Review Questions for Solving Nonlinear Equations

1. How do you use methods for root-finding to solve a nonlinear equation for some value other than the root?
2. For a given nonlinear equation (1D), you should be able to run a couple steps of:\(\\\) 1) Bisection method \(\\\) 2) Secant method \(\\\) 3) Newton's method
3. How many function evaluations are required per iteration for bisection?
4. What is the convergence rate of bisection method? Will it always converge?
5. Using the bisection method, given a specific initial interval \([a,b]\) and a given tolerance \(tol\), how many iterations would be required for the approximate root to be accurate to the given tolerance?
6. How many function evaluations are required per iteration for 1D Newton's method for root-finding? Which functions must be evaluated?
7. What is the convergence rate of Newton's method for 1D rootfinding?
8. How many function evaluations are required per iteration for secant method?
9. What is the convergence rate of secant method? Will it always converge?
10. What are the advantages and disadvantages of bisection, Newton's method, and secant method? (e.g. why would you use one over the other?)
11. For a given vector-valued function \(\mathbf{f}(\mathbf{x})\), what is the Jacobian (both in general and evaluated at a specific point).
12. For a simple system of nonlinear equations, you should be able to run one step of \(n\)-dimensional Newton's method.
13. What is the convergence rate of Newton's method for rootfinding in \(n\) dimensions? Will it always converge?
14. What operations are required per iteration for Newton's method in \(n\) dimensions?