# Review Questions for Solving Nonlinear Equations

- How do you use methods for root-finding to solve a nonlinear equation for some value other than the root?
- 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
- How many function evaluations are required per iteration for bisection?
- What is the convergence rate of bisection method? Will it always converge?
- 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?
- How many function evaluations are required per iteration for 1D Newton's method for root-finding? Which functions must be evaluated?
- What is the convergence rate of Newton's method for 1D rootfinding?
- How many function evaluations are required per iteration for secant method?
- What is the convergence rate of secant method? Will it always converge?
- What are the advantages and disadvantages of bisection, Newton's method, and secant method? (e.g. why would you use one over the other?)
- For a given vector-valued function \(\mathbf{f}(\mathbf{x})\), what is the Jacobian (both in general and evaluated at a specific point).
- For a simple system of nonlinear equations, you should be able to run one step of \(n\)-dimensional Newton's method.
- What is the convergence rate of Newton's method for rootfinding in \(n\) dimensions? Will it always converge?
- What operations are required per iteration for Newton's method in \(n\) dimensions?