## Abstract

The high speed of \(x_{k}\rightarrow x^\ast\in{\mathbb R}\) is usually measured using the *C*-, *Q-* or *R*-orders:

\begin{equation}\tag{$C$}

\lim \frac {|x^\ast – x_{k+1}|}{|x^\ast – x_k|^{p_0}}\in(0,+\infty),

\end{equation}

\begin{equation}\tag{$Q$}

\lim \frac {\ln |x^\ast – x_{k+1}|}{\ln |x^\ast – x_k|} =q_0,

\qquad \mbox{resp.}

\end{equation}

\begin{equation}\tag{$R$}

\lim \big\vert\ln |x^\ast – x_k| \big \vert ^\frac1k =r_0.

\end{equation}

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: \(\limsup \)\(|x^\ast – x_{k+1}|/|x^\ast – x_k|= +\infty\).

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.

## Authors

## Keywords

(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

## About this paper

##### Print ISSN

0036-1445

##### Online ISSN

1095-7200

## Github

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

## Notes

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)

- M. Grau-Sánchez, M. Noguera, A. Grau, J.R. Herrero,
*On new computational local orders of convergence*, Appl. Math. Lett. 25 (2012) 2023–2030, doi: 10.1016/j.aml.2012.04.012 - I. Pavaloiu,
*Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences*, Calcolo 32 (1–2) (1995) 69–82, doi: 10.1007/BF02576543. - I. Pavaloiu,
*On the convergence of one step and multistep iterative methods*, Unpublished manuscript, http://ictp.acad.ro/convergence-one-step-multistep-iterative-methods/.

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

- N. Bicanic and K. H. Johnson,
*Who was ’-Raphson’?*, Intl. J. Numerical Methods Engrg., 14 (1979), pp. 148–152, https://doi.org/10.1002/nme.1620140112. - A. Melman,
*Geometry and convergence of Euler’s and Halley’s methods*, SIAM Rev., 39 (1997), pp. 728–735, https://doi.org/10.1137/S0036144595301140. - S. Nash,
*Who was -Raphson? (an answer)*, NA Digest, 92 (1992).

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: October 30, 2021.

[1] Advanpix team, *Advanpix*, Version 4.7.0.13642, 2020, https://www.advanpix.com/ (accessed 2020/03/22). (Cited on pp. 4, 26)

[2] I. K. Argyros and S. George, *On a result by Dennis and Schnabel for Newton’s method: Further improvements*, Appl. Math. Lett., 55 (2016), pp. 49-53, https://doi.org/10.1016/ j.aml.2015.12.003. (Cited on p. 28)

[3] D. F. Bailey, *A historical survey of solution by functional iteration*, Math. Mag., 62 (1989), pp. 155-166, https://doi.org/10.1080/0025570X.1989.11977428. (Cited on pp. 7, 21, 34)

[4] W. A. Beyer, B. R. Ebanks, and C. R. Qualls, *Convergence rates and convergence-order profiles for sequences*, Acta Appl. Math., 20 (1990), pp. 267-284, https://doi.org/10. 1007/bf00049571. (Cited on pp. 8, 9, 11, 14, 17, 19, 21)

[5] J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, *Julia: A fresh approach to numerical computing*, SIAM Rev., 59 (2017), pp. 65-98, https://doi.org/10.1137/141000671. (Cited on p. 4)

[6] R. P. Brent, *Algorithms for Minimization without Derivatives*, Prentice-Hall, Englewood Cliffs, NJ, 1973. (Cited on p. 11)

[7] R. P. Brent, S. Winograd, and P. Wolfe*, Optimal iterative processes for root-finding*, Numer. Math., 20 (1973), pp. 327-341, https://doi.org/10.1007/bf01402555. (Cited on pp. 13, 15)

[8] C. Brezinski, *Comparaison des suites convergentes*, Rev. Franҫaise Inform. Rech. Oper., 5 (1971), pp. 95-99. (Cited on pp. 13, 14)

[9] C. Brezinski, *Acceleration de la Convergence en Analyse Numerique*, Springer-Verlag, Berlin, 1977. (Cited on pp. 2, 13, 14)

[10] C. Brezinski, *Limiting relationships and comparison theorems for sequences*, Rend. Circolo Mat. Palermo Ser. II, 28 (1979), pp. 273-280, https://doi.org/10.1007/bf02844100. (Cited on p. 19)

