## Concurrent Scientific ComputingMathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific dis ciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathe matics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface A successful concurrent numerical simulation requires physics and math ematics to develop and analyze the model, numerical analysis to develop solution methods, and computer science to develop a concurrent implemen tation. No single course can or should cover all these disciplines. Instead, this course on concurrent scientific computing focuses on a topic that is not covered or is insufficiently covered by other disciplines: the algorith mic structure of numerical methods. |

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### Contents

Exercises | 38 |

Exercises | 65 |

Exercises | 95 |

Exercises | 123 |

Tridiagonal Solvers | 141 |

The Fast Fourier Transform | 155 |

Poisson Solvers | 183 |

Multigrid Methods | 217 |

Domain Decomposition | 241 |

Particle Methods 265 | 264 |

Computer Dependency | 283 |

310 | |

319 | |

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### Common terms and phrases

algorithm arithmetic coarse-grid coefficient matrix column-distributed communication components conjugate-gradient method corresponding declare defined Dirichlet-boundary discrete Fourier transform domain decomposition duplicated efficiency Equation execution Exercise Figure floating-point operations function ghost-boundary exchange Givens rotations global indices grid points hypercube index set inner product integer interior-boundary vector iteration step Lemma linear logarithmic execution-time plot LU-decomposition matrix–vector memory Möbius transformation multicomputer implementation multicomputer program multigrid multigrid methods notation number of floating-point number of nodes obtained operation count optimal orthogonal Orthogonal matrix parameters performance analysis permutation pivot row Poisson problem positive-definite matrix process column process identifier process mesh process placement processors program Vector-Sum-1 quantification real assign real initially recursive doubling recursive-doubling procedure reduce relaxation methods representation requires right-hand side Section sequence sequential computation solution solve split subdomains successive over-relaxation symmetric positive-definite Theorem tion tridiagonal solver trigonometric polynomial values variables vector operations