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arxiv: 1906.11326 · v1 · pith:IE2ULMLTnew · submitted 2019-06-26 · 🧮 math.NA · cs.NA

Approximating the pth Root by Composite Rational Functions

Evan S. Gawlik , Yuji Nakatsukasa This is my paper

Pith reviewed 2026-05-25 15:07 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords rational approximationpth rootcomposite rational functionsZolotarev functionsexponential convergencenumerical analysis
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The pith

Composite rational functions approximate x to the 1/p with root-exponential accuracy in degree and doubly exponential accuracy in the number of degrees of freedom.

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

The paper examines composite rational functions built by nesting simpler rationals recursively to approximate the pth root of x on the unit interval. It establishes that these nested forms deliver approximately pth-root exponential error decay as the degree grows, while the error drops doubly exponentially when counted against the total number of free parameters. The construction is motivated by the known recursive optimality of Zolotarev rational functions for the square-root case. Although the composites no longer coincide with the minimax rational approximant once p reaches 3 or higher, the stated convergence rates still hold. This parameter efficiency extends the method to related functions such as the absolute value and the sector function.

Core claim

Composite rational functions of the form r_k(x, r_{k-1}(x, r_{k-2}(⋯(x,r_1(x,1))⋯))) achieve approximately pth-root exponential convergence with respect to the degree and doubly exponential convergence with respect to the number of degrees of freedom for approximating x^{1/p} on [0,1].

What carries the argument

The nested composite rational function r_k(x, r_{k-1}(x, ⋯ r_1(x,1)⋯)) that builds the approximant recursively from the constant 1.

If this is right

  • Approximations to x^{1/p} become available through a recursive construction whose cost grows only with the number of layers rather than total degree.
  • The same nested form yields efficient approximations to the absolute-value function and the sector function.
  • High accuracy remains attainable even though the composite class excludes the true minimax rational for p greater than or equal to 3.

Where Pith is reading between the lines

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

  • The double-exponential dependence on parameter count suggests the method could support very high-precision root computations with modest storage.
  • Composite nesting may extend usefully to other algebraic functions whose singularities limit single-stage rational approximation.
  • Explicit error bounds derived from the construction could guide adaptive choice of depth and inner degrees in practice.

Load-bearing premise

The recursive optimality property known for Zolotarev functions on the square root extends sufficiently to the composite construction for general p.

What would settle it

Numerical computation of the actual maximum error of the composite rational approximant for a chosen p greater than or equal to 3 at successively larger depths, followed by checking whether the observed decay matches the predicted pth-root exponential rate in degree.

Figures

Figures reproduced from arXiv: 1906.11326 by Evan S. Gawlik, Yuji Nakatsukasa.

Figure 1
Figure 1. Figure 1: Error curves fek(x) − x 1/p. Note that the error on [0, αp ] is bounded by that on [α p , 1], which is  = 2α in both cases. Putting these inequalities together, we conclude that k = log log 2  log( p p−1 ) + ek2 + log log 2 p log 2 (27) recursions are enough to yield accuracy , where ek2 is an integer satisfying k2 − log log 2 p 1+α ∗ 1−α∗ log 2 ≤ ek2 ≤ k2 − log log 2 p 1+α ∗ 1−α∗ log 2 + 1. Since k re… view at source ↗
Figure 2
Figure 2. Figure 2: Error history maxx∈[0,1] |fek(x) − x 1/p| for varying k for p ∈ {2, 5, 31}, along with linear fits shown as dashed lines. 0.2 0.4 0.6 0.8 1 10-15 10-10 10-5 100 0.2 0.4 0.6 0.8 1 10-3 10-2 10-1 100 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Error |gek(z) − sectp(z)| on [α, 1] for α = 0.1, p = 3 (left) and p = 31 (right). The fact that the plots do not appear to go down to 0 between equioscillation points is simply an artifact of the plotting scheme, which is based on 104 equispaced sample points. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
read the original abstract

A landmark result from rational approximation theory states that $x^{1/p}$ on $[0,1]$ can be approximated by a type-$(n,n)$ rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate approximating $x^{1/p}$ by composite rational functions of the form $r_k(x, r_{k-1}(x, r_{k-2}( \cdots (x,r_1(x,1)) )))$. While this class of rational functions ceases to contain the minimax (best) approximant for $p\geq 3$, we show that it achieves approximately $p$th-root exponential convergence with respect to the degree. Moreover, crucially, the convergence is doubly exponential with respect to the number of degrees of freedom, suggesting that composite rational functions are able to approximate $x^{1/p}$ and related functions (such as $|x|$ and the sector function) with exceptional efficiency.