[11] C. Brezinski, *Vitesse de convergence d’une suite*, Rev. Roumaine Math. Pures Appl., 30 (1985), pp. 403-417. (Cited on pp. 11, 13, 18)

[12] P. N. Brown, *A local convergence theory for combined inexact-Newton/finite-difference projection methods*, SIAM J. Numer. Anal., 24 (1987), pp. 407-434, https://doi.org/10.1137/ 0724031. (Cited on p. 29)

[13] F. Cajori, *Historical note on the Newton-Raphson method of approximation*, Amer. Math. Monthly, 18 (1911), pp. 29-32, https://doi.org/10.2307/2973939. (Cited on pp. 22, 23)

[14] A. Cauchy, *Sur la determination approximative des racines d’une equation algebrique ou transcendante*, Oeuvres Complete (II) 4, Gauthier-Villars, Paris, 1899, 23 (1829), pp. 573-609, http://iris.univ-lille1.fr/pdfpreview/bitstream/handle/1908/4026/41077-2- 4.pdf?sequence=1 (accessed 2019/02/01). (Cited on pp. 21, 23)

[15] J.-L. Chabert, ed., *A History of Algorithms. From the Pebble to the Microchip*, Springer Verlag, Berlin, 1999. (Cited on pp. 21, 22, 23, 28, 30, 34)

[16] J. Chen and I. K. Argyros, *Improved results on estimating and extending the radius of an attraction ball*, Appl. Math. Lett., 23 (2010), pp. 404-408, https://doi.org/10.1016/j.aml. 2009.11.007. (Cited on p. 35)

[17] A. R. Conn, N. I. M. Gould, and P. L. Toint, *Trust-Region Methods*, SIAM, Philadelphia, PA, 2000, https://doi.org/10.1137/1.9780898719857. (Cited on pp. 14, 17)

[18] E. Catinas, *Inexact perturbed Newton methods and applications to a class of Krylov solvers*, J. Optim. Theory Appl., 108 (2001), pp. 543-570, https://doi.org/10.1023/a: 1017583307974. (Cited on p. 29)

[19] E. Catinas, *On accelerating the convergence of the successive approximations method*, Rev. Anal. Numer. Theor. Approx., 30 (2001), pp. 3-8, https://ictp.acad.ro/acceleratingconvergence-successive-approximations-method/ (accessed 20/10/10). (Cited on p. 34)

[20] E. Catinas, *On the superlinear convergence of the successive approximations method*, J. Optim. Theory Appl., 113 (2002), pp. 473-485, https://doi.org/10.1023/a:1015304720071. (Cited on p. 35)

[21] E. Catinas, *The inexact, inexact perturbed and quasi-Newton methods are equivalent models*, Math. Comp., 74 (2005), pp. 291-301, https://doi.org/10.1090/s0025-5718-04-01646-1. (Cited on p. 29)

[22] E. Catinas, *Newton and Newton-type Methods in Solving Nonlinear Systems in RN *, Risoprint, Cluj-Napoca, 2007. (Cited on p. 35)

[23] E. Catinas*, Estimating the radius of an attraction ball*, Appl. Math. Lett., 22 (2009), pp. 712-714, https://doi.org/10.1016/j.aml.2008.08.007. (Cited on p. 35)

[24] E. Catinas, *A survey on the high convergence orders and computational convergence orders of sequences*, Appl. Math. Comput., 343 (2019), pp. 1-20, https://doi.org/10.1016/j.amc. 2018.08.006. (Cited on pp. 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21)

[25] E. Catinas, *How Many Steps Still Left to x*? Part II. Newton iterates without f and f’*, manuscript, 2020. (Cited on pp. 24, 35)

[26] G. Dahlquist and A. Bjork, *Numerical Methods, *Dover, Mineola, NY, 1974. (Cited on p. 33)

[27] R. S. Dembo, S. C. Eisenstat, and T. Steihaug, *Inexact Newton methods*, SIAM J. Numer. Anal., 19 (1982), pp. 400-408, https://doi.org/10.1137/0719025. (Cited on p. 29)

[28] J. W. Demmel, *Applied Numerical Linear Algebra*, SIAM, Philadelphia, PA, 1997, https: //doi.org/10.1137/1.9781611971446. (Cited on pp. 3, 29)

