### Applications of number theory to numerical analysis

1. Algebraic Number Fields and Rational Approximation.- 1.1. The units of algebraic number fields.- 1.2. The simultaneous Biophantine approximation of an integral basis.- 1.3. The real eyelotomie field.- 1.4. The units of a eyelotomie field.- 1.5. Continuation.- 1.6. The Drriehlet field.- 1.7. The cubic field.- Notes.- 2. Recurrence Relations and Rational Approximation.- 2.1. The recurrence formula for the elementary symmetric fonction.- 2.2. The generalization of Sn.- 2.3. PV numbers.- 2.4. The roots of the equation F(x) = 0.- 2.5. The roots of the equation G(x) = 0.- 2.6. The roots of the equation E(x) = 0.- 2.7. The irreducibility of a polynomial.- 2.8. The rational approximations of ?, ?, ?.- Notes.- 3. Uniform Distribution.- 3.1. Uniform distribution.- 3.2. Vinogradov's lemma.- 3.3. The exponential sum and the discrepancy.- 3.4. The number of solutions to the congruence.- 3.5. The solutions of the congruence and the discrepancy.- 3.6. The partial summation formula.- 3.7. The comparison of discrepancies.- 3.8. Eational approximation and the solutions of the congruence.- 3.9. The rational approximation and the discrepancy.- 3.10. The lower estimate of discrepancy.- Notes.- 4. Estimation of Discrepancy.- 4.1. The set of equi-distribution.- 4.2. The Halton theorem.- 4.3. The p set.- 4.4. The gp set.- 4.5. The eonstruetion of good points.- 4.6. The ?s set.- 4.7. The ? set.- 4.8. The ease s = 2.- 4.9. The glp set.- Notes.- 5. Uniform Distribution and Numerical Integration.- 5.1. The function of bounded variation.- 5.2. Uniform distribution and numerical integration.- 5.3. The lower estimation for the error term of quadrature formula.- 5.4. The quadrature formulas.- Notes.- 6. Periodic Functions.- 6.1. The classes of functions.- 6.2. Several lemmas.- 6.3. The relations between Hs?(C), Qs?(C) and Es?(C).- 6.4. Periodic functions.- 6.5. Continuation.- Notes.- 7. Numerical Integration of Periodic Functions.- 7.1. The set of equi-distribution and numerical integration.- 7.2. The p set and numerical integration.- 7.3. The gp set and numerical integration.- 7.4. The lower estimation of the error term for the quadrature formula.- 7.5. The solutions of congruences and numerical integration.- 7.6. The glp set and numerical integration.- 7.7. The Sarygin theorem.- 7.8. The mean error of the quadrature formula.- 7.9. Continuation.- Notes.- 8. Numerical Error for Quadrature Formula.- 8.1. The numerical error.- 8.2. The comparison of good points.- 8.3. The computation of the ? set.- 8.4. The computation of the ?s set.- 8.5. Examples of other F s sets.- 8.6. The computation of a glp set.- 8.7. Several remarks.- 8.8. Tables.- 8.9. Some examples.- Notes.- 9. Interpolation.- 9.1. Introduction.- 9.2. The set of equi-distribution and interpolation.- 9.3. Several lemmas.- 9.4. The approximate formula of the function of E?s(C).- 9.5. The approximate formula of the function of Q?s(C).- 9.6. The Bernoulli polynomial and the approximate polynomial.- 9.7. The ? results.- Notes.- 10. Approximate Solution of Integral Equations and Differential Equations.- 10.1. Several lemmas.- 10.2. The approximate solution of the Fredholm integral equation of second type.- 10.3. The approximate solution of the Volterra integral equation of second type.- 10.4. The eigenvalue and eigenfunction of the Fredholm equation.- 10.5. The Cauehy problem of the partial differential equation of the parabolic type.- 10.6. The Diriehlet problem of the partial differential equation of the elliptic type.- 10.7. Several remarks.- Notes.- Appendix Tables.