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 manuscript claims that composite rational functions of the nested form r_k(x, r_{k-1}(x, …, r_1(x,1)…)) approximate x^{1/p} on [0,1] with approximately pth-root exponential error decay in total degree and doubly exponential decay in the number of degrees of freedom. The construction is motivated by the recursive optimality of Zolotarev rational functions for the square-root case, while acknowledging that for p≥3 the composite class is strictly smaller than the full minimax class.

Significance. If the stated rates are rigorously established, the result would demonstrate an unusually efficient rational approximation scheme for pth-root and related functions (|x|, sector function) that exploits composition to obtain double-exponential convergence in parameters, offering a concrete computational advantage over single rational approximants of comparable total degree.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the central claim that the composite class retains approximately pth-root exponential convergence for p≥3 rests on an extension of recursive optimality from the Zolotarev (p=2) case, yet the manuscript supplies no explicit error bound or equioscillation argument that avoids invoking minimax uniqueness; without such an independent analysis the rate remains unanchored.
  2. [Numerical results / main theorem] The doubly-exponential claim in the number of degrees of freedom is a direct extrapolation from the p=2 case; the manuscript must exhibit at least one concrete error table or theorem for p=3 or p=4 that confirms the exponent does not degrade when the composite class ceases to contain the minimax approximant.
minor comments (2)
  1. [Introduction] Notation for the composite nesting depth k versus total degree should be clarified in the introduction to avoid ambiguity when comparing convergence plots.
  2. [Abstract] The abstract mentions “approximately pth-root exponential convergence”; a precise statement of the observed or proved exponent (e.g., exp(−c n^{1/p})) should appear in the theorem statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the convergence analysis for p≥3 can be strengthened. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the central claim that the composite class retains approximately pth-root exponential convergence for p≥3 rests on an extension of recursive optimality from the Zolotarev (p=2) case, yet the manuscript supplies no explicit error bound or equioscillation argument that avoids invoking minimax uniqueness; without such an independent analysis the rate remains unanchored.

    Authors: We agree that the current write-up of the error analysis for p≥3 would benefit from greater independence from the p=2 minimax case. The manuscript already derives a recursive error estimate by tracking how the approximation error at each composition level propagates through the pth-root function, but this estimate is presented only briefly. In the revision we will expand §2 to include a self-contained inductive bound on the composite error that relies solely on the local approximation properties of each rational factor and standard properties of the pth-root function on [0,1], without any appeal to uniqueness or equioscillation of the global minimax approximant. revision: yes

  2. Referee: [Numerical results / main theorem] The doubly-exponential claim in the number of degrees of freedom is a direct extrapolation from the p=2 case; the manuscript must exhibit at least one concrete error table or theorem for p=3 or p=4 that confirms the exponent does not degrade when the composite class ceases to contain the minimax approximant.

    Authors: Section 4 already reports numerical results for p=3 and p=4 that illustrate the observed convergence. To respond directly to the request, the revised manuscript will add an explicit theorem (new Theorem 3.2) stating a double-exponential bound in the number of degrees of freedom for the composite class when p=3, together with a dedicated table that tabulates maximum error against total degrees of freedom for both p=3 and p=4. The table will also record the effective exponent extracted from successive refinements, confirming that the rate does not degrade relative to the p=2 case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; convergence claims rest on explicit analysis of the composite construction

full rationale

The paper motivates the composite form from the known recursive optimality of Zolotarev functions in the square-root case but then states that it shows the pth-root exponential rate and doubly exponential rate in degrees of freedom directly for the composite class. No step reduces a claimed prediction to a fitted parameter, a self-citation that is itself unverified, or a definition that already encodes the target rate. The abstract explicitly distinguishes the p≥3 case (where the composite class is strictly smaller than the minimax class) yet asserts an independent demonstration of the rates, keeping the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a landmark rational-approximation result for single rational functions and on the recursive optimality property of Zolotarev functions for the square-root case; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption A landmark result states that x^{1/p} on [0,1] can be approximated by a type-(n,n) rational function with root-exponential accuracy.
    Invoked in the first sentence of the abstract as the baseline that the composite construction improves upon.
  • domain assumption Zolotarev functions possess a recursive optimality property for the square root and sign functions.
    Cited as the motivation for investigating the composite form for general p.

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Reference graph

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