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The update rule is conceptually very simple, using the projections to the vector subspaces generated by eigenvectors of positive (correspondingly negative) eigenvalues of the Hessian. The main result of this paper roughly says that if a sequence <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{x_n\\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>}<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> constructed by the method from a random initial point <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula><jats:bold>converges<\/jats:bold>, then the limit point is a critical point and not a saddle point, and the convergence rate is the same as that of Newton\u2019s method. A subsequent work has recently been successful incorporating Backtracking line search to New Q-Newton\u2019s method, thus resolving the global convergence issue observed for some (non-smooth) functions. An application to quickly find zeros of a univariate meromorphic function is discussed, accompanied with an illustration on basins of attraction.\n<\/jats:p>","DOI":"10.1007\/s10957-023-02270-9","type":"journal-article","created":{"date-parts":[[2023,7,18]],"date-time":"2023-07-18T19:02:35Z","timestamp":1689706955000},"page":"805-830","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Fast and Simple Modification of Newton\u2019s Method Avoiding Saddle Points"],"prefix":"10.1007","volume":"199","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9103-0923","authenticated-orcid":false,"given":"Tuyen Trung","family":"Truong","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tat Dat","family":"To","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hang-Tuan","family":"Nguyen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Thu Hang","family":"Nguyen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hoang Phuong","family":"Nguyen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maged","family":"Helmy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,18]]},"reference":[{"issue":"2","key":"2270_CR1","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1137\/040605266","volume":"16","author":"P-A Absil","year":"2005","unstructured":"Absil, P.-A., Mahony, R., Andrews, B.: Convergence of the iterates of descent methods for analytic cost functions. 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