Abstract
We have previously established delicate connections between the Voronoi diagram of polynomial roots and their basins of attraction with respect to the Basic Family of iteration functions. We have also previously defined polynomiography, visualization techniques in solving a polynomial equation, resulting in a medium with multidisciplinary applications. Here we describe several novel results and survey our recent work on polynomials and polynomiography and their extensions and applications. These include, (i) extension of root-finding methods to analytic functions and the analysis of infinite Voronoi diagram and basins of attraction of their roots; (ii) an application of polynomiography in the visualization of least square polynomial approximations; (iii) a Voronoi diagram property of roots of a cubic equation in connection to its critical points and its algorithmic application; (iv) association of polynomiography to special matrices such as permutation matrices, Latin Squares and Sudoku solutions, and alternating sign matrices. We will consider the application of the above and present corresponding polynomiographies.
| Original language | English (US) |
|---|---|
| Title of host publication | Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |
| Pages | 32-40 |
| Number of pages | 9 |
| DOIs | |
| State | Published - Oct 3 2012 |
| Event | 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 - Piscataway, NJ, United States Duration: Jun 27 2012 → Jun 29 2012 |
Publication series
| Name | Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |
|---|
Other
| Other | 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012 |
|---|---|
| Country | United States |
| City | Piscataway, NJ |
| Period | 6/27/12 → 6/29/12 |
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All Science Journal Classification (ASJC) codes
- Geometry and Topology
Keywords
- Basins of attraction
- Complex polynomials
- Dynamical systems
- Iteration functions
- Newton's method
- Voronoi diagrams
- Zeros
Cite this
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Voronoi diagram properties in polynomials with polynomiography applications and extensions. / Kalantari, Bahman.
Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012. 2012. p. 32-40 6257654 (Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012).Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
TY - GEN
T1 - Voronoi diagram properties in polynomials with polynomiography applications and extensions
AU - Kalantari, Bahman
PY - 2012/10/3
Y1 - 2012/10/3
N2 - We have previously established delicate connections between the Voronoi diagram of polynomial roots and their basins of attraction with respect to the Basic Family of iteration functions. We have also previously defined polynomiography, visualization techniques in solving a polynomial equation, resulting in a medium with multidisciplinary applications. Here we describe several novel results and survey our recent work on polynomials and polynomiography and their extensions and applications. These include, (i) extension of root-finding methods to analytic functions and the analysis of infinite Voronoi diagram and basins of attraction of their roots; (ii) an application of polynomiography in the visualization of least square polynomial approximations; (iii) a Voronoi diagram property of roots of a cubic equation in connection to its critical points and its algorithmic application; (iv) association of polynomiography to special matrices such as permutation matrices, Latin Squares and Sudoku solutions, and alternating sign matrices. We will consider the application of the above and present corresponding polynomiographies.
AB - We have previously established delicate connections between the Voronoi diagram of polynomial roots and their basins of attraction with respect to the Basic Family of iteration functions. We have also previously defined polynomiography, visualization techniques in solving a polynomial equation, resulting in a medium with multidisciplinary applications. Here we describe several novel results and survey our recent work on polynomials and polynomiography and their extensions and applications. These include, (i) extension of root-finding methods to analytic functions and the analysis of infinite Voronoi diagram and basins of attraction of their roots; (ii) an application of polynomiography in the visualization of least square polynomial approximations; (iii) a Voronoi diagram property of roots of a cubic equation in connection to its critical points and its algorithmic application; (iv) association of polynomiography to special matrices such as permutation matrices, Latin Squares and Sudoku solutions, and alternating sign matrices. We will consider the application of the above and present corresponding polynomiographies.
KW - Basins of attraction
KW - Complex polynomials
KW - Dynamical systems
KW - Iteration functions
KW - Newton's method
KW - Voronoi diagrams
KW - Zeros
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UR - http://www.scopus.com/inward/citedby.url?scp=84866785613&partnerID=8YFLogxK
U2 - 10.1109/ISVD.2012.10
DO - 10.1109/ISVD.2012.10
M3 - Conference contribution
AN - SCOPUS:84866785613
SN - 9780769547244
T3 - Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012
SP - 32
EP - 40
BT - Proceedings of the 2012 9th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2012
ER -