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**Random number generation and quasi-Monte Carlo methods.**
*(English)*
Zbl 0761.65002

CBMS-NSF Regional Conference Series in Applied Mathematics. 63. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. vi, 241 p. (1992).

The author deals with the application of random numbers to numerical analysis. The other main field of applications (cryptology) is not the subject of the present book.

It can be considered as an expanded written record of a series of ten talks presented by the author as the principal speaker at the Conference of Random Number Generation and Quasi-Monte Carlo Methods which was held at the University Of Alaska, Fairbanks from August 13–17, 1990. The book contains 10 Chapters and 2 Appendices. One can see from the headings of the Chapters (given below) that all aspects of the subject are covered.

1. Monte Carlo methods and quasi-Monte Carlo methods.

2. Quasi-Monte Carlo methods for numerical integration.

3. Low-discrepancy point sets and sequences.

4. Nets and \((t,s)\)-sequences.

5. Lattice rules for numerical integration.

6. Quasi-Monte Carlo methods for optimization.

7. Random numbers and pseudorandom numbers.

8. Nonlinear congruential pseudorandom numbers.

9. Shift-register pseudorandom numbers.

10. Pseudorandom vector generation.

Appendix A. Finite fields and linear recurring sequences.

Appendix B. Continued fractions.

Reviewer’s remarks: The author is one of the most competent people of the subject who wrote more than 60 papers on random number generation and quasi-Monte Carlo methods. He has a special skill to collect the relevant literature. In the bibliography of this book one can find 371 books and papers. In spite of that the author’s approach is that “The bibliography is not meant to be comprehensive, but lists only those references that are cited in the text”, it is unfortunate that in Chapter 9 the treatise of S. W. Golomb’s book [Shift register sequences. San Francisco etc.: Holden-Day (1967; Zbl 0267.94022)] is not mentioned.

The reviewer thinks that his hope is shared with many others, since the author is a very competent person in cryptology, he will be willing to publish a book on the application of random numbers to cryptology.

It can be considered as an expanded written record of a series of ten talks presented by the author as the principal speaker at the Conference of Random Number Generation and Quasi-Monte Carlo Methods which was held at the University Of Alaska, Fairbanks from August 13–17, 1990. The book contains 10 Chapters and 2 Appendices. One can see from the headings of the Chapters (given below) that all aspects of the subject are covered.

1. Monte Carlo methods and quasi-Monte Carlo methods.

2. Quasi-Monte Carlo methods for numerical integration.

3. Low-discrepancy point sets and sequences.

4. Nets and \((t,s)\)-sequences.

5. Lattice rules for numerical integration.

6. Quasi-Monte Carlo methods for optimization.

7. Random numbers and pseudorandom numbers.

8. Nonlinear congruential pseudorandom numbers.

9. Shift-register pseudorandom numbers.

10. Pseudorandom vector generation.

Appendix A. Finite fields and linear recurring sequences.

Appendix B. Continued fractions.

Reviewer’s remarks: The author is one of the most competent people of the subject who wrote more than 60 papers on random number generation and quasi-Monte Carlo methods. He has a special skill to collect the relevant literature. In the bibliography of this book one can find 371 books and papers. In spite of that the author’s approach is that “The bibliography is not meant to be comprehensive, but lists only those references that are cited in the text”, it is unfortunate that in Chapter 9 the treatise of S. W. Golomb’s book [Shift register sequences. San Francisco etc.: Holden-Day (1967; Zbl 0267.94022)] is not mentioned.

The reviewer thinks that his hope is shared with many others, since the author is a very competent person in cryptology, he will be willing to publish a book on the application of random numbers to cryptology.

Reviewer: József Dénes (Budapest)

### MSC:

65C10 | Random number generation in numerical analysis |

65-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to numerical analysis |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

11K45 | Pseudo-random numbers; Monte Carlo methods |

11K38 | Irregularities of distribution, discrepancy |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

65C05 | Monte Carlo methods |

65D32 | Numerical quadrature and cubature formulas |