Direct methods for sparse matrices
This book provides practical approaches to the efficient use of sparsity--a key to solving large problems in many fields, including computational science and engineering, where mathematical models give rise to very large systems of linear equations. The emphasis is on practicality, with conclusions based on concrete experience. Non-numeric computing techniques have been included as well as frequent illustrations in an attempt to bridge the gap between the written word and the working computer code. Exercises have been included to strengthen understanding of the material as well as to extend it for students and researchers in engineering, mathematics, and computer science.
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storage schemes and simple operations
the algebraic problem
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active submatrix ANALYSE arithmetic array back-substitution band matrix block triangular form Chapter components computation condition number consider corresponding data structure depth-first search diagonal blocks diagonal entries digraph Duff and Reid eigenvalues entry in row equations Erisman example Exercise fill-in Fortran forward substitution full-length vector Gaussian elimination George and Liu graph Harwell ill-conditioning illustrated in Figure implementation inequality integers involves IROWST level sets linear linked list loop lower triangular LU factorization matrix of Figure minimum degree algorithm multifrontal multiple needed nested dissection node number of entries number of nonzeros numerical stability off-diagonal entries partitioned path performed permutation matrix perturbation pivotal sequence pointers positive-definite matrix reduced reordering right-hand side row count rows and columns shown in Figure solution solving sparse matrix sparsity pattern SPARSPAK stability step storage stored strategy submatrix symmetric matrix Table techniques tridiagonal unsymmetric upper triangular variable-band variables X X X X X X zero