[29] J. E. Dennis, *On Newton-like methods*, Numer. Math., 11 (1968), pp. 324-330, https://doi. org/10.1007/bf02166685. (Cited on p. 17)

[30] J. E. Dennis, Jr., and J. J. More, *A characterization of superlinear convergence and its application to quasi-Newton methods*, Math. Comp., 28 (1974), pp. 549-560, https://doi. org/10.1090/s0025-5718-1974-0343581-1. (Cited on pp. 19, 29)

[31] J. E. Dennis, Jr., and J. J. More, *Quasi-Newton methods, motivation and theory*, SIAM Rev., 19 (1977), pp. 46-89, https://doi.org/10.1137/1019005. (Cited on pp. 27, 29)

[32] J. E. Dennis, Jr., and R. B. Schnabel, *Numerical Methods for Unconstrained Optimization and Nonlinear Equations*, SIAM, Philadelphia, PA, 1996, https://doi.org/10.1137/ 1.9781611971200. (Cited on pp. 15, 28)

[33] G. Deslauries and S. Dubuc, *Le calcul de la racine cubique selon Heron*, Elem. Math., 51 (1996), pp. 28-34, http://eudml.org/doc/141587 (accessed 2020/02/18). (Cited on p. 30)

[34] P. Deuflhard, *Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms*, Springer-Verlag, Heidelberg, 2004. (Cited on pp. 23, 24)

[35] P. Deuflhard and F. A. Potra, *Asymptotic mesh independence of Newton-Galerkin methods via a refined Mysovskii theorem*, SIAM J. Numer. Anal., 29 (1992), pp. 1395-1412, https://doi.org/10.1137/0729080. (Cited on p. 28)

[36] P. Diez, *A note on the convergence of the secant method for simple and multiple roots*, Appl. Math. Lett., 16 (2003), pp. 1211-1215, https://doi.org/10.1016/s0893-9659(03)90119-4. (Cited on pp. 27, 31)

[37] F. Dubeau and C. Gnang, *Fixed point and Newton’s methods for solving a nonlinear equation: From linear to high-order convergence*, SIAM Rev., 56 (2014), pp. 691-708, https://doi.org/10.1137/130934799. (Cited on pp. 25, 34)

[38] J. Fourier, *Question d’analyse algebrique*, Bull. Sci. Soc. Philo., 67 (1818), pp. 61-67, http: //gallica.bnf.fr/ark:/12148/bpt6k33707/f248.item (accessed 2018-02-23). (Cited on pp. 7, 28)

[39] W. Gautschi, *Numerical Analysis,* 2nd ed., Birkhauser/Springer, New York, 2011, https: //doi.org/10.1007/978-0-8176-8259-0. (Cited on p. 23)

[40] A. Greenbaum and T. P. Chartier, *Numerical Methods. Design, Analysis, and Computer Implementation of Algorithms,* Princeton University Press, Princeton, NJ, 2012. (Cited on pp. 23, 27, 31)

[41] M. T. Heath, *Scientific Computing. An Introductory Survey, *2nd ed., SIAM, Philadelphia, PA, 2018, https://doi.org/10.1137/1.9781611975581. (Cited on pp. 9, 29)

[42] M. Heinkenschloss, *Scientific Computing. An Introductory Survey*, Rice University, Houston, TX, 2018. (Cited on pp. 3, 15, 19)

[43] J. Herzberger and L. Metzner, *On the Q-order of convergence for coupled sequences arising in iterative numerical processes*, Computing, 57 (1996), pp. 357-363, https: //doi.org/10.1007/BF02252254. (Cited on p. 13)

[44] N. J. Higham, *Accuracy and Stability of Numerical Algorithms, *2nd ed., SIAM, Philadelphia, PA, 2002, https://doi.org/10.1137/1.9780898718027. (Cited on p. 3)

[45] M. Igarashi and T. J. Ypma, *Empirical versus asymptotic rate of convergence of a class of methods for solving a polynomial equation*, J. Comput. Appl. Math., 82 (1997), pp. 229-237, https://doi.org/10.1016/s0377-0427(97)00077-0. (Cited on p. 25)

