Pith. sign in

REVIEW 4 major objections 5 minor 43 references

AutoKnots: Adaptive Knot Allocation for Spline Interpolation

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Adaptive spline knots meet a user-set tolerance automatically

desk verdict Useful, well-tested adaptive spline scheme; the precision guarantee is overstated and the pseudocode has a fixable bug. read the letter →

arxiv 2412.13423 v1 pith:IR3GHGIC submitted 2024-12-18 astro-ph.IM

Sandro D. P. Vitenti , Fernando de Simoni , Mariana Penna-Lima , Eduardo J. Barroso This is my paper · ORCID
classification astro-ph.IM MSC 65D0765D0565D3041A15 PACS 02.60.Ed02.60.Jh
keywords adaptivesplineinterpolationknotplacementcubicerrortolerancenumericalprecisionSimpson'srulecosmologysurfacemassdensity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces an adaptive algorithm that decides where to put the knots of a cubic spline interpolation so that the approximation meets a user-supplied error tolerance. The core claim is that two local checks—the error at each interval midpoint and the difference between Simpson's rule and the spline's own integral—are enough to certify an interval, letting the method refine only where needed. If this works, cosmology codes that currently require hand-tuned knot grids can instead ask for a precision, and the spline will adjust itself as model parameters move during a statistical analysis. The paper also documents that plateau-like functions can fool the midpoint check, and adds a statistical refinement heuristic that reduces, but does not eliminate, those failures.

What carries the argument

The engine is the pair of convergence criteria (2.8a)-(2.8b): the pointwise error $\Delta^a f_t(\bar{x}_i)$ at the interval midpoint and the integral discrepancy $\Delta^a I^t_i$ between Simpson's 1/3 rule approximation and the integral of the spline. Both are compared against $\delta(|f|+\varepsilon)$ and $\delta(|\tilde{I}^e_i|+\varepsilon h_i)$ respectively, where $\delta$ is the user's relative tolerance and $\varepsilon$ is a scale parameter that prevents runaway refinement near zeros. The refinement heuristic (2.14) then re-opens any interval whose width exceeds the mean plus a chosen number of standard deviations, catching sharp features that hide inside otherwise flat intervals.

What would settle it

Take a function that is constant everywhere except for a narrow Gaussian pulse placed asymmetrically inside one starting interval, so that all sampled midpoints lie in the flat region; with refinement disabled, AutoKnots will declare every interval converged while the pulse is missed. Measure the maximum relative error on a fine grid and compare it with $\delta$; a single such example where the error exceeds $\delta$ by orders of magnitude shows the certification criterion is only heuristic.

Watch

Extended reading notes

Core claim

The central discovery is a deterministic, self-tuning knot-allocation rule for not-a-knot cubic splines. Starting from six uniformly spaced knots, the algorithm evaluates the function at interval midpoints, checks condition (2.8a) for the pointwise error and condition (2.8b) for the difference between Simpson's 1/3 integral and the spline integral, and subdivides any interval that fails. Because both checks scale with the fourth derivative and $h^4$ (or $h^5$ for the integral proxy), the method concentrates knots where the function varies quickly and spreads them where it is smooth. The paper's evidence is a battery of tests, including ten million random realizations of smooth, oscillatory, and intermediate functions, plus analytic NFW and Hernquist halo profiles, all showing that the requested relative tolerance is met, typically within one order of magnitude.

Load-bearing premise

The entire accuracy guarantee rests on the assumption that checking the error at each interval's midpoint, together with comparing Simpson's rule to the spline integral, is enough to certify that the whole interval is within tolerance, even though no other point in the interval is sampled.

