Nonrecursive Equivalent of the Conjugate Gradient Method without the Need to Restart

Dvornik, Josip; Lazarević, Damir; Jaguljnjak-Lazarević, Antonia; Demšić, Marija

doi:10.1155/2019/7527590

Cited by 7 publications

(9 citation statements)

References 13 publications

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“…For such a small example, an exact check of the algorithm stability is possible. Because the number of operations is small and the rounding error is negligible, both methods are intentionally perturbed by δ = 1 [4], added (for example) to the seventh component of p (1) (at the end of the first step): p 7,(1) ← p 7,(1) + δ…”

Section: Check Of the Algorithm Stabilitymentioning

confidence: 99%

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Exact arithmetic as a tool for convergence assessment of the IRM-CG method

Dvornik

Jaguljnjak-Lazarević

Lazarević

et al. 2020

Heliyon

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Exact arithmetic as a tool for convergence assessment of the IRM-CG method This is an author produced version of a paper submitted to a peer-review Exact arithmetic as a tool for convergence assessment of the IRM-CG method AbstractUsing exact computer arithmetic, it is possible to determine the (exact) solution of a numerical model without rounding error. For such purposes, a corresponding system of equations should be exactly defined, either directly or by rationalisation of numerically given input data. In the latter case there is an initial roundoff error, but this does not propagate during the solution process. If this system is first exactly solved, then by the floating-point arithmetic, convergence of the numerical method is easily followed. As one example, IRM-CG, a special case of the more general Iterated Ritz method and interesting replacement for a standard or preconditioned CG, is verified. Further, because the computer demands and execution time grow enourmously with the number of unknowns using this strategy, the possibilities for larger systems are also provided.

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Section: Check Of the Algorithm Stabilitymentioning

confidence: 99%

“…As a consequence, restarting of IRM-CG is not needed. Further, preconditioning-like techniques [11] can be adopted easily [4]. One more advantage of this formulation is natural adoption of the relaxation factor.…”

mentioning

confidence: 99%

Exact arithmetic as a tool for convergence assessment of the IRM-CG method

Dvornik

Jaguljnjak-Lazarević

Lazarević

et al. 2020

Heliyon

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“…Bez obzira na k(A), tijek točnoga proračuna podudara se za oba postupka i završava nakon 188 koraka. Riječ je o broju nepoznanica umanjenom za četiri, koliko je "neaktivnih" l j , onih kojima pripada v j • b jednak (u egzaktnoj aritmetici cjelobrojnoj) nuli [25,26]. Ili drugačije, potreban broj koraka jednak je broju "aktivnih" vlastitih vrijednosti, za koje je v j • b ≠ 0.…”

Section: Rezultati Proračuna Praktičnih Modelaunclassified

Iterated Ritz Method: fundamentals, current state and future development

Dvornik¹,

Lazarević²

2019

Numerički Postupci. Zbornik Radova Mini - Simpozija

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Sažetak U članku je opisan razvoj novoga iteracijskog postupka za rješavanje sustava linearnih algebarskih jednadžbi, utemeljen na primjeni diskretnoga Ritzova postupka u svakom koraku. Pogodan je za izrazito velike sustave slabo popunjenih, čak loše uvjetovanih, matrica. Osim vlastitih obilježja posjeduje i svojstvo općenitosti, jer su mnogi iteracijski postupci samo poseban slučaj ovoga pristupa. To pomaže drugačijoj interpretaciji tih postupaka, što doprinosi razumijevanju njihovih prednosti i nedostataka, a time i zamislima poboljšanja. Algoritam je realiziran samostalno, a potom je pridružen programu otvorena koda FEAP. Provedene su raznolike provjere, posebice na praktičnim modelima. Postupak je tek djelomice istražen, ali već pokazuje dobre rezultate.

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“…Primjena biblioteke Eigen omogućava gotovo doslovan prijepis pseudokôda (iz, primjerice, [4]) u programski kôd:…”

Section: Realizacija U Programskom Jeziku C++unclassified

On programme code opimisation for Iterated Ritz Method

Fresl¹,

Šamec²,

Gidak³

2019

Numerički Postupci. Zbornik Radova Mini - Simpozija

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Sažetak U radu su prikazana tri primjera optimizacije kôda u realizaciji iteriranoga Ritzova postupka (IRM) i njegove inačice istovrijedne metodi konjugiranih gradijenata (IRM-CG) u programskom jeziku C++. Nakon opisa osnovnih shema pohranjivanja rijetko popunjenih matrica, uključujući posebnosti simetričnih matrica, prikazano je izračunavanje umnoška rijetko popunjene matrice i vektora, izračunavanje umnoška rijetko popunjene i pune matrice te ispitivanje zadovoljenja uvjeta za prekid iteracije. Optimizacije su provedene radi ublažavanja "uskih grla" u algoritmima IRM-a i IRM-CG-a.

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Nonrecursive Equivalent of the Conjugate Gradient Method without the Need to Restart

Cited by 7 publications

References 13 publications

Exact arithmetic as a tool for convergence assessment of the IRM-CG method

Exact arithmetic as a tool for convergence assessment of the IRM-CG method

Iterated Ritz Method: fundamentals, current state and future development

On programme code opimisation for Iterated Ritz Method

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