![]() Fortunately, there are many heuristics to get an ordering for reducing the fill-in significantly, including the minimum degree (Tinney and Walker, 1967) and the nested dissection (George, 1973) heuristics and their variants. Minimizing the fill-in in Cholesky factorization has been shown to be NP-complete by Yannakakis (1981). Using this equivalence, ordering methods can be viewed as ordering the vertices of the graph \(G\) so that the same ordering determines the pivotal sequence for the matrix factorization. , n-1\) is used to reduce all entries below the \(i\)th diagonal entry to zero, corresponds to making the neighbours of vertex \(i\) a clique, and then removing the vertex \(i\) as described by Parter (1961) and Rose (1970). ![]() Below some tasks to be performed during the symbolic analysis are discussed.Ĭonsider the sparse symmetric positive definite matrixĪnd its Cholesky factor \(L_A\) is given by (up to four units of accuracy) The symbolic analysis can usually be viewed using the graph models for sparse matrices. Meeting these requirements usually entails first performing a symbolic analysis on the matrix in order to predict and reduce the memory and run time requirements for the subsequent numerical factorization. The added difficulty in the sparse case is that the factorization should try to avoid operating on the zeros of the matrix and should keep the factors as sparse as possible. The factorizations discussed above and their use in solving the systems are mathematically equivalent to their dense counterparts. Those with \(L\) are solved by forward substitution, those with \(U\) are solved by back substitution, those with \(D\) are solved by multiplying by the explicit inverse of the \(1\times 1\) and \(2\times 2\) blocks, and those with \(Q\) are solved by multiplying the right-hand side by \(Q^T\). The linear systems with \(L\), \(U\), \(D\), and \(Q\), as defined above, are easy to solve. Here, \(Q\) is an \(m\times m\) orthogonal matrix, and \(R\) is an \(m\times n\) upper trapezoidal matrix.
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