Applications of Number Theory to Numerical Analysis

1. Front Matter

   Pages i-ix

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2. Algebraic Number Fields and Rational Approximation

   • Hua Loo Keng, Wang Yuan

   Pages 1-27

3. Recurrence Relations and Rational Approximation

   • Hua Loo Keng, Wang Yuan

   Pages 28-47

4. Uniform Distribution

   • Hua Loo Keng, Wang Yuan

   Pages 48-69

5. Estimation of Discrepancy

   • Hua Loo Keng, Wang Yuan

   Pages 70-98

6. Uniform Distribution and Numerical Integration

   • Hua Loo Keng, Wang Yuan

   Pages 99-112

7. Periodic Functions

   • Hua Loo Keng, Wang Yuan

   Pages 113-130

8. Numerical Integration of Periodic Functions

   • Hua Loo Keng, Wang Yuan

   Pages 131-158

9. Numerical Error for Quadrature Formula

   • Hua Loo Keng, Wang Yuan

   Pages 159-182

10. Interpolation

    • Hua Loo Keng, Wang Yuan

    Pages 183-203

11. Approximate Solution of Integral Equations and Differential Equations

    • Hua Loo Keng, Wang Yuan

    Pages 204-223

12. Back Matter

    Pages 224-244

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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.

• Applications
• Numerische Mathematik
• Zahlentheorie
• calculus
• computer
• number theory
• numerical analysis

• Institute of Mathematics, Academia Sinica, Beijing, The People’s Republic of China

  Hua Loo Keng, Wang Yuan

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