Editorial extensions

If this is right

  • Users can set a single relative tolerance and receive a spline whose knot density adapts to the function's local complexity, removing the manual 'how many knots and where' step.
  • Because every function evaluation is reused in the final spline, the knot-generation cost is dominated by the cost of sampling the function itself, not by interpolation table recomputation.
  • The scale parameter $\varepsilon$ bounds the absolute error floor in regions where the function is tiny, so tails of Gaussians and other multi-scale functions do not trigger wasteful over-refinement.
  • With the recommended refinement setting, the failure rate over $10^7$ random realizations drops to $0.002\%$, and the maximum relative errors fall from thousands of times the tolerance to within a few dozen times it.
  • For the NFW and Hernquist surface mass density applications, the relative difference between spline and analytic results stays roughly an order of magnitude below the requested tolerance across the whole $R$ range.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The midpoint-only probe means the method gives a heuristic, not a certified, error bound; a narrow feature fully contained between a knot and a midpoint can escape detection, which is presumably why the plateau tests retain a small failure tail.
  • A natural testable extension is to prove or disprove a worst-case error bound by checking a third probe point (e.g., the quarter point) and measuring how the failure rate and knot count respond.
  • The same two-criterion principle could be transplanted to higher-order splines or to two-dimensional interpolation, as long as an integral proxy with compatible error order is available.
  • The ten-million-realization benchmark suggests a practical recipe: run the algorithm on randomly parameterized versions of a target function before using it in an MCMC, and use the worst-case relative error to choose $\delta$ conservatively.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper presents AutoKnots, a deterministic adaptive algorithm for allocating knots in cubic spline interpolation with the not-a-knot boundary condition. Starting from six uniformly spaced knots, the algorithm repeatedly evaluates the interpolation error at interval midpoints and compares a Simpson-rule integral estimate to the spline integral, refining intervals until the error criteria (2.8a-b) are satisfied for sconv consecutive iterations. An optional refinement heuristic (2.14) targets unusually wide intervals after the main loop. The method is validated on ln(x), an oscillatory-plateau function Id(x), a Gaussian, and on 10^7 random realizations of three function families (P6, Cs, Es), with reported failure rates. The paper also applies the method to NFW and Hernquist surface mass density calculations and describes the NumCosmo implementation. The central claim is that the algorithm automatically produces a spline meeting user-defined precision with near-optimal knot counts.

Significance. If appropriately scoped, this is a useful practical contribution. The algorithm is already implemented in an open-source, widely used library (NumCosmo), and the paper reports an unusually extensive empirical evaluation, including 10^7 random realizations for three function families and a comparison against a homogeneous knot distribution. The authors are candid about the failure rates of the method, which is a strength. The novelty over prior adaptive spline methods is modest, but the specific combination of the two local criteria (pointwise and integral) and the refine heuristic appears to work well in practice for many functions. However, the advertised guarantee of meeting a user-defined precision is not established by the local criteria or by the grid-based validation, so the paper's central claim needs to be either substantially strengthened or explicitly softened.

