COMPUTER TOOLS FOR SOLVING MATHEMATICAL PROBLEMS: A REVIEW

Ivan M. Petković, Đorđe Herceg

DOI Number
https://doi.org/10.22190/FUMI201203017P
First page
205
Last page
236

Abstract


The rapid development of digital computer hardware and software has had a dramatic influence on mathematics, and contrary. The advanced hardware and modern sophistical software such as computer visualization, symbolic computation, computerassisted proofs, multi-precision arithmetic and powerful libraries, have provided resolving many open problems, a huge very difficult mathematical problems, and discovering new patterns and relationships, far beyond a human capability. In the first part of the paper we give a short review of some typical mathematical problems solved by computer tools. In the second part we present some new original contributions, such as intriguing consequence of the presence of roundoff errors, distribution of zeros of random polynomials, dynamic study of zero-finding methods, a new three-point family of methods for solving nonlinear equations and two algorithms for the inclusion of a simple complex zero of a polynomial.

Keywords

Experimental mathematics, computer graphics, symbolic computation, visualization of iterative processes, interval arithmetic, roundoff error

Full Text:

PDF

References


bibitem{mancosu} {sc P. Mancosu}: textit{The Philosophy of Mathematical Practice}, Oxford University Press, 2008.

bibitem{hales-AMS} {sc T. C Hales}: textit{Cannonballs and honeycombs}.

Notices Amer. Math. Soc. {bf 47} (April 2000), 440--449.

bibitem{AMS} {sc M. S. Petkovi'c}: textit{Famous Puzzles of Great Mathematicians}. American Mathematical Society, 2008.

bibitem{zarnke} {sc H. C. Williams, R. A. German {rm and} C. R. Zarnke}:

textit{Solution of the cattle problem of Archimedes}. Math. Comp. {bf 10} (1965), 671--674.

bibitem{apel-haken-1} {sc K. Appel {rm and} W. Haken}: textit{Every planar map is four colorable, Part I, Discharging}. Illinois J. Math. {bf 21} (1977), 429--490.

bibitem{apel-haken-2} {sc K. Appel {rm and} W. Haken}, textit{Every planar map is four colorable, Part I, Reducibility}. Illinois J. Math. {bf 21} (1977), 491--567.

bibitem{gontie} {sc G. Gonthier}: textit{Formal proof -- The Four-Color Theorem}. Notices Amer. Math. Soc. {bf 55} (Dec. 2008), 1382--1393.

bibitem{hales} {sc T. C. Hales}: textit{A proof of the Kepler conjecture,

(unabridged)}. Discrete Comput. Geom. {bf 36} (2006),

--265.

bibitem{rump-2} {sc B. M. Brown, D. K. R. McCormack {rm and} A. Zettl}: textit{On the existence of an eigenvalue below the essential spectrum}. Proc. R. Soc. Lond. {bf 455} (1999), 2229--2234.

bibitem{rump-4} {sc J.-P. Eckmann {rm and} P. Wittwer}: textit{Computer Methods and Borel Summability Applied Feigenbaum's Equation}. Lecture Notes in Physics, Vol. 227, Springer Verlag, Berlin-Heidelberg-New York-Tokyo, 1985.

bibitem{rump-7} {sc J. Hass, M. Hutchings {rm and} R. Schlafy}: textit{The Double Bubble Conjecture}. Elec. Res. Announcement of the Am. Math. Soc. {bf 1} (1995), 98--102.

bibitem{rump-10} {sc K. Mischaikow {rm and} M. Mrozek}: textit{Chaos in the Lorenz equations: A computer assisted proof. Part II: Details}. Math. Comput. {bf 67} (1998), 1023--1046.

bibitem{rump-13} {sc A. Neumaier {rm and}

Th. Rage}: textit{Rigorous chaos verification in discrete dynamical systems}. Physica D, {bf 67} (1993), 327--346.

