{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T17:18:31Z","timestamp":1773422311909,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,6,17]],"date-time":"2020-06-17T00:00:00Z","timestamp":1592352000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,6,17]],"date-time":"2020-06-17T00:00:00Z","timestamp":1592352000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2021,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The aim of this paper is to visually investigate the dynamics and stability of the process in which the classic derivative is replaced by the fractional Riemann\u2013Liouville or Caputo derivatives in the standard Newton root-finding method. Additionally, instead of the standard Picard iteration, the Mann, Khan, Ishikawa and S iterations are used. This process when applied to polynomials on complex plane produces images showing basins of attractions for polynomial zeros or images representing the number of iterations required to achieve any polynomial root. The images are called polynomiographs. In this paper, we use the colouring according to the number of iterations which reveals the speed of convergence and dynamic properties of processes visualised by polynomiographs. Moreover, to investigate the stability of the methods, we use basins of attraction. To compare numerically the modified root-finding methods among them, we demonstrate their action for polynomial<jats:italic>z<\/jats:italic><jats:sup>3<\/jats:sup>\u2212\u20091 on a complex plane.<\/jats:p>","DOI":"10.1007\/s11075-020-00919-4","type":"journal-article","created":{"date-parts":[[2020,6,17]],"date-time":"2020-06-17T07:02:49Z","timestamp":1592377369000},"page":"953-1010","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["Newton\u2019s method with fractional derivatives and various iteration processes via visual analysis"],"prefix":"10.1007","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9434-9307","authenticated-orcid":false,"given":"Krzysztof","family":"Gdawiec","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5920-5161","authenticated-orcid":false,"given":"Wies\u0142aw","family":"Kotarski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0492-772X","authenticated-orcid":false,"given":"Agnieszka","family":"Lisowska","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,6,17]]},"reference":[{"key":"919_CR1","first-page":"205","volume":"2","author":"N Abel","year":"1823","unstructured":"Abel, N.: Opl\u00f8sning af et par opgaver ved hj\u00e6lp af bestemte integraler. Magazin for Naturvidenskaberne 2, 205\u2013215 (1823)","journal-title":"Magazin for Naturvidenskaberne"},{"issue":"1","key":"919_CR2","first-page":"61","volume":"8","author":"R Agarwal","year":"2007","unstructured":"Agarwal, R., O\u2019Regan, D., Sahu, D. R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex A. 8 (1), 61\u201379 (2007)","journal-title":"J. Nonlinear Convex A."},{"key":"919_CR3","doi-asserted-by":"publisher","first-page":"344","DOI":"10.1016\/j.aml.2019.06.028","volume":"98","author":"A Akg\u00fcl","year":"2019","unstructured":"Akg\u00fcl, A., Cordero, A., Torregrosa, J.: A fractional Newton method with 2\u03b1 th-order of convergence and its stability. Appl. Math. Lett. 98, 344\u2013351 (2019). https:\/\/doi.org\/10.1016\/j.aml.2019.06.028","journal-title":"Appl. Math. Lett."},{"issue":"2","key":"919_CR4","doi-asserted-by":"crossref","first-page":"135","DOI":"10.37193\/CMI.2016.02.03","volume":"25","author":"G Ardelean","year":"2016","unstructured":"Ardelean, G., Balog, L.: A qualitative study of Agarwal others. iteration procedure for fixed points approximation. Creative Math. Inf. 25(2), 135\u2013139 (2016)","journal-title":"Creative Math. Inf."