major comments (4)
  1. [§2.2 and §2.4, eqs. (2.8a-b), Table 2] The convergence criteria (2.8a-b) are evaluated only at interval midpoints and only against the spline from the previous iteration, before the current batch of midpoints is inserted. Because the not-a-knot spline is a global interpolant, adding knots in one interval changes the approximant in every other interval; intervals that reached sconv in an earlier iteration are never re-checked against the final spline. Consequently, these local conditions do not certify that the final spline satisfies the user's tolerance over the whole interval. This is not merely a theoretical gap: Table 2 reports failure rates of 0.3% for P6(x) without refinement and 0.002% for Cs(x)/Es(x) with the recommended (refine=1, refine_ns=1) configuration, with absolute maximum relative errors up to 50. The abstract's statements that the algorithm is "designed to meet user-defined precision requirements" and "ensures ... the desired precision consistently" are stronger than the algorithm's own tests support. Please either add a post-hoc certification test on the final spline or revise the abstract and conclusions to state that the method is a heuristic that usually, but not always, meets the tolerance.
  2. [§2.5, eq. (2.14), Table 2] The refine heuristic flags only intervals whose width exceeds mean(H) + refine_ns*std(H). This targets wide intervals, but premature convergence can also occur in intervals of typical width that contain a sharp feature (e.g., a narrow peak inside an otherwise smooth region). The heuristic therefore cannot reliably detect the problematic regions; it only reduces the failure rate, as the 0.002% residual failure rate under the recommended settings shows. The text claims that refine "ensures that regions with sharp features are better resolved," but the mechanism does not identify features—it selects large intervals. Please temper this claim and discuss the heuristic's limitation explicitly in Section 2.5, noting that it is a screening rule, not a guarantee.
  3. [§3.4, Table 2] The statistical validation declares a realization as "failed" only if the tolerance is exceeded on a uniform grid of 10,000 nodes. This is a finite sample and does not bound the maximum error over the continuous interval. In fact, the same midpoint-sampling limitation that affects the convergence criteria also affects the validation: a narrow feature between two grid nodes can be missed entirely, so the reported failure rates are lower bounds on the true probability that the tolerance is exceeded somewhere in the interval. Please state this caveat explicitly, or provide an argument (e.g., a derivative bound on the test functions) that the grid resolves all features of interest.
  4. [§2.5 and Algorithm 1] There is a discrepancy between the refine condition in the text, eq. (2.14), which reads hi > mean(H) + refine_ns*std(H), and the pseudocode in Algorithm 1, which computes mu <- mean(Ht) and sigma <- std(Ht) but then uses the condition `if hi > refine_ns * sigma` (i.e., without adding the mean). If the pseudocode reflects the actual implementation, the threshold is substantially lower for typical distributions (e.g., with mean 1 and std 0.1, refine_ns=1 would flag intervals only above 0.1, not above 1.1). If it is a typo, please correct the pseudocode. This is load-bearing because the default refine_ns=1 is recommended, and the two versions have very different behaviors.
minor comments (5)
  1. [Abstract and §2.2] The abstract says the method "simplifies configuration, often requiring only a single parameter," but Section 2 defines five user-controlled parameters (delta, epsilon, sconv, refine, refine_ns) in addition to the optional max_nodes. While most have defaults, the sentence is misleading and should be rephrased, e.g., to "requiring a single tolerance parameter in the most common usage."
  2. [§3.2 and Table 1] The text in Section 3.2 refers to "the third and fourth rows of table 1" for the two Id(x) configurations, but Table 1 lists ln(x) in the first row, Id(x) in the second and third rows, and N(x) in the fourth through seventh rows. Please renumber the table rows or adjust the references to match.
  3. [§2.4, eq. (2.12)] The derivation of eq. (2.12) silently combines two error bounds that use the same |f^(4)|_i but with different constants (1/2880 and 1/180). The difference of the two leading terms is indeed 1/192, but this assumes the same maximum fourth derivative over the interval for both estimates. Please add a sentence noting this is a heuristic estimate and not a rigorous bound.
  4. [Appendix A.1.1] The appendix repeatedly uses "knot-a-knot" where the paper elsewhere correctly writes "not-a-knot" (e.g., in the phrase "knot-a-knot condition"). Please correct the typo throughout Appendix A.1.1.
  5. [§2.2 and Algorithm 1] The algorithm description in the text says the new sub-intervals are assigned status s_i+1 if the criteria are satisfied, but Algorithm 1 does not explicitly show this status update in the pseudocode. Adding a line such as "s_j <- s_j+1" after the insertion would make the pseudocode consistent with the text.

Circularity Check

1 steps flagged · score 2.0 of 10

Core spline-error derivation is self-contained; minor in-sample tuning of the recommended refine/refine_ns configuration slightly inflates the reported validation.

  1. fitted input called prediction [Section 3.4.4 (Refinement Analysis) and Table 2]
    "The default configuration used in NumCosmo is to apply one refinement with refine_ns = 1, as this provides the best results across all functions tested."

