| Abstract | Prilikom rješavanja sustava linearnih jednadžbi Ax = b, pri čemu za A ∈ \(M_{mn}\left ( \mathbb{R} \right )\) i b ∈ \(M_{m1}\left ( \mathbb{R} \right )\) nastojimo odrediti \(x\in M_{mn}\left ( \mathbb{R} \right )\) razlikujemo dva pristupa rješavanja sustava. Prvi način na koji se može odrediti traženi vektor x jesu direktne metode. Direktne se metode koriste uglavnom
za rješavanje manjih sustava linearnih jednadžbi te se pomoću njih dobiva egzaktno rješenje
sustava. Direktne metode jesu primjerice Cramerovo pravilo i Gaussove eliminacije. S druge
strane, pristup rješavanja sustava koji ćemo proučavati u ovom radu jesu iterativne metode
koje uvelike olašavaju rješavanje sustava velikih dimenzija zahtjevne vremenske i prostorne
složenosti. Iterativne metode su nizovi matematičkih postupaka čijim se uzastopnim ponavljanjem dolazi do aproksimacije rješenja sustava. No, takvim metodama generalno ne dolazimo do točnog rješenja sustava u konačno mnogo koraka, nego se svakim korakom odstupanje od točnog rješenja smanjuje. Ukoliko pripadna iterativna metoda konvergira, konstruiramo niz
\(x^{k }\rightarrow x\) za koji vrijedi \(x^{k }\rightarrow x\), za \( k\rightarrow x\) . U ovom radu proučit ćemo neke načešće
korištene iterativne metode kao što su: Jacobijeva metoda, Gauss-Seidelova metoda, metoda
najbržeg silaska, metoda konjugiranih gradijenata.
M_{mn}\left ( \mathbb{R} \right ) |
| Abstract (english) | While solving system of linear equations Ax = b, where for we A ∈ \(M_{mn}\left ( \mathbb{R} \right )\) and b ∈ \(M_{m1}\left ( \mathbb{R} \right )\)
try to \(x\in M_{mn}\left ( \mathbb{R} \right )\) , we distinguish two approaches. The first approach for finding the
vector x is using direct methods. Direct methods are mostly used for solving simpler systems
of smaller dimensions, giving the exact solution to the system. Such methods are, for example,
Cramer’s rule and Gaussian elimination. On the other hand, the approach for solving systems
of equations we will examine are iterative methods, which greatly simplify the process of solving
linear systems of big dimensions and challenging time and space complexity. Iterative methods
are series of mathematical procedures which, when applied repeatedly, give an approximated
solution of the system. However, in these methods, the exact solution is generally not found,
but rather the deviation from the exact solution decrease with each iteration. Iterative methods
construct the sequence \(x^{k }\rightarrow x\), where \(x^{k }\rightarrow x\), for \( k\rightarrow x\) .
In this paper we will study
some of the most commonly used iterative methods, such as the Jacobi method, Gauss-Seidel
method, steepest descent method, conjugate gradient method. |
