Nonlinear Equations

• Only need the first derivatives
• Equal amounts of equations and unknowns

$$\text{UNSURE ABOUT THIS: }\text{Find}\mathbf{x}s.t.\mathbf{r}(\mathbf{x})=0$$

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Given an optimal point $x^{\ast }$ which satisfies $r(x^{\ast })=0$, we say that the solution is

• Degenerate if $J(x^{\ast })$ is singular
• Nondegenerate otherwise Where $J$ is the Jacobian, and singular singular means $\det (J)=0$

Newton’s Method for Nonlinear Equations