# Applications of Number Theory to Numerical Analysis

• Hua Loo Keng
• Wang Yuan
Book

1. Front Matter
Pages i-ix
2. Hua Loo Keng, Wang Yuan
Pages 1-27
3. Hua Loo Keng, Wang Yuan
Pages 28-47
4. Hua Loo Keng, Wang Yuan
Pages 48-69
5. Hua Loo Keng, Wang Yuan
Pages 70-98
6. Hua Loo Keng, Wang Yuan
Pages 99-112
7. Hua Loo Keng, Wang Yuan
Pages 113-130
8. Hua Loo Keng, Wang Yuan
Pages 131-158
9. Hua Loo Keng, Wang Yuan
Pages 159-182
10. Hua Loo Keng, Wang Yuan
Pages 183-203
11. Hua Loo Keng, Wang Yuan
Pages 204-223
12. Back Matter
Pages 224-244

### Introduction

Owing to the developments and applications of computer science, ma­ thematicians began to take a serious interest in the applications of number theory to numerical analysis about twenty years ago. The progress achieved has been both important practically as well as satisfactory from the theoretical view point. It'or example, from the seventeenth century till now, a great deal of effort was made in developing methods for approximating single integrals and there were only a few works on multiple quadrature until the 1950's. But in the past twenty years, a number of new methods have been devised of which the number theoretic method is an effective one. The number theoretic method may be described as follows. We use num­ ber theory to construct a sequence of uniformly distributed sets in the s­ dimensional unit cube G , where s ~ 2. Then we use the sequence to s reduce a difficult analytic problem to an arithmetic problem which may be calculated by computer. For example, we may use the arithmetic mean of the values of integrand in a given uniformly distributed set of G to ap­ s proximate the definite integral over G such that the principal order of the s error term is shown to be of the best possible kind, if the integrand satis­ fies certain conditions.

### Keywords

Applications Numerische Mathematik Zahlentheorie calculus computer number theory numerical analysis

#### Authors and affiliations

• Hua Loo Keng
• 1
• Wang Yuan
• 1
1. 1.Institute of MathematicsAcademia SinicaBeijingThe People’s Republic of China

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-67829-5
• Copyright Information Springer-Verlag Berlin Heidelberg 1981
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-67831-8
• Online ISBN 978-3-642-67829-5
• Buy this book on publisher's site
Industry Sectors