[46] L. O. Jay, *A note on Q-order of convergence*, BIT, 41 (2001), pp. 422-429, https://doi.org/ 10.1023/A:1021902825707. (Cited on pp. 5, 11, 12, 17)

[47] T. A. Jeeves, *Secant modification of Newton’s method*, Commun. ACM, 1 (1958), pp. 9-10, https://doi.org/10.1145/368892.368913. (Cited on pp. 11, 30)

[48] V. J. Katz, *A History of Mathematics. An Introduction, *2nd ed., Addison-Wesley, 1998. (Cited on p. 21)

[49] C. T. Kelley, *Iterative Methods for Linear and Nonlinear Equations*, SIAM, Philadelphia, PA, 1995, https://doi.org/10.1137/1.9781611970944. (Cited on pp. 5, 10, 17)

[50] C. T. Kelley, *Iterative Methods for Optimization*, SIAM, Philadelphia, PA, 1999, https: //doi.org/10.1137/1.9781611970920. (Cited on pp. 15, 17)

[51] E. S. Kennedy and W. R. Transue, *A medieval iterative algorism*, Amer. Math. Monthly, 63 (1956), pp. 80-83, https://doi.org/10.1080/00029890.1956.11988762. (Cited on p. 34)

[52] D. E. Knuth, *Big Omicron and big Omega and big Theta*, ACM SIGACT News, 8 (1976), pp. 18-24, https://doi.org/10.1145/1008328.1008329. (Cited on pp. 2, 14)

[53] D. E. Knuth, *The TeXbook, *Addison-Wesley, 1984. (Cited on p. 4)

[54] N. Kollerstrom, *Thomas Simpson and Newton’s method of approximation: an enduring myth*, British J. History Sci., 25 (1992), pp. 347-354, https://doi.org/10.1017/ s0007087400029150. (Cited on pp. 22, 23)

[55] K. Liang, *Homocentric convergence ball of the secant method*, Appl. Math. Chin. Univ., 22 (2007), pp. 353–365, https://doi.org/10.1007/s11766-007-0313-3. (Cited on p. 33)

[56] D. Luca and I. Pavaloiu, *On the Heron’s method for approximating the cubic root of a real number,* Rev. Anal. Numer. Theor. Approx., 26 (1997), pp. 103-108, https://ictp.acad. ro/jnaat/journal/article/view/1997-vol26-nos1-2-art15 (accessed 2020-03-21). (Cited on p. 30)

[57] M. Grau-Sanchez, M. Noguera, and J. M. Gutierrez, *On some computational orders of convergence*, Appl. Math. Lett., 23 (2010), pp. 472-478, https://doi.org/10.1016/j.aml. 2009.12.006. (Cited on p. 20)

[58] Mathworks*, MATLAB, Version 2019b*, 2019, http://www.mathworks.com (accessed 2019/12/12). (Cited on pp. 4, 26)

[59] J. M. McNamee, *Numerical Methods for Roots of Polynomials, Part I, *Elsevier, Amsterdam, 2007. (Cited on p. 23)

[60] J. M. McNamee and V. Y. Pan, *Numerical Methods for Roots of Polynomials, Part II*, Elsevier, Amsterdam, 2013. (Cited on pp. 28, 33)

[61] U. C. Merzbach and C. B. Boyer, *A History of Mathematics*, 3rd ed., John Wiley \& Sons, 2011. (Cited on p. 30)

[62] J.-M. Muller, N. Brunie, F. de Dinechin, C.-P. Jeannerod, M. Joldes, V. Lefevre, G. Melquiond, N. Revol, and S. Torres, *Handbook of Floating-Point Arithmetic, *2nd ed., Springer, New York, 2018, https://doi.org/10.1007/978-3-319-76526-6. (Cited on pp. 22, 24)

[63] A. S. Nemirovsky and D. B. Yudin, *Problem Complexity and Method Efficiency in Optimization*, Wiley, 1983. (Cited on p. 15)

[64] A. Neumaier, *Introduction to Numerical Analysis*, Cambridge University Press, 2001. (Cited on pp. 31, 33)

[65] J. Nocedal and S. J. Wright, *Numerical Optimization, *2nd ed.*,* Springer, New York, 2006, https://doi.org/10.1007/978-0-387-40065-5. (Cited on pp. 8, 17)

