Author | Keng, Hua Loo. author |
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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 |
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. Vinogradovโs 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