},{"issue":"3","key":"919_CR5","doi-asserted-by":"crossref","first-page":"277","DOI":"10.37193\/CJM.2016.03.03","volume":"32","author":"G Ardelean","year":"2016","unstructured":"Ardelean, G., Cosma, O., Balog, L.: A comparison of some fixed point iteration procedures by using the basins of attraction. Carpathian J. Math. 32(3), 277\u2013284 (2016)","journal-title":"Carpathian J. Math."},{"key":"919_CR6","doi-asserted-by":"publisher","first-page":"358","DOI":"10.1016\/j.cam.2014.09.020","volume":"291","author":"D Babajee","year":"2016","unstructured":"Babajee, D., Cordero, A., Torregrosa, J.: Study of iterative methods through the Cayley quadratic test. J. Comput. Appl. Math. 291, 358\u2013369 (2016). https:\/\/doi.org\/10.1016\/j.cam.2014.09.020","journal-title":"J. Comput. Appl. Math."},{"key":"919_CR7","unstructured":"Brambila-Paz, F., Torres-Hernandez, A.: Fractional newton\u2013Raphson method. arXiv:1710.07634 (2017)"},{"key":"919_CR8","unstructured":"Brambila-Paz, F., Torres-Hernandez, A., Iturrar\u00e1n-Viveros, U.: Caballero-cruz: Fractional newton-Raphson method accelerated with Aitken\u2019s method. arXiv:1804.08445 (2018)"},{"key":"919_CR9","volume-title":"Numer. Analysis","author":"R Burden","year":"2011","unstructured":"Burden, R., Faires, J.: Numer. Analysis, 9th edn. Brooks\u2013Cole, Boston (2011)","edition":"9th edn."},{"key":"919_CR10","unstructured":"Candelario, G., Cordero, A., Torregrosa, J.: A fractional Traub method with (2\u03b1 +\u20091)th-order of convergence and its stability. arXiv:1909.09076v1 (2019)"},{"key":"919_CR11","doi-asserted-by":"publisher","first-page":"762","DOI":"10.1016\/j.amc.2015.10.092","volume":"274","author":"C Chun","year":"2016","unstructured":"Chun, C., Neta, B.: Comparison of several families of optimal eighth order methods. Appl. Math. Comput. 274, 762\u2013773 (2016). https:\/\/doi.org\/10.1016\/j.amc.2015.10.092","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"919_CR12","doi-asserted-by":"publisher","first-page":"636","DOI":"10.1007\/s11075-015-0048-9","volume":"72","author":"C Chun","year":"2016","unstructured":"Chun, C., Neta, B.: On the new family of optimal eighth order methods developed by Lotfi others. Numer. Algorithms 72(2), 636\u2013376 (2016). https:\/\/doi.org\/10.1007\/s11075-015-0048-9","journal-title":"Numer. Algorithms"},{"key":"919_CR13","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.matcom.2015.08.012","volume":"119","author":"A Cordero","year":"2016","unstructured":"Cordero, A., Ferrero, A., Torregrosa, J.: Damped Traub\u2019s method: Convergence and stability. Math. Comput. Simul. 119, 57\u201368 (2016). https:\/\/doi.org\/10.1016\/j.matcom.2015.08.012","journal-title":"Math. Comput. Simul."},{"issue":"8","key":"919_CR14","doi-asserted-by":"publisher","first-page":"Article 1017","DOI":"10.3390\/sym11081017","volume":"11","author":"A Cordero","year":"2019","unstructured":"Cordero, A., Girona, I., Torregrosa, J.: A variant of Chebyshev\u2019s method with 3\u03b1 th-order of convergence by using fractional derivatives. Symmetry 11(8), Article 1017 (2019). https:\/\/doi.org\/10.3390\/sym11081017","journal-title":"Symmetry"},{"issue":"4","key":"919_CR15","doi-asserted-by":"publisher","first-page":"2457","DOI":"10.1007\/s11071-017-3813-6","volume":"90","author":"K Gdawiec","year":"2017","unstructured":"Gdawiec, K.: Fractal patterns from the dynamics of combined polynomial root finding methods. Nonlinear Dyn. 90(4), 2457\u20132479 (2017). https:\/\/doi.org\/10.1007\/s11071-017-3813-6","journal-title":"Nonlinear Dyn."},{"issue":"4","key":"919_CR16","doi-asserted-by":"publisher","first-page":"2235","DOI":"10.1007\/s11071-016-3186-2","volume":"87","author":"K Gdawiec","year":"2017","unstructured":"Gdawiec, K.: Switching processes in polynomiography. Nonlinear Dyn. 87(4), 2235\u20132249 (2017). https:\/\/doi.org\/10.1007\/s11071-016-3186-2","journal-title":"Nonlinear Dyn."