Applications of Number Theory to Numerical Analysis

1. Algebraic Number Fields and Rational Approximation.- x1.1. The units of algebraic number fields.- x1.2. The simultaneous Biophantine approximation of an integral basis.- x1.3. The real eyelotomie field.- x1.4. The units of a eyelotomie field.- x1.5. Continuation.- x1.6. The Drriehlet field.- x1.7. The cubic field.- Notes.- 2. Recurrence Relations and Rational Approximation.- x2.1. The recurrence formula for the elementary symmetric fonction.- x2.2. The generalization of Sn.- x2.3. PV numbers.- x 2.4. The roots of the equation F(x) = 0.- x2.5. The roots of the equation G(x) = 0.- x2.6. The roots of the equation E(x) = 0.- x2.7. The irreducibility of a polynomial.- x2.8. The rational approximations of ?, ?, ?.- Notes.- 3. Uniform Distribution.- x3.1. Uniform distribution.- x3.2. Vinogradov's lemma.- x3.3. The exponential sum and the discrepancy.- x3.4. The number of solutions to the congruence.- x3.5. The solutions of the congruence and the discrepancy.- x3.6. The partial summation formula.- x3.7. The comparison of discrepancies.- x3.8. Eational approximation and the solutions of the congruence.- x3.9. The rational approximation and the discrepancy.- x3.10. The lower estimate of discrepancy.- Notes.- 4. Estimation of Discrepancy.- x4.1. The set of equi-distribution.- x4.2. The Halton theorem.- x4.3. The p set.- x4.4. The gp set.- x4.5. The eonstruetion of good points.- x4.6. The ?s set.- x4.7. The ? set.- x4.8. The ease s = 2.- x4.9. The glp set.- Notes.- 5. Uniform Distribution and Numerical Integration.- x5.1. The function of bounded variation.- x5.2. Uniform distribution and numerical integration.- x5.3. The lower estimation for the error term of quadrature formula.- x5.4. The quadrature formulas.- Notes.- 6. Periodic Functions.- x6.1. The classes of functions.- x6.2. Several lemmas.- x6.3. The relations between Hs?(C), Qs?(C) and Es?(C).- x6.4. Periodic functions.- x 6.5. Continuation.- Notes.- 7. Numerical Integration of Periodic Functions.- x7.1. The set of equi-distribution and numerical integration.- x7.2. The p set and numerical integration.- x7.3. The gp set and numerical integration.- x7.4. The lower estimation of the error term for the quadrature formula.- x7.5. The solutions of congruences and numerical integration.- x7.6. The glp set and numerical integration.- x7.7. The Sarygin theorem.- x7.8. The mean error of the quadrature formula.- x7.9. Continuation.- Notes.- 8. Numerical Error for Quadrature Formula.- x8.1. The numerical error.- x8.2. The comparison of good points.- x8.3. The computation of the ? set.- x8.4. The computation of the ?s set.- x8.5. Examples of other F s sets.- x8.6. The computation of a glp set.- x8.7. Several remarks.- x8.8. Tables.- x 8.9. Some examples.- Notes.- 9. Interpolation.- x9.1. Introduction.- x9.2. The set of equi-distribution and interpolation.- x9.3. Several lemmas.- x9.4. The approximate formula of the function of E?s(C).- x9.5. The approximate formula of the function of Q?s(C).- x9.6. The Bernoulli polynomial and the approximate polynomial.- x9.7. The ? results.- Notes.- 10. Approximate Solution of Integral Equations and Differential Equations.- x10.1. Several lemmas.- x 10.2. The approximate solution of the Fredholm integral equation of second type.- x 10.3. The approximate solution of the Volterra integral equation of second type.- x10.4. The eigenvalue and eigenfunction of the Fredholm equation.- x 10.5. The Cauehy problem of the partial differential equation of the parabolic type.- x 10.6. The Diriehlet problem of the partial differential equation of the elliptic type.- x 10.7. Several remarks.- Notes.- Appendix Tables.