Solving Transcendental Equations

Preface; Notation; Part I. Introduction and Overview: 1. Introduction: key themes in rootfinding; Part II. The Chebyshev-Proxy Rootfinder and Its Generalizations. 2. The Chebyshev-proxy/companion matrix rootfinder; 3. Adaptive Chebyshev interpolation; 4. Adaptive Fourier interpolation and rootfinding; 5. Complex zeros: interpolation on a disk, the Delves–Lyness algorithm, and contour integrals; Part III. Fundamentals: Iterations, Bifurcation, and Continua: 6. Newton iteration and its kin; 7. Bifurcation theory; 8. Continuation in a parameter; Part IV. Polynomials: 9. Polynomial equations and the irony of Galois theory; 10. The quadratic equation; 11. Roots of a cubic polynomial; 12. Roots of a quartic polynomial; Part V. Analytical methods: 13. Methods for explicit solutions; 14. Regular perturbation methods for roots; 15. Singular perturbation methods: fractional powers, logarithms, and exponential asymptotics; Part VI. Classics, Special Functions, Inverses, and Oracles: 16. Classic methods for solving one equation in one unknown; 17. Special algorithms for special functions; 18. Inverse functions of one unknown; 19. Oracles: theorems and algorithms for determining the existence, nonexistence, and number of zeros; Part VII. Bivariate Systems: 20. Two equations in two unknowns; Part VIII. Challenges: 21. Past and future; Appendix A. Companion matrices; Appendix B. Chebyshev interpolation and quadrature; Appendix C. Marching triangles; Appendix D. Imbricate-Fourier series and the Poisson summation formula.