},{"key":"919_CR17","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/j.amc.2017.02.038","volume":"307","author":"K Gdawiec","year":"2017","unstructured":"Gdawiec, K., Kotarski, W.: Polynomiography for the polynomial infinity norm via Kalantari\u2019s formula and nonstandard iterations. Appl. Math. Comput. 307, 17\u201330 (2017). https:\/\/doi.org\/10.1016\/j.amc.2017.02.038","journal-title":"Appl. Math. Comput."},{"key":"919_CR18","doi-asserted-by":"publisher","first-page":"Article ID 797,","DOI":"10.1155\/2015\/797594","volume":"2015","author":"K Gdawiec","year":"2015","unstructured":"Gdawiec, K., Kotarski, W., Lisowska, A.: Polynomiography based on the non-standard Newton-like root finding methods. Abstract and Applied Analysis 2015, Article ID 797, 594 (2015). https:\/\/doi.org\/10.1155\/2015\/797594","journal-title":"Abstract and Applied Analysis"},{"issue":"9","key":"919_CR19","doi-asserted-by":"publisher","first-page":"Article ID 1143","DOI":"10.3390\/sym11091143","volume":"11","author":"K Gdawiec","year":"2019","unstructured":"Gdawiec, K., Kotarski, W., Lisowska, A.: Visual analysis of the Newton\u2019s method with fractional order derivatives. Symmetry 11(9), Article ID 1143 (2019). https:\/\/doi.org\/10.3390\/sym11091143","journal-title":"Symmetry"},{"issue":"3","key":"919_CR20","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1142\/S0218348X01000737","volume":"9","author":"W Gilbert","year":"2001","unstructured":"Gilbert, W.: Generalizations of Newton\u2019s method. Fractals 9(3), 251\u2013262 (2001). https:\/\/doi.org\/10.1142\/S0218348X01000737","journal-title":"Fractals"},{"key":"919_CR21","doi-asserted-by":"publisher","DOI":"10.1142\/8934","volume-title":"Fractional Calculus: an Introduction for Physicists","author":"R Herrmann","year":"2014","unstructured":"Herrmann, R.: Fractional Calculus: an Introduction for Physicists, 2nd edn. World Scientific, Singapore (2014)","edition":"2nd edn."},{"key":"919_CR22","doi-asserted-by":"publisher","DOI":"10.1142\/3779","volume-title":"Applications of Fractional Calculus in Physics","author":"R Hilfer","year":"2000","unstructured":"Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)"},{"issue":"1","key":"919_CR23","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1090\/S0002-9939-1974-0336469-5","volume":"44","author":"S Ishikawa","year":"1974","unstructured":"Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44(1), 147\u2013150 (1974). https:\/\/doi.org\/10.1090\/S0002-9939-1974-0336469-5","journal-title":"Proc. Am. Math. Soc."},{"key":"919_CR24","unstructured":"Ishteva, M.: Properties and Applications of the Fractional Caputo Operator. Master\u2019s Thesis, Department of Mathematics, Universit\u00e4t Karlsruhe (2005)"},{"issue":"3","key":"919_CR25","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1162\/0024094054029010","volume":"38","author":"B Kalantari","year":"2005","unstructured":"Kalantari, B.: Polynomiography: from the fundamental theorem of algebra to art. Leonardo 38(3), 233\u2013238 (2005). https:\/\/doi.org\/10.1162\/0024094054029010","journal-title":"Leonardo"},{"key":"919_CR26","doi-asserted-by":"crossref","unstructured":"Kalantari, B.: Polynomial Root-Finding and Polynomiography, World Scientific, Singapore (2009)","DOI":"10.1142\/6265"},{"key":"919_CR27","doi-asserted-by":"publisher","first-page":"Article 69","DOI":"10.1186\/1687-1812-2013-69","volume":"2013","author":"S Khan","year":"2013","unstructured":"Khan, S.: A picard-Mann hybrid iterative process. Fixed Point Theory Appl. 2013, Article 69 (2013). https:\/\/doi.org\/10.1186\/1687-1812-2013-69","journal-title":"Fixed Point Theory Appl."