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Generating roots of cubic polynomials by Cardano's approach on correspondence analysis
Cardano's formula is among the most popular cubic formula to solve any third-degree polynomial equation. In this paper, we propose the Cardano's approach as the alternative solution to generate the roots of the cubic characteristic polynomial analytically. In the context of correspondence...
Autores principales: | , , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Elsevier
2020
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7300094/ https://www.ncbi.nlm.nih.gov/pubmed/32577543 http://dx.doi.org/10.1016/j.heliyon.2020.e03998 |
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author | Lestari, Karunia E. Pasaribu, Udjianna S. Indratno, Sapto W. Garminia, Hanni |
author_facet | Lestari, Karunia E. Pasaribu, Udjianna S. Indratno, Sapto W. Garminia, Hanni |
author_sort | Lestari, Karunia E. |
collection | PubMed |
description | Cardano's formula is among the most popular cubic formula to solve any third-degree polynomial equation. In this paper, we propose the Cardano's approach as the alternative solution to generate the roots of the cubic characteristic polynomial analytically. In the context of correspondence analysis, these roots referred to eigenvalues, which play an important role in assessing the quality of the correspondence plot. Considering the correspondence analysis on the [Formula: see text] contingency table for [Formula: see text] and [Formula: see text] , we obtained a cubic characteristic polynomial (since zero is one of its eigenvalues). Therefore, Cardano's formula allows us to obtain the eigenvalues directly without involving numerical processes, e.g., using singular value decomposition. We note several advantages of using Cardano's approach, such as (1) it produces the roots with the same result as singular value decomposition, as well more precise because without errors involving, (2) the algorithm is simpler and does not depend on initial guess, hence the computation time becomes shorter than numerical process, and (3) the manual calculation is easy because it uses a formula. The results show that the matrix operations on correspondence analysis can be replaced by a formula for determining eigenvalues and eigenvectors of the standard residual matrix directly. Some mathematical results are also presented. |
format | Online Article Text |
id | pubmed-7300094 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2020 |
publisher | Elsevier |
record_format | MEDLINE/PubMed |
spelling | pubmed-73000942020-06-22 Generating roots of cubic polynomials by Cardano's approach on correspondence analysis Lestari, Karunia E. Pasaribu, Udjianna S. Indratno, Sapto W. Garminia, Hanni Heliyon Article Cardano's formula is among the most popular cubic formula to solve any third-degree polynomial equation. In this paper, we propose the Cardano's approach as the alternative solution to generate the roots of the cubic characteristic polynomial analytically. In the context of correspondence analysis, these roots referred to eigenvalues, which play an important role in assessing the quality of the correspondence plot. Considering the correspondence analysis on the [Formula: see text] contingency table for [Formula: see text] and [Formula: see text] , we obtained a cubic characteristic polynomial (since zero is one of its eigenvalues). Therefore, Cardano's formula allows us to obtain the eigenvalues directly without involving numerical processes, e.g., using singular value decomposition. We note several advantages of using Cardano's approach, such as (1) it produces the roots with the same result as singular value decomposition, as well more precise because without errors involving, (2) the algorithm is simpler and does not depend on initial guess, hence the computation time becomes shorter than numerical process, and (3) the manual calculation is easy because it uses a formula. The results show that the matrix operations on correspondence analysis can be replaced by a formula for determining eigenvalues and eigenvectors of the standard residual matrix directly. Some mathematical results are also presented. Elsevier 2020-06-11 /pmc/articles/PMC7300094/ /pubmed/32577543 http://dx.doi.org/10.1016/j.heliyon.2020.e03998 Text en © 2020 The Authors http://creativecommons.org/licenses/by/4.0/ This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). |
spellingShingle | Article Lestari, Karunia E. Pasaribu, Udjianna S. Indratno, Sapto W. Garminia, Hanni Generating roots of cubic polynomials by Cardano's approach on correspondence analysis |
title | Generating roots of cubic polynomials by Cardano's approach on correspondence analysis |
title_full | Generating roots of cubic polynomials by Cardano's approach on correspondence analysis |
title_fullStr | Generating roots of cubic polynomials by Cardano's approach on correspondence analysis |
title_full_unstemmed | Generating roots of cubic polynomials by Cardano's approach on correspondence analysis |
title_short | Generating roots of cubic polynomials by Cardano's approach on correspondence analysis |
title_sort | generating roots of cubic polynomials by cardano's approach on correspondence analysis |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7300094/ https://www.ncbi.nlm.nih.gov/pubmed/32577543 http://dx.doi.org/10.1016/j.heliyon.2020.e03998 |
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