Nonlinear Equations 3

• What were the two methods we discussed for solving multiple simultaneous nonlinear equations?
• Write the linear approximation to a function using a Taylor series.
  • Do the same thing for two functions of two variables: $f(x,y)$, $g(x,y)$.
• Derive the Newton’s method iteration using a Taylor series
  • Do the same thing for the 2-D Newton’s method
• If a matrix has rows $i$ and columns $j$, what are the elements of the Jacobian matrix?
  • (how are you going to think about this to keep it straight?)
• Write the Jacobian elements in terms of a finite difference approximation
  • What should you use for $\Delta x$?
• When writing the multi-dimensional Newton’s method, we write a vector of functions that depends on a vector of unknowns $\vec{F}(\vec{x})=\vec{0}$.
  • If $\vec{x}$ has $n$ components, how many evaluations of $\vec{F}$ are needed to compute the Jacobian matrix?
  • Think about how you would code a function to compute a numerical Jacobian.