},{"key":"919_CR28","doi-asserted-by":"publisher","DOI":"10.1119\/1.13295","volume-title":"The Fractal Geometry of Nature","author":"B Mandelbrot","year":"1983","unstructured":"Mandelbrot, B.: The Fractal Geometry of Nature. W.H. Freeman and Company, New York (1983)"},{"issue":"3","key":"919_CR29","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1090\/S0002-9939-1953-0054846-3","volume":"4","author":"W Mann","year":"1953","unstructured":"Mann, W.: Mean value methods in iteration. Proc. Am. Math. Soc. 4(3), 506\u2013510 (1953). https:\/\/doi.org\/10.1090\/S0002-9939-1953-0054846-3","journal-title":"Proc. Am. Math. Soc."},{"issue":"1","key":"919_CR30","first-page":"Article 6","volume":"6","author":"J Munkhammar","year":"2005","unstructured":"Munkhammar, J.: Fractional calculus and the taylor-Riemann series. Ros-Hulman Undergraduate Math. J. 6(1), Article 6 (2005)","journal-title":"Ros-Hulman Undergraduate Math. J."},{"key":"919_CR31","volume-title":"The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order Mathematics in Science and Engineering, vol. 111","author":"K Oldham","year":"1974","unstructured":"Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order Mathematics in Science and Engineering, vol. 111. Academic Press, San Diego (1974)"},{"issue":"11","key":"919_CR32","doi-asserted-by":"publisher","first-page":"4174","DOI":"10.1016\/j.cnsns.2011.02.022","volume":"16","author":"M Ortigueira","year":"2011","unstructured":"Ortigueira, M., Rodr\u00edguez-Germ\u00e1, L., Trujillo, J.: Complex gr\u00fcnwald-Letnikov, Liouville, riemann-Liouville, and Caputo derivatives for analytic functions. Commun. Nonlinear Sci. Numer. Simul. 16(11), 4174\u20134182 (2011). 10.1016\/j.cnsns.2011.02.022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"4","key":"919_CR33","first-page":"145","volume":"6","author":"E Picard","year":"1890","unstructured":"Picard, E.: M\u00e9moire sur la th\u00e9orie des \u00e9quations aux d\u00e9riv\u00e9es partielles et la m\u00e9thode des approximations successives. J Math. Pure. Appl. 6(4), 145\u2013210 (1890)","journal-title":"J Math. Pure. Appl."},{"key":"919_CR34","volume-title":"Fractional Integrals and Derivatives: Theory and Applications","author":"S Samko","year":"1993","unstructured":"Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Amsterdam (1993)"},{"key":"919_CR35","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.cnsns.2018.04.019","volume":"64","author":"H Sun","year":"2018","unstructured":"Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213\u2013231 (2018). https:\/\/doi.org\/10.1016\/j.cnsns.2018.04.019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"919_CR36","doi-asserted-by":"publisher","first-page":"Article ID 169,","DOI":"10.1155\/2011\/169421","volume":"169","author":"L V\u00e1zquez","year":"2011","unstructured":"V\u00e1zquez, L.: From Newton\u2019s equation to fractional diffusion and wave equations. Adv Differ Equ 169, Article ID 169,421 (2011). https:\/\/doi.org\/10.1155\/2011\/169421","journal-title":"Adv Differ Equ"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00919-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-020-00919-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00919-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,28]],"date-time":"2022-10-28T23:28:51Z","timestamp":1666999731000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-020-00919-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,17]]},"references-count":36,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,3]]}},"alternative-id":["919"],"URL":"https:\/\/doi.org\/10.1007\/s11075-020-00919-4","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,17]]},"assertion":[{"value":"23 December 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 March 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 June 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}]}}