Electronic Theses and Dissertation

Penelitian ini menjabarkan secara teoretis rumus Metode Durand–Kerner berdasarkan Teorema Fundamental Aljabar, serta Metode Aberth berdasarkan Teorema Vieta, bentuk Euler, dan binomial Newton. Selanjutnya, dilakukan pencarian akar-akar polinomial kompleks berderajat tiga (P_3(z)=z^3+1) dan berderajat empat (P_4(z)=z^4+16), serta polinomial kompleks berderajat dua dengan variasi sifat akar (kembar, real berbeda, kompleks berbeda, dan konjugat). Langkah awal, digunakan metode Aberth untuk menentukan titik awal. Lalu langkah selanjutnya dilakukan iterasi menggunakan metode Durand-Kerner untuk pencarian akar-akar polinomial kompleks. Hasilnya menunjukkan titik awal yang diperoleh menyebar secara merata di kuadran sesuai derajat polinomial kompleks. Hasil iterasi untuk polinomial kompleks P_3(z) dan P_4(z) dibutuhkan sebanyak 6 iterasi dengan galat toleransi $10^{-10}$. Sementara pada polinomial kompleks berderajat dua yaitu P_2(z)=z^2-(2+2i)z+2i membutuhkan sebanyak 21 iterasi, polinomial kompleks P_2(z)=z^2+z+1 membutuhkan sebanyak 7 iterasi, polinomial kompleks P_2(z)=z^2-1 membutuhkan sebanyak 5 iterasi dan polinomial kompleks P_2(z)=z^2-4z+3, P_2(z)=z^2-z-2, P_2(z)=z^2-z+1, P_2(z)=z^2+4z-3 membutuhkan sebanyak 6 iterasi dengan galat toleransi 10^{-10}.

This study presents a theoretical explanation of the Durand–Kerner Method formula based on the Fundamental Theorem of Algebra, as well as the Aberth Method based on Vieta's Theorem, Euler form, and Newton's Binomial. Subsequently, root-finding is conducted for complex polynomials of degree three (P_3(z) = z^3 + 1) and degree four (P_4(z) = z^4 + 16), as well as quadratic complex polynomials with various types of roots (repeated, distinct real, distinct complex, and conjugate pairs). In the initial step, the Aberth method is employed to determine the starting points. The next step involves iterative computation using the Durand–Kerner method to find the roots of complex polynomials. The results show that the initial points are evenly distributed across the quadrants according to the degree of the polynomial. The iteration results for the complex polynomials P_3(z) and P_4(z) required 6 iterations with a tolerance error of 10^{-10}. Meanwhile, for the quadratic complex polynomial P_2(z) = z^2 - (2 + 2i)z + 2i, 21 iterations were needed; for P_2(z) = z^2 + z + 1, 7 iterations; for P_2(z) = z^2 - 1, 5 iterations; and for P_2(z) = z^2 - 4z + 3, P_2(z) = z^2 - z - 2, P_2(z) = z^2 - z + 1, and P_2(z) = z^2 + 4z - 3, 6 iterations were required, all with a tolerance error of 10^{-10}.