Titlebook: Algorithms for Sparse Linear Systems; Jennifer Scott,Miroslav T?ma Book‘‘‘‘‘‘‘‘ 2023 The Editor(s) (if applicable) and The Author(s) 2023

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https://doi.org/10.1007/978-3-476-03748-0This chapter considers the LU factorization of a general nonsymmetric nonsingular sparse matrix .. In practice, numerical pivoting for stability and/or ordering of . to limit fill-in in the factors is often needed and the computed factorization is then of a permuted matrix .. Pivoting is discussed in Chapter . and ordering algorithms in Chapter ..

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Sparse Matrices and Their Graphs,Many sparse matrix algorithms exploit the close relationship between matrices and graphs. We make no assumption regarding the reader’s prior knowledge of graph theory. The purpose of this chapter is to summarize basic concepts from graph theory that will be exploited later and to establish the notation and terminology that will be used throughout.

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Introduction to Matrix Factorizations,This chapter introduces the basic concepts of Gaussian elimination and its formulation as a matrix factorization that can be expressed in a number of mathematically equivalent but algorithmically different ways.

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Sparse Cholesky Solver: The Symbolic Phase,This chapter focuses on the symbolic phase of a sparse Cholesky solver. The sparsity pattern . of the symmetric positive definite (SPD) matrix . is used to determine the nonzero structure of the Cholesky factor . without computing the numerical values of the nonzeros.

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Sparse LU Factorizations,This chapter considers the LU factorization of a general nonsymmetric nonsingular sparse matrix .. In practice, numerical pivoting for stability and/or ordering of . to limit fill-in in the factors is often needed and the computed factorization is then of a permuted matrix .. Pivoting is discussed in Chapter . and ordering algorithms in Chapter ..

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Jennifer Scott,Miroslav T?maThis book is open access, which means that you have free and unlimited access.This monograph presents factorization algorithms for solving large sparse linear systems of equations.It unifies the study

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https://doi.org/10.1007/978-3-658-01483-4em we are interested in is that of solving linear systems of equations .?=?., where the square sparse matrix . and the vector . are given and the solution vector . is required. Such systems arise in a huge range of practical applications, including in areas as diverse as quantum chemistry, computer

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https://doi.org/10.1007/978-3-663-20180-9orage and the work involved in the computation of the factors and in their use during the solve phase it is generally necessary to reorder (permute) the matrix before the factorization commences. The complexity of the most critical steps in the factorization is highly dependent on the amount of fill