Definition
Direct methods for solving linear systems of the form Ax= b are based on computing A= LU, where L and U are lower and upper triangular, respectively. Computing the triangular factors of the coefficient matrix A is also known as LU decomposition. Following the factorization, the original system is trivially solved by solving the triangular systems Ly= b and Ux= y. If A is symmetric, then a factorization of the form A= LL T or A= LDL T is computed via Cholesky factorization, where L is a lower triangular matrix (unit lower triangular in the case of A= LDL T factorization) and D is a diagonal matrix. One set of common formulations of LU decomposition and Cholesky factorization for dense matrices are shown in Figs. 1 and 2, respectively. Note that other mathematically equivalent formulations are possible by rearranging the loops in these algorithms. These algorithms must be adapted for sparse...
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Gupta, A. (2011). Sparse Direct Methods. In: Padua, D. (eds) Encyclopedia of Parallel Computing. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09766-4_507
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