    The recommended settings (refine=1, refine_ns=1) are selected because they provide the best results on the same functions (P6, Cs, Es) whose failure rates are then reported in Table 2. The reported 0% and 0.002% failure rates are therefore in-sample: the configuration was tuned on the very benchmark suite used as validation. This is a mild circularity in the performance claim for the default parameters, not in the spline interpolation derivation itself, which rests on standard error bounds and external analytical benchmarks.

full rationale

The paper's central derivation is not circular. The adaptive algorithm is constructed from standard cubic-spline error estimates, with convergence criteria (2.8a-b) comparing local midpoint errors and Simpson/spline integral estimates. The final accuracy is checked against exact analytical functions (ln, NFW, Hernquist) and 10^7 random realizations, providing external, independent evidence. No load-bearing self-citation chain or imported uniqueness theorem appears; citations to NumCosmo are contextual, not logical premises. The only notable issue is that the recommended refine and refine_ns values are chosen using the same test families later used to report success rates, making part of the statistical validation in-sample. This weakens the strength of the claim that the default configuration 'ensures' the desired precision consistently, but it does not make the algorithm's derivation equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central algorithm depends on user-tuned hyperparameters and on the assumption that local midpoint/integral checks suffice to certify global precision. The latter is explicitly violated by plateau-like functions, motivating the ad hoc refine heuristic. No new physical entities are introduced.

free parameters (5)
  • delta (relative tolerance) = 10^-8 in tests
    User-specified target precision; not fitted, but its choice controls knot density and is an input to the algorithm.
  • epsilon (scale parameter) = 0 (default), 1, 10, 100 in tests
    User-chosen absolute scale to avoid over-refinement near zeros; the choice affects knot distribution and accuracy, as shown in Section 3.3.
  • sconv = 1 (default)
    Number of consecutive passes required before an interval is accepted; authors tested values and chose 1 to avoid unnecessary knots (Section 2.2).
  • refine = 1 (recommended)
    Number of statistical refinement iterations; chosen based on test performance across function families (Sections 3.4.4 and 5).
  • refine_ns = 1 (recommended)
    Threshold in standard deviations for flagging large intervals for refinement; tuned on the same test suite used for validation.
assumptions (4)
  • standard math The interpolation error for cubic splines with not-a-knot conditions is O(h^4) and the Simpson/spline integral error difference is O(h^5), requiring a bounded fourth derivative.
    Used to justify the local error criteria and the refine behavior in Section 2.4.
  • domain assumption Satisfying the midpoint function error and Simpson/spline integral difference within a sub-interval certifies that the spline error across the entire sub-interval is below the tolerance.
    Core assumption of the adaptive stop rule; false for functions with sharp features inside an interval, as the paper itself demonstrates in Section 2.5 and Table 2.
  • domain assumption The function f(x) is smooth enough to be well represented by a cubic spline and has a bounded fourth derivative on [a,b].
    Required for the error bounds in eqs. (2.11)-(2.12) and for the adaptive refinement to converge in practice; the authors target cosmological functions of this type.
  • ad hoc to paper The refine heuristic, which flags sub-intervals wider than mean(H) + refine_ns * std(H), identifies intervals suffering premature convergence.
    A statistical heuristic without proof; it targets large intervals but can add unnecessary knots and does not eliminate all failures (Section 2.5 and Table 2).

how reviews work

0 comments
Cite this review

Pith. "Pith review of AutoKnots: Adaptive Knot Allocation for Spline Interpolation." pith.science (2026). https://pith.science/paper/IR3GHGIC

@misc{pith2026241213423,
  author       = {Pith},
  title        = {Pith review of: AutoKnots: Adaptive Knot Allocation for Spline Interpolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IR3GHGIC}},
  note         = {Machine review of arXiv:2412.13423}
}
abstract

