## Mathematical Foundations of Parallel ComputingParallel implementation of algorithms involves many difficult problems. In particular among them are round-off analysis, the way to convert sequential programs and algorithms into the parallel mode, the choice of appropriate or optimal computer architect and so on. To solve these problems, it is necessary to know very well the structure of algorithms. This book deal with the mathematical mechanism that permits us to investigate structures of both sequential and parallel algorithms. This mechanism allows us to recognize and explain the relations between different methods of constructing parallel algorithms, the methods of analysing round-off errors, the methods of optimizing memory traffic, the methods of working out the fastest implementation for a given parallel computer and other methods attending the joint investigation of algorithms and computers. |

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### Contents

Chapter | 1 |

Graph of algorithm | 16 |

Chapt | 24 |

Topological sorting | 26 |

Schedules and graph machine | 34 |

Examples | 43 |

32 | 50 |

Algorithm Execution Time | 55 |

33 | 136 |

Sectioning of memory | 143 |

Matrix Investigation of Algorithm Structure | 155 |

Recovering the linear functional | 164 |

Roundoff error analysis | 174 |

Examples | 186 |

Functional Investigation of Algorithm Structure | 193 |

Regular graphs | 203 |

Number semirings and other sets | 67 |

Minimax properties of schedules | 80 |

Optimal and highspeed schedules | 89 |

Examples | 100 |

Algorithms and Computer Memory | 107 |

Total required memory size | 120 |

Hierarchical memory | 135 |

35 | 208 |

Afterword | 215 |

333 | |

339 | |

341 | |

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### Common terms and phrases

adjacency matrix algorithm execution algorithm graph building algorithm graph nodes algorithm implementation algorithm notation algorithm structure assignment statement assume backward error analysis bearing nested loops class R.,(s components computer system Consider control transfer coordinates defined delay vector depend described determined directed cuts domain dot product elementary graphs entries equal equations equivalent perturbations example exists expanded graph external memory follows FORTRAN gorithm grad t x grad t(x gradient graph of algorithm high-speed schedules hold homomorphic hyperplanes infinite regular graph initial conditions vector input data integer investigation jth node layers level surfaces lexicographic order linear algebraic loop indexes macrograph macronodes memory computer nonnegative nonzero number of arcs obtain output p-vectors parallel computers parallel form parallelepipeds parameters partial algorithms points polyhedron problem properties relations rithm semiring set of schedules solution solve space-time schedules specified subgraph subset Suppose tion topological sorting total number values zero