Author	Keng, Hua Loo. author
Title	Applications of Number Theory to Numerical Analysis [electronic resource] / by Hua Loo Keng, Wang Yuan
Imprint	Berlin, Heidelberg : Springer Berlin Heidelberg, 1981
Connect to	http://dx.doi.org/10.1007/978-3-642-67829-5
Descript	X, 244 p. online resource

Owing to the developments and applications of computer science, maยญ 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 numยญ ber theory to construct a sequence of uniformly distributed sets in the sยญ 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 apยญ 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 satisยญ fies certain conditions

1. Algebraic Number Fields and Rational Approximation -- ยง1.1. The units of algebraic number fields -- ยง1.2. The simultaneous Biophantine approximation of an integral basis -- ยง1.3. The real eyelotomie field -- ยง1.4. The units of a eyelotomie field -- ยง1.5. Continuation -- ยง1.6. The Drriehlet field -- ยง1.7. The cubic field -- Notes -- 2. Recurrence Relations and Rational Approximation -- ยง2.1. The recurrence formula for the elementary symmetric fonction -- ยง2.2. The generalization of Sn -- ยง2.3. PV numbers -- ยง 2.4. The roots of the equation F(x) = 0 -- ยง2.5. The roots of the equation G(x) = 0 -- ยง2.6. The roots of the equation E(x) = 0 -- ยง2.7. The irreducibility of a polynomial -- ยง2.8. The rational approximations of ?, ?, ? -- Notes -- 3. Uniform Distribution -- ยง3.1. Uniform distribution -- ยง3.2. Vinogradovโ{128}{153}s lemma -- ยง3.3. The exponential sum and the discrepancy -- ยง3.4. The number of solutions to the congruence -- ยง3.5. The solutions of the congruence and the discrepancy -- ยง3.6. The partial summation formula -- ยง3.7. The comparison of discrepancies -- ยง3.8. Eational approximation and the solutions of the congruence -- ยง3.9. The rational approximation and the discrepancy -- ยง3.10. The lower estimate of discrepancy -- Notes -- 4. Estimation of Discrepancy -- ยง4.1. The set of equi-distribution -- ยง4.2. The Halton theorem -- ยง4.3. The p set -- ยง4.4. The gp set -- ยง4.5. The eonstruetion of good points -- ยง4.6. The ?s set -- ยง4.7. The ? set -- ยง4.8. The ease s = 2 -- ยง4.9. The glp set -- Notes -- 5. Uniform Distribution and Numerical Integration -- ยง5.1. The function of bounded variation -- ยง5.2. Uniform distribution and numerical integration -- ยง5.3. The lower estimation for the error term of quadrature formula -- ยง5.4. The quadrature formulas -- Notes -- 6. Periodic Functions -- ยง6.1. The classes of functions -- ยง6.2. Several lemmas -- ยง6.3. The relations between Hs?(C), Qs?(C) and Es?(C) -- ยง6.4. Periodic functions -- ยง 6.5. Continuation -- Notes -- 7. Numerical Integration of Periodic Functions -- ยง7.1. The set of equi-distribution and numerical integration -- ยง7.2. The p set and numerical integration -- ยง7.3. The gp set and numerical integration -- ยง7.4. The lower estimation of the error term for the quadrature formula -- ยง7.5. The solutions of congruences and numerical integration -- ยง7.6. The glp set and numerical integration -- ยง7.7. The Sarygin theorem -- ยง7.8. The mean error of the quadrature formula -- ยง7.9. Continuation -- Notes -- 8. Numerical Error for Quadrature Formula -- ยง8.1. The numerical error -- ยง8.2. The comparison of good points -- ยง8.3. The computation of the ? set -- ยง8.4. The computation of the ?s set -- ยง8.5. Examples of other F s sets -- ยง8.6. The computation of a glp set -- ยง8.7. Several remarks -- ยง8.8. Tables -- ยง 8.9. Some examples -- Notes -- 9. Interpolation -- ยง9.1. Introduction -- ยง9.2. The set of equi-distribution and interpolation -- ยง9.3. Several lemmas -- ยง9.4. The approximate formula of the function of E?s(C) -- ยง9.5. The approximate formula of the function of Q?s(C) -- ยง9.6. The Bernoulli polynomial and the approximate polynomial -- ยง9.7. The ? results -- Notes -- 10. Approximate Solution of Integral Equations and Differential Equations -- ยง10.1. Several lemmas -- ยง 10.2. The approximate solution of the Fredholm integral equation of second type -- ยง 10.3. The approximate solution of the Volterra integral equation of second type -- ยง10.4. The eigenvalue and eigenfunction of the Fredholm equation -- ยง 10.5. The Cauehy problem of the partial differential equation of the parabolic type -- ยง 10.6. The Diriehlet problem of the partial differential equation of the elliptic type -- ยง 10.7. Several remarks -- Notes -- Appendix Tables