Soustavy lineárních rovnic s hierarchickými maticemi řešené pomocí rozkladů
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Při počítání praktických úloh např. z fyziky se často setkáváme s potřebouřešit soustavy rovnic s velkými řídkými maticemi. Tyto se mohou řešit klasicky např. pomocí iteračních metod, kdy matice soustavy zůstává po celou dobu výpočtu nezměněná a využívá se pouze jejího součinu s vektorem. Druhou možností jsou metody založené na rozkladech matic, kde je ovšem potřeba pracovat chytře se zaplněním matice v průběhu úprav. Moderní přístup k těmto metodám mohou zprostředkovat tzv. hierarchické formáty uložení matice.Tato diplomová práce si klade za cíl stručně zrekapitulovat základní manipulacis hierarchickými maticemi a ukázat jak lze řešit soustavu rovnic s takovoumaticí pomocí rozkladových metod. Zaměříme se zejména na Gaussovu eliminaci,přesněji řečeno na její variantu pro symetrickou pozitivně definitní matici, tzv. Choleského rozklad.
While dealing with practical real-world problems in, e.g., physics, it is often required to solve a large system of linear equations with a sparse matrix. Such linear problems can be solved in a classical way: By using iterative methods. In such methods, the matrix stays unchanged during the whole calculation, and it is employed repetitively only in evaluation matrix-vector products. The other way is represented by a bunch of methods based on matrix decompositions. During such decompositions, however, the matrix is changing and we have to use clever methods to avoid its ll-in with nonzeros. Hierarchical format for storing matrices is one of the modern approaches to do that. The goals of this diploma thesis are to summarize basic arithmetics of hierarchical matrices, and to show how to solve a linear system with such matrix by using decompositions. We focus in particular on the Gau ian elimination method, more speci cally, on its variant applicable to symmetric positive de nite matrices, socalled Cholesky decomposition. Finally, we also show how to employ hierarchical approach to explicitely assemble an inverse to a large sparse matrix.
While dealing with practical real-world problems in, e.g., physics, it is often required to solve a large system of linear equations with a sparse matrix. Such linear problems can be solved in a classical way: By using iterative methods. In such methods, the matrix stays unchanged during the whole calculation, and it is employed repetitively only in evaluation matrix-vector products. The other way is represented by a bunch of methods based on matrix decompositions. During such decompositions, however, the matrix is changing and we have to use clever methods to avoid its ll-in with nonzeros. Hierarchical format for storing matrices is one of the modern approaches to do that. The goals of this diploma thesis are to summarize basic arithmetics of hierarchical matrices, and to show how to solve a linear system with such matrix by using decompositions. We focus in particular on the Gau ian elimination method, more speci cally, on its variant applicable to symmetric positive de nite matrices, socalled Cholesky decomposition. Finally, we also show how to employ hierarchical approach to explicitely assemble an inverse to a large sparse matrix.
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hierarchická matice, symetrická pozitivně de nitní matice; LU rozklad; Gaussovaeliminace; Choleského rozklad; maticové rovnice, hierarchical matrix, symmetric positive de nite matrix, LU decomposition, Gaussianelimination, Cholesky decomposition, matrix equations