In astrophysical and cosmological analyses, the increasing quality and volume of astronomical data demand efficient and precise computational tools. This work introduces a novel adaptive algorithm for automatic knots (AutoKnots) allocation in spline interpolation, designed to meet user-defined precision requirements. Unlike traditional methods that rely on manually configured knot distributions with numerous parameters, the proposed technique automatically determines the optimal number and placement of knots based on interpolation error criteria. This simplifies configuration, often requiring only a single parameter. The algorithm progressively improves the interpolation by adaptively sampling the function-to-be-approximated, $f(x)$, in regions where the interpolation error exceeds the desired threshold. All function evaluations contribute directly to the final approximation, ensuring efficiency. While each resampling step involves recomputing the interpolation table, this process is highly optimized and usually computationally negligible compared to the cost of evaluating $f(x)$. We show the algorithm's efficacy through a series of precision tests on different functions. However, the study underscores the necessity for caution when dealing with certain function types, notably those featuring plateaus. To address this challenge, a heuristic enhancement is incorporated, improving accuracy in flat regions. This algorithm has been extensively used and tested over the years. NumCosmo includes a comprehensive set of unit tests that rigorously evaluate the algorithm both directly and indirectly, underscoring its robustness and reliability. As a practical application, we compute the surface mass density $\Sigma(R)$ and the average surface mass density $\overline{\Sigma}(<R)$ for Navarro-Frenk-White and Hernquist halo density profiles, which provide analytical benchmarks. (abridged)

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 27 canonical work pages

  1. [1]

    Levi, L.E

    M. Levi, L.E. Allen, A. Raichoor, C. Baltay, S. BenZvi, F. Beutler et al.,The Dark Energy Spectroscopic Instrument (DESI), inBulletin of the American Astronomical Society, vol. 51, p. 57, Sept., 2019, DOI [1907.10688]

    arXiv 2019

  2. [2]

    Benitez, R

    N. Benitez, R. Dupke, M. Moles, L. Sodre, J. Cenarro, A. Marin-Franch et al.,J-PAS: The Javalambre-Physics of the Accelerated Universe Astrophysical Survey, arXiv e-prints (2014) arXiv:1403.5237 [1403.5237]

    arXiv 2014

  3. [3]

    Mellier, Abdurro’uf, J.A

    Euclid Collaboration, Y. Mellier, Abdurro’uf, J.A. Acevedo Barroso, A. Achúcarro, J. Adamek et al.,Euclid. I. Overview of the Euclid mission, arXiv e-prints (2024) arXiv:2405.13491 [2405.13491]. – 30 –

    arXiv 2024

  4. [4]

    LSST Dark Energy Science Collaboration,Large Synoptic Survey Telescope: Dark Energy Science Collaboration, arXiv e-prints (2012) arXiv:1211.0310 [1211.0310]

    arXiv 2012

  5. [5]

    O. Dore, C. Hirata, Y. Wang, D. Weinberg, T. Eifler, R.J. Foley et al.,WFIRST: The Essential Cosmology Space Observatory for the Coming Decade, BAAS 51 (2019) 341 [1904.01174]

    arXiv 2019

  6. [6]

    Di Valentino, O

    E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri et al.,In the realm of the Hubble tension-a review of solutions, Classical and Quantum Gravity38 (2021) 153001 [2103.01183]

    arXiv 2021

  7. [7]

    Kamionkowski and A.G

    M. Kamionkowski and A.G. Riess,The Hubble Tension and Early Dark Energy, Annual Review of Nuclear and Particle Science73 (2023) 153 [2211.04492]

    arXiv 2023

  8. [8]

    Riess and L

    A.G. Riess and L. Breuval,The Local Value of H0, inIAU Symposium, R. de Grijs, P.A. Whitelock and M. Catelan, eds., vol. 376 ofIAU Symposium, pp. 15–29, Jan., 2024, DOI [2308.10954]

    arXiv 2024

Show all 43 references
  1. [9]

    CAMB: Code for Anisotropies in the Microwave Background

    A. Lewis and A. Challinor, “CAMB: Code for Anisotropies in the Microwave Background.” Astrophysics Source Code Library, record ascl:1102.026, Feb., 2011

    2011

  2. [10]

    Lesgourgues,The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview, arXiv e-prints (2011) arXiv:1104.2932 [1104.2932]

    J. Lesgourgues,The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview, arXiv e-prints (2011) arXiv:1104.2932 [1104.2932]

    2011 arXiv

  3. [11]

    Diemer,COLOSSUS: A Python Toolkit for Cosmology, Large-scale Structure, and Dark Matter Halos, The Astrophysical Journal Supplement Series239 (2018) 35 [1712.04512]

