{"title":"斐波那契-曼德勃洛特多项式和矩阵","authors":"Eunice Y. S. Chan, Robert M Corless","doi":"10.1145/3055282.3055288","DOIUrl":null,"url":null,"abstract":"We explore a family of polynomials similar to the Mandelbrot polynomials called the Fibonacci-Mandelbrot polynomials defined by <i>q</i><sub>0</sub>(<i>z</i>) = 0, <i>q</i><sub>1</sub>(<i>z</i>) = 1, and <i>q<sub>n</sub></i>(<i>z</i>) = <i>zq</i><sub><i>n</i>−1</sub><i>q</i><sub><i>n</i>−2</sub> + 1. We compute the roots of the Fibonacci-Mandelbrot polynomials using two methods. One method uses a recursively constructed matrix, where elements are 0, 1, or −1, whose eigenvalues are the roots of <i>q<sub>n</sub></i>(<i>z</i>). The other method uses a special-purpose homotopy continuation method, where the solution of the differential equation, [EQUATION], in which the initial condition are 0, and the roots of <i>q</i><sub><i>n</i>−1</sub> and <i>q</i><sub><i>n</i>−2</sub>, are also the roots of the Fibonacci-Mandelbrot polynomials.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"92 1","pages":"155-157"},"PeriodicalIF":0.0000,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Fibonacci-mandelbrot polynomials and matrices\",\"authors\":\"Eunice Y. S. Chan, Robert M Corless\",\"doi\":\"10.1145/3055282.3055288\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We explore a family of polynomials similar to the Mandelbrot polynomials called the Fibonacci-Mandelbrot polynomials defined by <i>q</i><sub>0</sub>(<i>z</i>) = 0, <i>q</i><sub>1</sub>(<i>z</i>) = 1, and <i>q<sub>n</sub></i>(<i>z</i>) = <i>zq</i><sub><i>n</i>−1</sub><i>q</i><sub><i>n</i>−2</sub> + 1. We compute the roots of the Fibonacci-Mandelbrot polynomials using two methods. One method uses a recursively constructed matrix, where elements are 0, 1, or −1, whose eigenvalues are the roots of <i>q<sub>n</sub></i>(<i>z</i>). The other method uses a special-purpose homotopy continuation method, where the solution of the differential equation, [EQUATION], in which the initial condition are 0, and the roots of <i>q</i><sub><i>n</i>−1</sub> and <i>q</i><sub><i>n</i>−2</sub>, are also the roots of the Fibonacci-Mandelbrot polynomials.\",\"PeriodicalId\":7093,\"journal\":{\"name\":\"ACM Commun. Comput. Algebra\",\"volume\":\"92 1\",\"pages\":\"155-157\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2017-02-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACM Commun. Comput. Algebra\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3055282.3055288\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Commun. Comput. Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3055282.3055288","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We explore a family of polynomials similar to the Mandelbrot polynomials called the Fibonacci-Mandelbrot polynomials defined by q0(z) = 0, q1(z) = 1, and qn(z) = zqn−1qn−2 + 1. We compute the roots of the Fibonacci-Mandelbrot polynomials using two methods. One method uses a recursively constructed matrix, where elements are 0, 1, or −1, whose eigenvalues are the roots of qn(z). The other method uses a special-purpose homotopy continuation method, where the solution of the differential equation, [EQUATION], in which the initial condition are 0, and the roots of qn−1 and qn−2, are also the roots of the Fibonacci-Mandelbrot polynomials.