THE NEWTON-RAPHSON METHOD OF REAL-VALUED FUNCTIONS IN DISCRETE METRIC SPACE

  • Indra Herdiana Universitas Jenderal Soedirman
  • Idha Sihwaningrum Universitas Jenderal Soedirman
  • Agus Sugandha Universitas Jenderal Soedirman

Abstract

ABSTRACT. This paper studies the Newton-Raphson method to approximate a root of a real-valued function in one-dimensional real discrete metric space. The method involves a derivative and is considered to be convergent very fast. However, the derivative is derived from the limit definition with respect to the Euclidean distance, different from that of the discrete metric space. This research investigates the Newton-Raphson method with respect to derivatives defined in discrete metric spaces by deriving the derivative first. The examined derivatives are absolute derivative and parameterised derivative on metric spaces. The results show that the constructed Newton-Raphson method can be an alternative root-finding method exemplified by some examples.
Keywords: Newton-Raphson method, discrete metric space, metric space derivative.


ABSTRAK. Artikel ini membahas metode Newton-Raphson untuk mengaproksimasi akar dari fungsi bernilai riil pada ruang metrik diskrit riil berdimensi satu. Metode tersebut melibatkan turunan fungsi dan cenderung konvergen dengan cepat. Namun, turunan fungsi tersebut didapatkan dari definisi limit untuk jarak Euclid yang berbeda dengan jarak pada ruang metrik diskrit. Penelitian ini menyelidiki metode Newton-Raphson untuk turunan fungsi pada ruang metrik diskrit diawali dengan mengonstruksi turunan tersebut. Turunan yang digunakan adalah turunan mutlak dan turunan berparameter pada ruang metrik. Hasil penelitian menunjukkan bahwa metode Newton-Raphson yang dibentuk dapat menjadi metode alternatif untuk pencarian akar fungsi menurut beberapa contoh.
Kata Kunci: metode Newton-Raphson, ruang metrik diskrit, turunan ruang metric.

Published
2024-01-15
How to Cite
HERDIANA, Indra; SIHWANINGRUM, Idha; SUGANDHA, Agus. THE NEWTON-RAPHSON METHOD OF REAL-VALUED FUNCTIONS IN DISCRETE METRIC SPACE. Jurnal Ilmiah Matematika dan Pendidikan Matematika (JMP), [S.l.], v. 15, n. 2, p. 113-128, jan. 2024. ISSN 2550-0422. Available at: <https://jos.unsoed.ac.id/index.php/jmp/article/view/8875>. Date accessed: 23 june 2024. doi: https://doi.org/10.20884/1.jmp.2023.15.2.8875.

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