bibitem{taker} {sc W. Tucker}: textit{A rigorous ODE solver and Smale's 14th problem}. Found. Comput. Math. {bf 2} (2002), 53--117.

bibitem{BBP} {sc D. Bailey, P. Borwein {rm and} S. Plouffe}: textit{On the rapid computation of various polylogarithmic constants}. Math. Comp. {bf 66} (1997), 903--913.

bibitem{MBE} {sc J. Borwein {rm and} D. Bailey}: textit{Mathematics by Experiment}. A K Peters, Wellesley, Massachusetts, 2008.

bibitem{rump} {sc S.M. Rump}: textit{INTLAB - INTerval LABoratory.} In: Proceeding of the Conference on Developement in Reliable Computing (Scendes, T., ed.), Kluwer Academic Publishers, 1999, pp. 77--104. {tt http://www.ti3.tu-harburg.de/rump/intlab/index.html.}

bibitem{kulis} {sc U. Kulisch}: textit{Computer Arithmetic and Validity}. Walter de Gruyter, Berlin-New York, 2008.

bibitem{EIM} {sc J. Borwein, D. Bailey {rm and} R. Girgensohn}: textit{Experimentation in Mathematics}. A K Peters, Natick, Massachusetts, 2004.

bibitem{EMA} {sc D. Bailey, J. M. Borwein, N. J. Calcin, R. Girgensohn, D. R. Luke {rm and} V. H. Moll}: textit{Experimental Mathematics in Action}. A K Peters, Wellesley, Massachusetts, 2007.

bibitem{dongara} {sc J. Dongarra {rm and} F. Sullivan}: textit{The top 10 algorithms}. Computing in Science and Engineering, Jan./Feb (2000), 22--23.

bibitem{IVAN-DD} {sc I. Petkovi'c}: textit{Analysis of Processing and Computing Iterations by Applying Modern Computer Arithmetics}. Ph. D. Thesis, Faculty of Electronic Engineering, University of Niv s, Niv s, 2012.

bibitem{random-knjiga} {sc A.T. Bharucha-Reid {rm and} M. Sambandham}, textit{Random Polynomials}. Academic Press, 1986.

bibitem{random-rad} {sc I. Ibragimov {rm and} D. Zaporozhets}: textit{On distribution of zeros of random polynomials in complex plane}. arXiv:1102.3517v1 [math.PR] 17 Feb 2011.]

bibitem{henrici} {sc P. Henrici}: textit{Applied and Computational Complex

Analysis,

Vol. I. John Wiley and Sons}. New York, 1974.

bibitem{stjuart-b} {sc B. D. Stewart}: textit{Attractor Basins of Various Root-finding Methods}. Master Thesis, Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA, June 2001.

bibitem{varona} {sc J. L. Varona}: textit{Graphic and numerical comparison between iterative methods}. Math. Intelligencer {bf 24} (2002), 37--46.

bibitem{amat1} {sc S. Amat, S. Busquier {rm and} S. Plaza}, textit{Review of some iterative root-finding methods from a dynamical point of view}. Scientia {bf 10} (2004), 3--35.

bibitem{amat2} {sc S. Amat, S. Busquier {rm and} S. Plaza}: textit{Dynamics of a family of third-order iterative methods that do not require using second derivatives}. Appl. Math. Comput. {bf 154} (2004), 735--746.

bibitem{amat3} {sc S. Amat, S. Busquier {rm and} S. Plaza}, textit{Dynamics of the King and Jarratt iterations}. Aequ. Math. {bf 69} (2005), 212--223.

bibitem{CN2} {sc M. Scott, B. Neta {rm and} C. Chun}: textit{Basin attractors for various methods}. Appl. Math. Comput. {bf 218} (2011), 2584--2599.

bibitem{CN7} {sc C. Chun {rm and} B. Neta}: textit{Basins of attraction for

Zhou-Chen-Song fourth order family of methods for multiple roots}. Math. Comput. Simul. {bf 109} (2015), 74--91.