    B. Diemer,COLOSSUS: A Python Toolkit for Cosmology, Large-scale Structure, and Dark Matter Halos, The Astrophysical Journal Supplement Series239 (2018) 35 [1712.04512]

    2018 arXiv

  4. [12]

    Chisari, D

    N.E. Chisari, D. Alonso, E. Krause, C.D. Leonard, P. Bull, J. Neveu et al.,Core Cosmology Library: Precision Cosmological Predictions for LSST, The Astrophysical Journal Supplement Series 242 (2019) 2 [1812.05995]

    2019 arXiv

  5. [13]

    NumCosmo: Numerical Cosmology

    S. Dias Pinto Vitenti and M. Penna-Lima, “NumCosmo: Numerical Cosmology.” Astrophysics Source Code Library, record ascl:1408.013, Aug., 2014

    2014

  6. [14]

    Powell,Approximation Theory and Methods, Cambridge University Press (1981)

    M.J.D. Powell,Approximation Theory and Methods, Cambridge University Press (1981)

    1981

  7. [15]

    Obradovic, L.N

    D. Obradovic, L.N. Mishra and V.N. Mishra,Interpolation and Approximation of Functions, Journal of Pure and Applied Mathematics5 (2021) 5

    2021

  8. [16]

    Quiñonero-Candela, C.E

    J. Quiñonero-Candela, C.E. Rasmussen and C.K.I. Williams,Approximation Methods for Gaussian Process Regression, inLarge-Scale Kernel Machines, The MIT Press (2007), DOI [https://direct.mit.edu/book/chapter-pdf/2286468/9780262255790_cai.pdf]

    2007

  9. [17]

    Quadrianto, K

    N. Quadrianto, K. Kersting and Z. Xu,Gaussian process, inEncyclopedia of Machine Learning, (Boston, MA), pp. 428–439, Springer US (2010), DOI

    2010

  10. [18]

    Lyche and K

    T. Lyche and K. Mørken,The sensitivity of a spline function to perturbations of the knots, BIT Numerical Mathematics 39 (1999) 305

    1999

  11. [19]

    Michel and A

    D. Michel and A. Zidna,A new deterministic heuristic knots placement for b-spline approximation, Mathematics and Computers in Simulation186 (2021) 91

    2021

  12. [20]

    Goepp, O

    V. Goepp, O. Bouaziz and G. Nuel,Spline regression with automatic knot selection, arXiv preprint arXiv:1808.01770 (2018)

    2018 arXiv

  13. [21]

    W. Li, S. Xu, G. Zhao and L.P. Goh,Adaptive knot placement in b-spline curve approximation, Computer-Aided Design 37 (2005) 791

    2005

  14. [22]

    Goldenthal and M

    R. Goldenthal and M. Bercovier,Spline curve approximation and design by optimal control over the knots, inGeometric Modelling, pp. 53–64, Springer (2004)

    2004

  15. [23]

    Gálvez, A

    A. Gálvez, A. Iglesias, A. Avila, C. Otero, R. Arias and C. Manchado,Elitist clonal selection algorithm for optimal choice of free knots in b-spline data fitting, Applied Soft Computing26 (2015) 90. – 31 –

    2015

  16. [24]

    Idais, M

    H. Idais, M. Yasin, M. Pasadas and P. González,Optimal knots allocation in the cubic and bicubic spline interpolation problems, Mathematics and Computers in Simulation164 (2019) 131

    2019

  17. [25]

    de Boor,Convergence of cubic spline interpolation with the not-a-knot condition, Tech

    C. de Boor,Convergence of cubic spline interpolation with the not-a-knot condition, Tech. Rep. 2876, Mathematics Research Cente, University of Wisconsin, 610 Walnut St.Madison WI 53705 (Oct., 1985)

    1985

  18. [26]

    Davis, P

    P. Davis, P. Rabinowitz and W. Rheinbolt,Methods of Numerical Integration, Computer Science and Applied Mathematics, Elsevier Science (2014)

    2014

  19. [27]

    Trefethen,Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics (2013)

    L.N. Trefethen,Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics (2013)

