walter Gautschi, ed., Birkh¨ auser, Boston, 1976

    work page 1976

  53. [53]

    Ruttan, A characterization of best complex rational appro ximants in a fundamental case, Constr

    A. Ruttan, A characterization of best complex rational appro ximants in a fundamental case, Constr. Approx. , 1 (1985), 287–296

    work page 1985

  54. [54]

    Saad, Iterative Methods for Sparse Linear Systems , 2nd edition, SIAM, Philadelphia, 2003

    Y. Saad, Iterative Methods for Sparse Linear Systems , 2nd edition, SIAM, Philadelphia, 2003

    work page 2003

  55. [55]

    E. B. Saff and R. S. Varga, Nonuniqueness of best approximatin g complex rational functions, Bull. Amer. Math. Soc. , 83 (1977), 375–377

    work page 1977

  56. [56]

    C. K. Sanathanan and J. Koerner, Transfer function synthe sis as a ratio of two complex polynomials, IEEE T. Automat. Contr. , 8 (1963), 56–58, URL https://doi.org/10.1109/TAC.1963.1105517. 25

    work page doi:10.1109/tac.1963.1105517 1963

  57. [57]

    Schneider and W

    C. Schneider and W. Werner, Some new aspects of rational inte rpolation, Math. Comp. , 47 (1986), 285–299

    work page 1986

  58. [58]

    G. D. Taylor and J. William, Existence questions of Chebyshev by in terpolating for the problem approximation rationals, Math. Comp. , 28 (1974), 1097–1103

    work page 1974

  59. [59]

    W. J. Taylor, Method of Lagrangian curvilinear interpolation, J. Res. Nat. Bur. Standards , 35 (1945), 151–155

    work page 1945

  60. [60]

    Thiran and M.-P

    J.-P. Thiran and M.-P. Istace, Optimality and uniqueness conditio ns in complex rational Chebyshev approximation with examples, Constr. Approx. , 9 (1993), 83–103

    work page 1993

  61. [61]

    L. N. Trefethen, Approximation Theory and Approximation Practice, Extende d Edition, SIAM, 2019

    work page 2019

  62. [62]

    L. N. Trefethen and H. D. Wilber, Computation of Zolotarev rat ional functions, 2024, URL https://arxiv.org/abs/2408.14092

    work page arXiv 2024

  63. [63]

    J. L. Walsh, On interpolation and approximation by rational func tions with preassigned poles, Trans. Amer. Math. Soc. , 34 (1932), 22–74

    work page 1932

  64. [64]

    Werner, Polynomial interpolation: Lagrange versus Newton , Math

    W. Werner, Polynomial interpolation: Lagrange versus Newton , Math. Comp. , 43 (1984), 205–217

    work page 1984

  65. [65]

    Williams, Numerical Chebyshev approximation in the complex plan e, SIAM J

    J. Williams, Numerical Chebyshev approximation in the complex plan e, SIAM J. Numer. Anal., 9 (1972), 638–649

    work page 1972

  66. [66]

    Williams, Characterization and computation of rational Cheby shev approximations in the complex plane, SIAM J

    J. Williams, Characterization and computation of rational Cheby shev approximations in the complex plane, SIAM J. Numer. Anal. , 16 (1979), 819–827

    work page 1979

  67. [67]

    D. E. Wulbert, On the characterization of complex rational app roximations, Illinois J. Math. , 24 (1980), 140–155

    work page 1980

  68. [68]

    Wuytack, On some aspects of the rational interpolation pro blem, SIAM J

    L. Wuytack, On some aspects of the rational interpolation pro blem, SIAM J. Numer. Anal. , 11 (1974), 52–60

    work page 1974

  69. [69]

    Yang, L.-H

    L. Yang, L.-H. Zhang and Y. Zhang, The Lq-weighted dual prog ramming of the linear Chebyshev approximation and an interior-point method, Adv. Comput. Math. , 50:80 (2024)

    work page 2024

  70. [70]

    Zhang and S

    L.-H. Zhang and S. Han, Convergence analysis of Lawson’s iteration for the polynomial and rational minimax approximations , Technical report, 2023, URL https://arxiv.org/pdf/2401.00778, Https://arxiv.org/pdf/2401.00778

    work page arXiv 2023

  71. [71]

    Zhang, L

    L.-H. Zhang, L. Yang, W. H. Yang and Y.-N. Zhang, A convex dua l problem for the rational minimax approximation and Lawson’s iteration, Math. Comp. , DOI: https://doi.org/10.1090/mcom/4021, to appear. 26

    work page doi:10.1090/mcom/4021