[66] J. M. Ortega, *Numerical Analysis. A Second Course*, SIAM, Philadelphia, PA, 1990, https: //doi.org/10.1137/1.9781611971323. (Cited on pp. 28, 34)

[67] J. M. Ortega and W. C. Rheinboldt, *Iterative Solution of Nonlinear Equations in Several Variables*, SIAM, Philadelphia, PA, 2000, https://doi.org/10.1137/1.9780898719468. (Cited on pp. 8, 10, 11, 14, 15, 16, 18, 21, 27, 34)

[68] A. M. Ostrowski, *Solution of Equations and Systems of Equations, *2nd ed.*,* Academic Press, New York, 1966; 3rd ed., 1972. (Cited on pp. 27, 30, 34)

[69] M. L. Overton, *Numerical Computing with IEEE Floating Point Arithmetic*, SIAM, Philadelphia, PA, 2001, https://doi.org/10.1137/1.9780898718072. (Cited on pp. 3, 29)

[70] J. M. Papakonstantinou and R. A. Tapia, *Origin and evolution of the secant method in one dimension*, Amer. Math. Monthly, 120 (2013), pp. 500-518, https://doi.org/10.4169/ amer.math.monthly.120.06.500. (Cited on p. 30)

[71] K. Plofker, *An example of the secant method of iterative approximation in a fifteenth century sanskrit text*, Historia Math., 23 (1996), pp. 246-256, https://doi.org/10.1006/ hmat.1996.0026. (Cited on pp. 30, 34)

[72] K. Plofker, *Use and transmission of iterative approximations in India and the Islamic World*, in From China to Paris: 2000 Years Transmission of Mathematical Ideas, Y. Dold-Samplonius, J. W. Dauben, M. Folkerts, and B. Van Dalen, eds., Franz Steiner Verlag Wiesbaden GmbH, Stuttgart, 2002, pp. 167-186. (Cited on pp. 21, 22, 30, 34)

[73] E. Polak, *Optimization. Algorithms and Consistent Approximations*, Springer-Verlag, New York, 1997. (Cited on pp. 5, 11, 15, 19)

[74] F. A. Potra, *On Q-order and R-order of convergence*, J. Optim. Theory Appl., 63 (1989), pp. 415-431, https://doi.org/10.1007/bf00939805. (Cited on pp. 8, 11, 13, 14, 15, 16, 17, 18, 21) [75] F. A. Potra, *Q-superlinear convergence of the iterates in primal-dual interior-point methods*, Math. Program. Ser. A, 91 (2001), pp. 99-115, https://doi.org/10.1007/s101070100230. (Cited on p. 15)

[76] F. A. Potra, *A superquadratic version of Newton’s method*, SIAM J. Numer. Anal., 55 (2017), pp. 2863-2884, https://doi.org/10.1137/17M1121056. (Cited on p. 13)

[77] F. A. Potra and V. Ptak, *Nondiscrete Induction and Iterative Processes*, Pitman, Boston, MA, 1984. (Cited on pp. 8, 11, 13, 15, 19)

[78] F. A. Potra and R. Sheng, *A path following method for LCP with superlinearly convergent iteration sequence*, Ann. Oper. Res., 81 (1998), pp. 97-11, https://doi.org/10.1023/A: 1018942131812. (Cited on p. 14)

[79] I. Pavaloiu, *Sur l’ordre de convergence des methodes d’iteration*, Mathematica, 23(46) (1981), pp. 261-272, https://doi.org/10.1007/BF02576543. (Cited on p. 14)

[80] I. Pavaloiu, *Optimal algorithms concerning the solving of equations by interpolation*, Ed. Srima, (1999), pp. 222-248, http://ictp.acad.ro/optimal-algorithms-concerning-solvingequations-interpolation/ (accessed 2018-03-22). (Cited on p. 13)

[81] A. Quarteroni, R. Sacco, and F. Saleri, *Numerical Mathematics*, Springer, Berlin, 2007. (Cited on pp. 28, 29)

[82] R. Rashed, *The Development of Arabic Mathematics: Between Arithmetic and Algebra*, Stud. Philos. Sci. 156, Springer, Boston, MA, 1994. (Cited on pp. 22, 30)

[83] M. Raydan, *Exact order of convergence of the secant method*, J. Optim. Theory Appl., 78 (1993), pp. 541-551, https://doi.org/10.1137/060670341. (Cited on pp. 25, 31)