    2013

  20. [28]

    de Boor,A Practical Guide to Spline, vol

    C. de Boor,A Practical Guide to Spline, vol. 27 ofApplied Mathematical Sciences, Springer (01, 1978), 10.2307/2006241

    1978 doi

  21. [29]

    M. Sun, L. Lan, C.-G. Zhu and F. Lei,Cubic spline interpolation with optimal end conditions, Journal of Computational and Applied Mathematics425 (2023) 115039

    2023

  22. [30]

    Behforooz,Approximation by integro cubic splines, Applied Mathematics and Computation 175 (2006) 8

    H. Behforooz,Approximation by integro cubic splines, Applied Mathematics and Computation 175 (2006) 8

    2006

  23. [31]

    Phythian and R

    J.E. Phythian and R. Williams,Direct cubic spline approximation to integrals with applications in nautical, International Journal for Numerical Methods in Engineering23 (1986) 305 [https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.1620230212]

    1986 doi

  24. [32]

    Hall,On error bounds for spline interpolation, Journal of Approximation Theory1 (1968) 209

    C. Hall,On error bounds for spline interpolation, Journal of Approximation Theory1 (1968) 209

    1968

  25. [33]

    Ujevic and A.J

    N. Ujevic and A.J. Roberts,A corrected quadrature formula and applications, Anziam Journal 45 (2003) 41

    2003

  26. [34]

    Burg,Derivative-based closed newton–cotes numerical quadrature, Applied Mathematics and Computation 218 (2012) 7052

    C.O. Burg,Derivative-based closed newton–cotes numerical quadrature, Applied Mathematics and Computation 218 (2012) 7052

    2012

  27. [35]

    Górski, E

    K.M. Górski, E. Hivon, A.J. Banday, B.D. Wandelt, F.K. Hansen, M. Reinecke et al., HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere, ApJ 622 (2005) 759 [arXiv:astro-ph/0409513]

    2005 arXiv

  28. [36]

    Seager, D.D

    S. Seager, D.D. Sasselov and D. Scott,A New Calculation of the Recombination Epoch, ApJ 523 (1999) L1 [astro-ph/9909275]

    1999 arXiv

  29. [37]

    Aguena, C

    M. Aguena, C. Avestruz, C. Combet, S. Fu, R. Herbonnet, A.I. Malz et al.,CLMM: a LSST-DESC cluster weak lensing mass modeling library for cosmology, MNRAS 508 (2021) 6092 [2107.10857]

    2021 arXiv

  30. [38]

    Navarro, C.S

    J.F. Navarro, C.S. Frenk and S.D.M. White,The Structure of Cold Dark Matter Halos, ApJ 462 (1996) 563 [astro-ph/9508025]

    1996 arXiv

  31. [39]

    Hernquist,An Analytical Model for Spherical Galaxies and Bulges, ApJ 356 (1990) 359

    L. Hernquist,An Analytical Model for Spherical Galaxies and Bulges, ApJ 356 (1990) 359

    1990

  32. [40]

    Knott,Interpolating cubic splines, vol

    G.D. Knott,Interpolating cubic splines, vol. 18 ofProgress in Computer Science and Applied Logic, Springer Science & Business Media (2000)

    2000

  33. [41]

    Tsai, J.-C

    T.-L. Tsai, J.-C. Yang and L.-H. Huang,Characteristics method using cubic–spline interpolation for advection–diffusion equation, Journal of Hydraulic Engineering130 (2004) 580

    2004

  34. [42]

    Ahlberg, J

    J. Ahlberg, J. Walsh, R. Bellman and E.N. Nilson,The theory of splines and their applications, Academic press (1967)

    1967

  35. [43]

    Angerson, Z

    E. Angerson, Z. Bai, J. Dongarra, A. Greenbaum, A. McKenney, J. Du Croz et al.,LAPACK: A portable linear algebra library for high-performance computers, inSupercomputing ’90:Proceedings of the 1990 ACM/IEEE Conference on Supercomputing, pp. 2–11, 1990, DOI. – 32 – A Algorithmi...

    1990

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.