## Iterative Methods for Sparse Linear Systems: Second EditionTremendous progress has been made in the scientific and engineering disciplines regarding the use of iterative methods for linear systems. The size and complexity of linear and nonlinear systems arising in typical applications has grown, meaning that using direct solvers for the three-dimensional models of these problems is no longer effective. At the same time, parallel computing, becoming less expensive and standardized, has penetrated these application areas. Iterative methods are easier than direct solvers to implement on parallel computers but require approaches and solution algorithms that are different from classical methods. This second edition gives an in-depth, up-to-date view of practical algorithms for solving large-scale linear systems of equations, including a wide range of the best methods available today. A new chapter on multigrid techniques has been added, whilst material throughout has been updated, removed or shortened. Numerous exercises have been added, as well as an updated and expanded bibliography. |

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### Contents

Discretization of Partial Differential Equations | 45 |

Sparse Matrices | 73 |

Basic Iterative Methods | 103 |

Projection Methods | 129 |

Krylov Subspace Methods Part I | 151 |

Krylov Subspace Methods Part II | 217 |

Methods Related to the Normal Equations | 245 |

Preconditioned Iterations | 261 |

### Common terms and phrases

applied approximate solution Arnoldi array associated Assume block CG algorithm Chapter Chebyshev polynomials coefficients column components Compute condition Consider convergence data structure defined denoted dot product eigenvalues eigenvector EndDo example exploited Figure fill-in finite element formula Gauss-Seidel Gaussian elimination GMRES GMRES algorithm grid Hermitian Hessenberg matrix ILU factorization implementation inner product inverse iterative methods Jacobi iteration Lanczos algorithm least-squares Lemma linear system matrix-by-vector product mesh minimizes nodes nonnegative nonsingular nonzero elements normal equations obtained operations orthogonal parallel partitioning PDEs performed permutation permuted matrix polynomial positive definite preconditioner preconditioning problem procedure processors projector Proof properties Proposition recurrence relation reordering residual norm residual vector result right-hand side satisfy scalar Schur complement Section sequence Show SIAM Journal solving sparse matrix stencil step subdomain symmetric techniques Theorem tridiagonal tridiagonal matrix updated upper triangular variables zero