John P. Boyd joined the faculty at the University of Michigan in 1977 and has been Professor of Atmospheric, Oceanic, and Space Science in the College of Engineering since 1988. Concurrently, he was the founding associate director of the Laboratory for Scientific Computation (now the Michigan Institute for Computational Discovery and Engineering). He has published 240 journal articles in atmospheric and oceanic dynamics, nonlinear waves, physics, and Chebyshev, Fourier and RBF spectral methods.
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Description
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.