bibitem{CN1} {sc C. Chun {rm and} B. Neta}: textit{Basins of attraction for several third order methods to find multiple roots of nonlinear equations}. Appl. Math. Comput. {bf 268} (2015), 129--137.

bibitem{NSC} {sc B. Neta, M. Scott {rm and} C. Chun}: textit{Basin of attractions for several methods to find simple roots of nonlinear equations}. Appl. Math. Comput. {bf 218} (2012), 10548--10556.

bibitem{argiros} {sc I. K. Argyros {rm and} 'A. A. Magre~{n}'an}: textit{n the convergence of an optimal fourth-order family of methods and its dynamics}. Appl. Math. Comput. {bf 252} (2015), 336--346.

bibitem{kalantari} {sc B. Kalantari}: textit{Polynomial Root-Finding and Polynomiography}. World Scientific, New Jersey, 2009.

bibitem{IVAN-JCAM-2016} {sc I. Petkovi'c {rm and} B. Neta}: textit{On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity}.

J. Comput. Appl. Math. {bf 308} (2016), 215--230.

bibitem{ivan-herceg} {sc I. Petkovi'c {rm and} D{D}. Herceg}: textit{Symbolic computation and computer graphics as tools for developing and studying new root-finding methods}. Appl. Math. Comput. {bf 295} (2017), 95--113.

bibitem{hajem} {sc N. Higham}: textit{Accuracy and Stability of Numerical Algorithms}. SIAM, Philadelpha, 2002.

bibitem{SA-2016} {sc J.R. Sharma {rm and} H. Arora}: textit{A new family of optimal eighth order methods with dynamics for nonlinear equations}. Appl. Math. Comput. {bf 273} (2016), 924--933.

bibitem{ivan-NUMA} {sc I. Petkovi'c {rm and} D{D}. Herceg}: textit{Computers in mathematical research: the study of three-point root-finding methods}. Numer. Algor. doi.org/10.1007/s11075-019-00796-6 (online version).

bibitem{JML} {sc J. Dv zuni'c, M.S. Petkovi'c {rm and} L.D.

Petkovi'c}: textit{A family of optimal three-point methods for

solving nonlinear equations using two parametric functions}. Appl.

Math. Comput. {bf 217} (2011), 7612--7619.

bibitem{CN-2017} {sc C. Chun {rm and} B. Neta}: textit{Comparative study of eighth-order methods for finding simple roots of nonlinear equations}. Numer. Algor. {bf 74} (2017), 1169--1201.

bibitem{KT} {sc H.T. Kung, J.F. Traub}: textit{Optimal order of one-point and multipoint iteration}. Journal of the ACM {bf 21} (1974), 643--651.

bibitem{mur1} {sc R.E. Moore}: textit{Interval Analysis}. Prentice Hall, Englewood Cliff, New Jersey, 1966.

bibitem{GH} {sc I. Gargantini, P. Henrici}: textit{Circular arithmetic and the determination of polynomial zeros}. Numer. Math. {bf 18} (1972), 305--320.

bibitem{herceg-rad} {sc Dj. Herceg}: textit{An algorithm for localization of polynomial zeros}. In: Proc. VIII International Conf. on Logic and Computer Science (R. Tov si'c, Z. Budimac, eds.), Institute of Mathematics, University of Novi Sad, Novi Sad, 1997, pp. 67--75.

bibitem{herceg-master} {sc Dj. Herceg}: textit{Computer Implementation and Analysis of Iterative Methods for Solving Nonlinear Equations} (in Serbian). Master Thesis, Faculty of Science, University of Novi Sad, Novi Sad, 1997.

bibitem{CAMWA} {sc M.S. Petkovi'c}: textit{Some interval iterations

for finding a zero of a polynomial with error bounds}. Comput.

Math. Appls. {bf 14} (1987), 479--495.




DOI: https://doi.org/10.22190/FUMI201203017P

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)