The high speed of \(x_{k}\rightarrow x^\ast\in{\mathbb R}\) is usually measured using the C-, Q- or R-orders:
\lim \frac {|x^\ast – x_{k+1}|}{|x^\ast – x_k|^{p_0}}\in(0,+\infty),
\lim \frac {\ln |x^\ast – x_{k+1}|}{\ln |x^\ast – x_k|} =q_0,
\qquad \mbox{resp.}
\lim \big\vert\ln |x^\ast – x_k| \big \vert ^\frac1k =r_0.
By connecting them to the natural, term-by-term comparison of the errors of two sequences, we find that the C-orders — including (sub)linear — are in agreement. Weird relations may appear though for the Q-orders: we expect
|x^\ast – x_k|=\mathcal{O}(|x^\ast – y_k|^\alpha), \qquad\forall\alpha>1,
to attract “\(\geq\)” for the Q-orders of \(\{x_{k}\}\) vs. \(\{y_{k}\}\); the contrary is shown by an example providing no vs. infinite Q-orders. The R-orders seem even worse: an \(\{x_{k}\}\) with infinite R-order may have unbounded nonmonotone errors: \(|x^\ast – x_{k+1}|/|x^\ast – x_k|\rightarrow +\infty\).

Such aspects motivate the study of equivalent definitions, computational variants, etc.

The orders are also the perspective from which we analyze the three basic iterative methods for nonlinear equations. The Newton method, widely known for its quadratic convergence, may in fact attain any C-order from \([1,+\infty]\) (including sublinear); we briefly recall such convergence results, along with connected aspects (historical notes, known asymptotic constants, floating point arithmetic issues, radius of attraction balls, etc.) and provide examples.

This approach leads to similar results for the successive approximations, while the secant method exhibits different behavior: it may not have high C-, but just Q-orders.


Emil Cătinaş
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy


(computational) convergence orders, iterative methods, Newton method, secant method, successive approximations, asymptotic rates.


freely available at the publisher: http://doi.org/10.1137/19M1244858

Paper coordinates:

E. Catinas, How many steps still left to x*?, SIAM Rev., 63 (2021) no. 3, pp. 585–624, doi: 10.1137/19M1244858

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Some small codes in LaTeX and Julia have been posted on Github in order to illustrate some aspects.

We will post soon in Python too.

The link is: https://github.com/ecatinas/conv-ord


Working with the SIAM staff was an amazing experience. I would like to thank to Louis Primus for his solicitude and professionalism.

Some references I have missed:

(these were included in the initial submission, but missing in the final version, as not cited in text, and we switched to BibTeX)

known to me, but still missing (nobody is perfect):

found after sending the revised version:

  • J. H. Wilkinson, The evaluation of the zeros of ill-conditioned polynomials. part ii, Numer. Math., 1 (1959), pp. 167–180, doi: 10.1007/BF01386382
  • D. F. Bailey and J. M. Borwein, Ancient indian square roots: an exercise in forensic paleo mathematics, Amer. Math. Monthly, 119 (2012), pp. 646–657, doi 10.4169/amer.math.monthly.119.08.646.
  • E. Halley, Methodus nova accurata & facilis inveniendi radices aeqnationum quarumcumque generaliter, sine praviae reductione, Phil. Trans. R. Soc., 18 (1694), pp. 136–148, https://doi.org/10.1098/rstl.1694.0029.
  • A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, 2nd ed., 1978.
  • M. A. Nordgaard, The origin and development of our present method of extracting the square and cube roots of numbers, The Mathematics Teacher, 17 (1924), pp. 223–238, https://doi.org/https://www.jstor.org/stable/27950616.

and others (to be completed)

Version: August 27, 2021.

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