[84] W. C. Rheinboldt, *An adaptive continuation process for solving systems of nonlinear equations, *in Mathematical Models and Numerical Methods, Banach Ctr. Publ. 3, Polish Academy of Science, 1977, pp. 129-142, https://doi.org/10.4064/-3-1-129-142. (Cited on p. 28)

[85] W. C. Rheinboldt, *Methods for Solving Systems of Nonlinear Equations, *2nd ed., SIAM, Philadelphia, PA, 1998, https://doi.org/10.1137/1.9781611970012. (Cited on pp. 11, 15)

[86] T. Sauer, *Numerical Analysis, *2nd ed., Pearson, Boston, MA, 2012. (Cited on pp. 21, 29)

[87] E. Schroder, *Ueber unendlich viele algorithmen zur auflosung der gleichungen*, Math. Ann., 2 (1870), pp. 317-365, https://doi.org/10.1007/bf01444024; available as On Infinitely Many Algorithms for Solving Equations, Tech reports TR-92-121 and TR-2990, Institute for Advanced Computer Studies, University of Maryland, 1992; translated by G. W. Stewart. (Cited on pp. 7, 34)

[88] H. Schwetlick, *Numerische Losung Nichtlinearer Gleichungen*, VEB, Berlin, 1979. (Cited on pp. 8, 13, 14, 16)

[89] L. F. Shampine, R. C. Allen, Jr., and S. Pruess, *Fundamentals of Numerical Computing*, John Wiley and Sons, New York, 1997. (Cited on pp. 29, 33)

[90] T. Steihaug, *Computational science in the eighteenth century. Test cases for the methods of Newton*, *Raphson, and Halley: 1685 to 1745*, Numer. Algor., 83 (2020), pp. 1259-1275, https://doi.org/10.1007/s11075-019-00724-8. (Cited on pp. 22, 23)

[91] E. Suli and D. Mayers, *An Introduction to Numerical Analysis*, Cambridge University Press, Cambridge, UK, 2003. (Cited on p. 33)

[92] T. Tantau, *The tikz and pgf Packages*. Manual for Version 3.1.5b-34-gff02ccd1, https:// github.com/pgf-tikz/pgf. (Cited on p. 4)

[93] R. A. Tapia, J. E. Dennis, Jr., and J. P. Schafermeyer,* Inverse, shifted inverse, and Rayleigh quotient iteration as Newton’s method*, SIAM Rev., 60 (2018), pp. 3-55, https: //doi.org/10.1137/15M1049956. (Cited on pp. 7, 16, 17, 18)

[94] R. A. Tapia and D. L. Whitley, *The projected Newton method has order 1 + √2 for the symmetric eigenvalue problem*, SIAM J. Numer. Anal., 25 (1989), pp. 1376-1382, https: //doi.org/10.1137/0725079. (Cited on p. 13)

[95] J. F. Traub, *Iterative Methods for the Solutions of Equations*, Prentice Hall, Englewood Cliffs, NJ, 1964. (Cited on pp. 13, 31, 34)

[96] E. E. Tyrtyshnikov, *A Brief Introduction to Numerical Analysis*, Birkhauser/Springer, New York, 1997. (Cited on p. 24)

[97] J. S. Vandergraft, *Introduction to Numerical Computations, *2nd ed., Academic Press, New York, 1983. (Cited on p. 33)

[98] H. F. Walker, *An approach to continuation using Krylov subspace methods*, in Computational Science in the 21st Century, J. Periaux, ed., John Wiley and Sons, 1997, pp. 72-81. (Cited on p. 19)

[99] S. J. Wright, *Primal-Dual Interior-Point Methods*, SIAM, Philadelphia, PA, 1997, https: //doi.org/10.1137/1.9781611971453. (Cited on p. 12)

[100] S. J. Wright, *Optimization algorithms for data analysis,* in The Mathematics of Data, IAS/Park City Math. Ser. 25, AMS, Providence, RI, 2018, pp. 49-97. (Cited on p. 15)

[101] T. J. Ypma, *Historical development of the Newton-Raphson method*, SIAM Rev., 37 (1995), pp. 531–551, https://doi.org/10.1137/1037125. (Cited on pp. 7, 21, 22, 23, 30, 34)